Abstract
Usually, the Weierstraß gap theorem is derived as a straightforward corollary of the Riemann–Roch theorem. Our main objective in this article is to prove the Weierstraß gap theorem by following an alternative approach based on “first principles”, which does not use the Riemann–Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case when the given compact Riemann surface is associated with the modular curve \(X_0(N)\).
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1 Main Objective
Various topical areas in the theory of partitions, such as congruences for partition numbers, are connected to modular functions for congruence subgroups of \({\mathrm {SL}}_2({\mathbb {Z}})\) as, for instance, \(\Gamma _0(N)\); see Sect. 15 for definitions. Such functions live on compact Riemann surfaces, for instance, on \(X_0(N)\) for \(\Gamma _0(N)\). Number theoretic aspects then relate to properties of certain subalgebras formed by these functions. In cases where the genus of such surfaces is zero like, for instance, for \(X_0(5)\) and \(X_0(7)\), these algebras essentially have a relatively simple structure. For positive genus g, for example, in the case of \(X_0(11)\), this changes. One explanation is this: when considering sets of meromorphic functions with poles only at one point p, the Weierstraß gap theorem says that one can obtain functions with all possible pole orders at p with exactly g exceptions.
Theorem 1.1
(Weierstraß gap theorem; e.g., Sect. III.5.3 in [6]). Let X be a compact Riemann surface having genus \(g\ge 1\). Then, for each \(p\in X\), there are precisely g integers \(n_j=n_j(p)\) with
such that there does not exist a meromorphic function on X which is holomorphic on \(X\backslash \{ p\}\) and which has a pole of pole order \(n_j\) at p.
We want to stress that “precisely” in the theorem means that for any positive integer n other than the g values \(n_j\), a meromorphic function with a pole of order n at p exists.
Usually, as in [6, III. 5.3], this theorem is derived as a straightforward corollary of the Riemann–Roch theorem. Our main objective in this article is to prove the Weierstraß gap theorem by following an alternative approach based on “first principles” which does not use the Riemann–Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case when the given compact Riemann surface X is associated with \(X_0(N)\). Some ingredients of our setting are related to ideas from the celebrated paper [3] by Dedekind and Weber; see [2] for an English translation together with an excellent introduction by John Stillwell.
2 Introduction
To exemplify the usage of Weierstraß’s gap theorem, we choose an example related to the classical Ramanujan congruences, which in further details are discussed in [14]. Following the definition given in Sect. 15, let
To keep this article as much self-contained as possible, we list basic definitions and properties of modular functions in a separate Appendix Sect. 15.
One standard way to construct modular functions is by eta quotients, i.e., products of the form:
Here, \(\mathbb {H}\) denotes the upper half of the complex plane, \(m\in \mathbb {Z}_{>0}\), \(r_d\) are chosen integers, and \(\eta \) denotes the Dedekind eta function defined as
Usually one writes q instead of \(q(\tau )\).
The case \(m=\ell \), \(\ell \ge 5\) a prime, gives rise to a simple but important class of eta quotients:
which are modular functions in \(M(\ell )\) with (e.g., [9, Chap. 7, Theorem 1])
Here, the notation \(z_\ell ^*\) is explained by the fact that, in general, every modular function \(f\in M(N)\) gives rise to an induced meromorphic function \(f^*:X_0(N)\rightarrow {\hat{\mathbb {C}}}:=\mathbb {C}\cup \{\infty \}\) which for \(x=[\tau ]_N\) is defined as
see Sect. 15. There one also finds the definition of \([\tau ]_N\) as the orbit of \(\tau \) under \(\Gamma _0(N)\), as well as definitions of basic notions like of \({{\,\mathrm{ord}\,}}_{[a/c]_N} f^*\), the order of \(f^*\) at a cusp \([a/c]_N\), \(a/c\in \mathbb {Q}\cup \{\infty \}\). Note that \([\infty ]_N= [1/0]_N\).
Example 2.1
[9, Chap. 7, Theorem 1]. Consider
We have \({{\,\mathrm{ord}\,}}_{[\infty ]_N}f^* := {{\,\mathrm{ord}\,}}_{q} f\), confirming that
Because of
which owing to \({{\,\mathrm{ord}\,}}_{[0]_N}f^* := {{\,\mathrm{ord}\,}}_{q^{1/N}} f(-1/\tau )\) confirms that
In general, for \(\ell \) a prime, \(X_0(\ell )\) has exactly two cusps \([\infty ]_\ell \) and \([0]_\ell \) with widths 1 and \(\ell \), respectively; see [9, Chap. 2, Sect. 2], resp. Sect. 15 for the definition of width. The q-series (2.5) and (2.8) are the local q-expansions of \(z_5^*\) at these cusps.
Being meromorphic, modular functions form fields. For example, a classical fact, e.g., [5, Proposition 7.5.1], is that \(M(N)=\mathbb {C}(j(\tau ), j(N\tau ))\), where j is the modular invariant (the Klein j function). The subset
obviously is not a field but a \(\mathbb {C}\)-algebra.Footnote 1
Example 2.2
By definition (2.5) together with (2.6) and (2.9), \(z_5\in M^{!}(5)\), because \(z_5^*\) has its only pole of pole order 1 at \([0]_5\).
An important \(\mathbb {C}\)-subalgebra, in particular, with regard to algorithms, is
By [15, Lemma 20], \(M^\infty (N)\) for each \(N\ge 1\) contains an eta quotient \(\mu _N\) of the form as in (2.1), such that \({{\,\mathrm{ord}\,}}_{[a/c]_N} \mu _N^* >0\) for all \(a/c\in \mathbb {Q}\) with \([a/c]_N\ne [\infty ]_N\). Hence, one can multiply with a suitable power of \(\mu _N\) to turn any given \(f\in M^{!}(N)\) into an element \(\mu _N^\alpha f\) in \(M^\infty (N)\).
Example 2.3
Choose \(f(\tau ):=j(\tau )\in M^{!}(N)\), the Klein j function, and \(\alpha \), such that \(\mu _N^\alpha j \in M^\infty (N)\). Let \(\beta :={{\,\mathrm{ord}\,}}_{[\infty ]_N} \mu _N\), then \({{\,\mathrm{ord}\,}}_{[\infty ]_N} \mu _N^\alpha j = \alpha \beta -1.\) In particular
which will be needed later. A description of how to construct such \(\mu _N\) is given in [15].
Since we will prove the gap theorem in the version of Theorem 12.2, where \(X=X_0(N)\), and with \(p=[\infty ]_N\), a key issue in our approach concerns the question of finding appropriate representations of \(M^\infty (N)\).
Example 2.4
Owing to (2.6) and (2.9), \(1/z_{5}=\frac{1}{q}-6+9q+\cdots \in M^{\infty }(5)\). Because of \({{\,\mathrm{ord}\,}}_{[\infty ]_5} (1/z_5)^*=-1\), each \(f\in M^\infty (5)\) can be written as a polynomial in \(1/z_5\); in short:
where \(\mathbb {C}[x]\) denotes the ring of polynomials in x with complex coefficients. One also has
but already for \(\ell =11\), the situation is quite different. For example, in [14], we proved (implicitly) that \(M^\infty (11)\) can be represented as a \(\mathbb {C}[1/z_{11}]\)-module which is freely generated by modular functions \(F_2\), \(F_3\), \(F_4\), and \(F_6\in M^\infty (11)\). More concretely
where the \(F_i\) are determined as follows [14, Sect. 9]: from the two functions \(f_2, f_3\in M^{!}(22)\):
and
one constructs the desired \(F_i\in M^\infty (11)\) by
where \(U_2\) is the special case \(\ell =2\) (“summing the even part”) of the standard U-operator:
In addition to \({{\,\mathrm{ord}\,}}_{[\infty ]_{11}}(1/z_{11})^*=-5\), one has
Thus the minimal pole order of the functions which in the sense of (2.10) generate \(M^\infty (11)\) is 2, not 1. Indeed, the gap at 1 is predicted by the Weierstraß gap theorem, Theorem 1.1, owing to the fact that the compact Riemann surface \(X:=X_0(11)\) has genus 1. A formula for the genus of \(X_0(N)\), if \(N=\ell \) is a prime, for instance, can be found in [5, Exercises 3.1.4(e)]; the genus for general N is determined in [5, Sect. 3.9].
In Sect. 12, we prove Theorem 12.2, a version of the gap Theorem 1.1 for the case \(X=X_0(N)\) and with the only pole put at \(\infty \), utilizing only first principles and avoiding the use of the Riemann–Roch formula. In particular, we avoid the use of any differentials. In addition, our approach provides new algebraic insight by consisting in a combination of module presentations of modular function algebras, integral bases, Puiseux series, and discriminants. For example, using our approach to prove the bound \(\le 2g-1\) stated in the Weierstraß gap theorem is reduced to an elementary combinatorial argument, see Sect. 12. Another by-product of our proof of the Weierstraß gap Theorem 12.2 is a natural explanation of the genus \(g=0\) case as a consequence of the reduction to an integral basis.
In view of various constructive aspects involved, we are planning to exploit the algorithmic content of our approach for computer algebra applications, for instance, for the effective computation of suitable module bases for modular functions. As already mentioned, some ideas we used trace back to the celebrated work [3] by Dedekind and Weber; see [2] for an English translation together with an excellent introduction by John Stillwell.
Finally, we remark that the history of Weierstraß’s gap theorem and related topics such as Weierstraß points somehow presents a challenge. The historical account [4] by Andrea Del Centina describes the scientific evolution of the gap theorem up to the 1970s. Concerning its beginnings Centina says, “The history of Weierstraß points is not marked by a precise starting date because it is not clear when Weierstraß stated and proved his Lückensatz (or “gap” theorem), but one can argue that probably it was in the early 1860s.”
The rest of our article is structured as follows. In Sect. 3, we introduce order-complete bases of modules over a polynomial ring \(\mathbb {C}[t]\) to describe modular function algebras. In Sect. 4, we describe how such bases can be stepwise modified to obtain an integral basis; i.e., an order-complete basis for the full algebra \(M^\infty (N)\). Under particular circumstances, one can keep track of the total number of such steps, which then gives a proof of the Weierstraß gap Theorem 12.2. To do this bookkeeping, one can use “order-reduction” polynomials discussed in Sect. 5. In Sect. 6, we explain how to obtain order-reduction polynomials computationally; Sect. 7 deals with important special cases. In Sects. 8 and 9, we derive important ingredients of our proof of Theorem 12.2; for example, a factorization property of the discriminant polynomial in Proposition 9.3. In Sects. and 11, we relate discriminant polynomials to order-reduction polynomials associated with integral bases. In Sect. 12, we use these results to prove the Weierstraß gap theorem in the version of Theorem 12.2. To prove the bound \(2g-1\) for the size of the maximal gap, our approach allows a purely combinatorial argument (a gap property of monoids) which we describe in Sect. 13. At various places, we require functions to have the separation property, as defined in Sect. 9. In Sect. 14, we prove the existence of such functions by giving an explicit construction.
The first Appendix Sect. 15 gives a short account on basic modular function facts needed; the second Appendix Sect. 16 recollects some fundamental facts about meromorphic functions on Riemann surfaces.
3 Modular Function Algebras as \(\mathbb {C}[t]\)-Modules
We already used (implicitly) the convention that if a meromorphic function f has a pole, then the pole order is defined as the negative order at this point, that is
If \(f\in M^\infty (N)\), we simplify notation using the convention for the pole order at infinity:
Definition 3.1
A tuple \((b_0, b_1,\ldots , b_{n-1})\), \(n\ge 1\), of modular functions in \(M^\infty (N)\) is called order-complete if
Slightly more generally, any tuple \((1,\beta _1,\ldots , \beta _{n-1})\) which is a reordering of an order-complete tuple \((1,b_1,\ldots , b_{n-1})\), that is,
is also called order-complete.
Example 3.2
The tuple \((1,F_6, F_2, F_3, F_4)\) with \(F_j\in M^\infty (11)\) as in (2.10) is order-complete.
Example 3.3
Let
The tuple \((1,f, f^2, f^3, f^4)\) is order-complete. Notice that \({{\,\mathrm{pord}\,}}f=4\). In [13], it is shown that the subalgebra \(\mathbb {C}[1/z_{11}, f]\) of \(M^\infty (11)\), which is generated by all bivariate polynomials in \(1/z_{11}\) and f, has a representation as a \(\mathbb {C}[1/z_{11}]\)-module as follows:
In view of these examples, we note that in contrast to (2.10), \(\mathbb {C}[1/z_{11},f]\ne M^\infty (11)\). For instance, it is obvious that this subalgebra does not contain any \(g\in M^\infty (11)\) with \({{\,\mathrm{pord}\,}}g = 3\). Nevertheless, both function tuples
form a basis of the corresponding \(\mathbb {C}[t]\)-module they generate, where \(t:=1/z_{11}\). Namely, since the generators have different pole-order modulo \({{\,\mathrm{pord}\,}}t =5\), each element contained in these modules can be represented as a unique linear combination of the module generators with coefficients being polynomials in t. This motivates the following definition.
Definition 3.4
For \(t\in M^\infty (N)\), let \(n:={{\,\mathrm{pord}\,}}t\ge 1\). Let, M be the \(\mathbb {C}[t]\)-module generated by an order-complete tuple in \(M^\infty (N)\), that is,
Then, we call \((1,b_1,\ldots , b_{n-1})\) an order-complete basis for M over \(\mathbb {C}[t]\). Slightly more generally, any tuple \((1,\beta _1,\ldots , \beta _{n-1})\) which is a reordering, in the sense of (3.1), of an order-complete basis \((1,b_1,\ldots , b_{n-1})\) for M is also called an order-complete basis for M.
Proposition 3.5
Let \(t, f\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t\ge 1\) and \(\gcd (n,{{\,\mathrm{pord}\,}}f)=1\). Then
where \((1,f,f^2,\ldots , f^{n-1})\) is an order-complete module basis.
Proof
If \({{\,\mathrm{pord}\,}}f^i \equiv {{\,\mathrm{pord}\,}}f^j \pmod {n}\), then \(n \mid (i-j){{\,\mathrm{pord}\,}}f\). This implies
In addition, as a consequence of Theorem 7.1 and Lemma 7.3 in [14], \(f^n\in \langle 1, f, f^2, \ldots , f^{n-1} \rangle _{\mathbb {C}[t]}\). Hence, \(\mathbb {C}[t,f]\subseteq \langle 1, f, f^2, \ldots , f^{n-1}\rangle _{\mathbb {C}[t]}\). The reverse direction of this inclusion is trivial, which completes the proof. \(\square \)
4 Integral Bases
In Example 3.3, we saw that \((1,f,\ldots ,f^4)\) is an order-complete basis of \(\mathbb {C}[1/z_{11},f]\) which is a proper subalgebra of \(M^\infty (11)\).Footnote 2 In this section, we shall see how such an order-complete basis can be step-wise modified to obtain an order-complete basis for the full algebra \(M^\infty (11)\).
Definition 4.1
Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t\ge 1\). An order-complete tuple \((1,b_1,\ldots ,b_{n-1})\), \(b_j\in M^\infty (N)\) is called an integral basis for \(M^\infty (N)\) over \(\mathbb {C}[t]\) if
The motivation for this terminology comes from
Lemma 4.2
Let \(f\in M(N)\) and \(t\in M^\infty (N)\) with \({{\,\mathrm{pord}\,}}t \ge 1\) and
Then, f satisfies an algebraic relation
with polynomials \(p_j(x)\in \mathbb {C}[x]\) (i.e., f is integral over \(\mathbb {C}[t]\)) if and only if
Moreover, if \(f\in M^\infty (N)\), then there exists an algebraic relation with \(n={{\,\mathrm{pord}\,}}t\).
Proof
The statement with the assumption \(f\in M^\infty (N)\) follows immediately from Proposition 3.5. For the other direction, assume that \(m:={{\,\mathrm{pord}\,}}_p f^* > 0\) for \(p\ne [\infty ]_N\). Then \({{\,\mathrm{pord}\,}}_p (f^n)^*= m n\), a contradiction to
\(\square \)
A crucial observation for the process to obtain an integral basis for \(M^\infty (N)\) from an order-complete basis is stated in the following.
Proposition 4.3
Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\). Let \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) be an order-complete basis of the \(\mathbb {C}[t]\)-module:
Then, for any \(f\in M^\infty (N)\), there exist polynomials q(x) and \(p_j(x)\) in \(\mathbb {C}[x]\), such that
Proof
For \(j\in \mathbb {Z}_{\ge 0}\), consider the sets
For each \(j\ge 0\), choose a non-zero \(g_j\in G_j\), such that \({{\,\mathrm{pord}\,}}g_j\) is minimal amongst all the elements in \(G_j\). By construction, using the convention \(n \mathbb {Z}_{\ge 0} := \{n k: k \in \mathbb {Z}_{\ge 0}\}\), we have for all \(j\ge 0\):
Obviously, S is an additive submonoid of \((\mathbb {Z}_{\ge 0},+)\). Moreover, \(\mathbb {Z}_{\ge 0}\backslash S\) has only finitely many elements; let k be the maximal element in this set. Then there exist \(c_j\in \mathbb {C}\), not all zero, such that
This is owing to the fact that equating the coefficients of non-positive powers in the q-expansions of both sides (which are functions in \(M^\infty (N)\)) gives \(k+1\) equations in \(k+2\) variables \(c_j\). Hence, the dimension of the \(\mathbb {C}\)-vector space G, which is generated by all the \(g_j\), \(j\ge 0\), is bounded by \(k+1\). Using \(g_j:=t^j f-h_j\) with \(h_j\in M\), (4.2) rewrites into the form:
The linear combination of \(h_j\) is in M; hence, this gives the desired relation for f with \(q(t)=c_0 + c_1 t + \cdots +c_{k+1} t^{k+1}\). \(\square \)
Corollary 4.4
Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\). Let \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) be an order-complete basis of the \(\mathbb {C}[t]\)-module:
If \(M\ne M^\infty (N)\), then there exist \(c_j\in \mathbb {C}\), not all zero, and \(\alpha \) in \(\mathbb {C}\), such that
In particular, there exists a uniquely determined \(k\in \{1,\ldots ,n-1\}\), such that
Proof
By Proposition 4.3, there exists an \(f\in M^\infty (N)\backslash M\) of the form (4.1), such that \(q(x)\not \mid p_i(x)\) for some \(i\in \{0,\ldots , n-1\}\). Hence, there exists \(\alpha \in \mathbb {C}\), such that \(x-\alpha \mid q(x)\), but \(x-\alpha \not \mid p_i(x)\). Consequently, \(q(t)/(t-\alpha )\in M^\infty (N)\) and thus
By division with remainder, there are polynomials \(q_j(x)\in \mathbb {C}[x]\) and \(c_j\in \mathbb {C}\), such that \(p_j(x)=(x-\alpha ) q_j(x) + c_j\), \(j=0,\ldots ,n-1\). Rewriting the representation of g and noting that \(c_i\ne 0\) proves the first part of the statement on \(h_\alpha \). To prove (4.4), consider
which implies
Let k be the index for which \({{\,\mathrm{pord}\,}}b_k\) becomes maximal with \(c_k\ne 0\). Recalling \({{\,\mathrm{pord}\,}}b_j\equiv j \, ({\mathrm {mod}}\, n)\), \(j=1,\ldots , n\), proves \({{\,\mathrm{pord}\,}}b_k\ge k + n\). Because otherwise \({{\,\mathrm{pord}\,}}b_k = k\) which owing to the choice of k would imply \({{\,\mathrm{pord}\,}}b_j = j\) for all \(j=1,\ldots , n\), and the given order-complete basis would be integral. This proves (4.4). \(\square \)
Corollary 4.4 motivates the following.
Definition 4.5
Let \(M= \langle 1,b_1,\ldots ,b_{n-1} \rangle _{\mathbb {C}[t]}\) and \(h_\alpha \in M^\infty (N)\) be as in Corollary 4.4; i.e., \(M \ne M^\infty (N)\) and \({{\,\mathrm{pord}\,}}h_\alpha = {{\,\mathrm{pord}\,}}b_k - n \ge k\). The replacement
of \(b_k\) by \(h_\alpha \) is called a pole-order-reduction step associated with \(\alpha \in \mathbb {C}\).
We summarize in the form of
Proposition 4.6
Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\). Let \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) be an order-complete basis of the \(\mathbb {C}[t]\)-module:
If \(M\ne M^\infty (N)\), then:
- (i)
By a finite sequence of pole-order-reduction steps the order-complete basis \((1,b_1,\ldots ,b_{n-1})\) can be transformed into an integral basis \((1,\beta _1,\ldots ,\beta _{n-1})\), such that
$$\begin{aligned} \langle 1,\beta _1,\ldots ,\beta _{n-1} \rangle _{\mathbb {C}[t]} = M^\infty (N). \end{aligned}$$ - (ii)
If \((1,\beta '_1,\ldots ,\beta '_{n-1})\) is any another integral basis, that is,
$$\begin{aligned} \langle 1,\beta '_1,\ldots ,\beta '_{n-1} \rangle _{\mathbb {C}[t]} = M^\infty (N), \end{aligned}$$then
$$\begin{aligned} \{{{\,\mathrm{pord}\,}}\beta _1,\ldots , {{\,\mathrm{pord}\,}}\beta _{n-1}\}= \{{{\,\mathrm{pord}\,}}\beta '_1,\ldots , {{\,\mathrm{pord}\,}}\beta '_{n-1}\}. \end{aligned}$$(4.5)
Proof
The proof of part (i) is an immediate consequence of Corollary 4.4. Namely, owing to (4.4), each step reduces the pole order of one of the basis elements by n. This guarantees termination in finitely many steps. To prove (ii), without loss of generality, we can assume that \({{\,\mathrm{pord}\,}}\beta _j \equiv {{\,\mathrm{pord}\,}}\beta '_j\equiv j \pmod {n}\) for all j. Suppose \({{\,\mathrm{pord}\,}}\beta _j\ne {{\,\mathrm{pord}\,}}\beta '_j\) for some \(j\in \{1,\ldots , n-1\}\), i.e., \({{\,\mathrm{pord}\,}}\beta '_j= {{\,\mathrm{pord}\,}}\beta _j + k n\) with \(k\ge 1\). However, this implies that \(\beta _j \not \in \langle 1,\beta '_1,\ldots ,\beta '_{n-1} \rangle _{\mathbb {C}[t]}\), because then, no element in this module can have the same pole order as \(\beta _j\), a contradiction. \(\square \)
5 Order-Reduction Polynomials
It was shown in the previous section that by applying a procedure using finitely many steps, any order-complete basis of a subalgebra of \(M^\infty (N)\) can be extended to an integral basis of \(M^\infty (N)\). Moreover, by (4.5), the pole orders of the integral basis functions are uniquely determined. It turns out that under particular circumstances, one can keep track of the number of order-reduction steps, which then gives a proof of the Weierstraß gap Theorem 12.2. To do this bookkeeping, one can use “order-reduction” polynomials. To our knowledge, for the first time such polynomials have been used by Dedekind and Weber [3], see [2] for Stillwell’s translation into English.
Throughout this section, \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\) and \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) is an order-complete basis of the \(\mathbb {C}[t]\)-module:
Owing to \(t(\tau )=\infty \) if and only if \([\tau ]_N = [\infty ]_N\), t is a holomorphic function on \(\mathbb {H}\). Moreover, the induced function \(t^*\), which is meromorphic on the compact Riemann surface \(X_0(N)\), has a pole only at \([\infty ]_N\).
Remark 5.1
(A basic notational convention). In general, every modular function \(f\in M(N)\) gives rise to an induced meromorphic function \(f^*:X_0(N)\rightarrow {\hat{\mathbb {C}}}\) which for \(x=[\tau ]_N\) is defined as
see Appendix Sect. 15. A central theme in what follows is to consider maps:
where \(U\subseteq X_0(N)\) and \(V\subseteq \mathbb {C}\) are open sets, such that
Hence, for \(v\in V\), the evaluations
have to be interpreted in the sense of (5.1), i.e., interpreting \(x=(t^*\mid U)^{-1}(v)\) as \(x=[\tau ]_N\) for some \(\tau \in {\hat{\mathbb {H}}}\).
Depending on the context, we will freely move between considering t as a function on \(\mathbb {H}\), resp. \({\hat{\mathbb {H}}}\), and its induced version \(t^*:X_0(N)\rightarrow {\hat{\mathbb {C}}}\).
Using the terminology explained in the Appendix Sect. 16, we assume that \(v_0\in \mathbb {C}\) is not a branch point of \(t^*\); in short, \(v_0\not \in {\mathrm {BranchPts}}(t^*)\). In this case, there are n pairwise distinct points \(x_j=[\tau _j]_N\in X_0(N)\) with \(\tau _j\in \mathbb {H}\), such that
In addition, there exists a neighborhood V of \(v_0\) and neighborhoods \(U_j\) of \(x_j\), such that
as a disjoint union of open sets, and such that for \(j=1,\ldots , n\), the restricted functions
are bi-holomorphic.
Let
Define
by
Taking the square of the determinant guarantees that the expression on the right side is symmetric with respect to any permutation of \(T_1,\ldots , T_n\). Consequently, \(D_t(1,b_1,\ldots ,b_{n-1})\) is a holomorphic function on V. Carrying out the same construction on neighborhoods V for all \(v_0\in \mathbb {C}\backslash {\mathrm {BranchPts}}(t^*)\), and gluing the resulting functions \(D_t(1,b_1,\ldots ,b_{n-1}): V\rightarrow \mathbb {C}\) together, gives a global holomorphic function:
Using the same arguments as in the proof of Theorem 8.2 in [7], this function can be extended to a meromorphic function:
with \(\infty \) as its only pole. Classical complex analysis tells that \({\mathcal {M}}({\hat{\mathbb {C}}}) = \mathbb {C}(z)\), i.e., the field of meromorphic functions on \({\hat{\mathbb {C}}}\) are rational functions with coefficients in \(\mathbb {C}\). Hence, we have the following.
Lemma 5.2
The meromorphic function \(D_t(1,b_1,\ldots ,b_{n-1})(v)\) constructed above is a polynomial function in v.
Definition 5.3
The polynomial \(D_t(1,b_1,\ldots ,b_{n-1})(x)\in \mathbb {C}[x]\) is called order-reduction polynomial for the order-complete basis \((1,b_1,\ldots ,b_{n-1})\), \(b_j\in M^\infty (N)\), of the \(\mathbb {C}[t]\)-module
where \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\).
Example 5.4
Taking
and
where \(F_j\in M^\infty (11)\) are as in Example 2.4, one obtains
Example 5.5
Taking t and the \(b_j\) as in Example 5.4, one obtains
Remark 5.6
How such polynomials are computed is explained in Sect. 6.
In Corollary 4.4, we proved that if \(M\ne M^\infty (N)\), then there exist \(c_j\in \mathbb {C}\), not all zero, and \(v_0\) in \(\mathbb {C}\), such that
Recall that we denoted the n pairwise distinct preimages of \(v_0\) as follows:
Relation (5.6) implies
As a necessary condition for the existence of \(c_j\in \mathbb {C}\) not all zero, the determinant
of the corresponding linear system has to be zero. In view of
the square of this determinant (taking the underlying matrix transposed) is \(D_t(1,b_1,\ldots ,b_{n-1})(v_0)\). Above, we used the fact that the definition for
extends to the polynomial function
This means, the case when \(v_0\in \mathbb {C}\) is a branch point of \(t^*\) is also covered by the same determinant condition:
However, if \(v_0\in \mathbb {C}\) is a branch point, this condition is automatically satisfied, because then at least the two rows
are equal for \(i\ne j\). Summarizing, this gives
Lemma 5.7
Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\). Let \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) be an order-complete basis of the \(\mathbb {C}[t]\)-module
Let \(v_0\in \mathbb {C}\) be such thatFootnote 3
for \(c_j\in \mathbb {C}\), not all zero. Then
If \(v_0\) is a branch point of \(t^*\), the condition (5.7) is automatically satisfied.
6 How to Compute Order-Reduction Polynomials
Next, we explain how to compute the order-reduction polynomials in (5.4) and (5.5).
To this end, it will be convenient to introduce the following notation:
Definition 6.1
If
is the q-expansion at infinity for some \(f\in M^\infty (N)\), we define
that is,
Returning to the setting (5.2), we again assume that \(v_0\in \mathbb {C}\) is not a branch point of \(t^*\). This means that there exists pairwise distinct \(x_j=[\tau _j]_N\in X_0(N)\) with \(\tau _j\in \mathbb {H}\cup \mathbb {Q}\) such that \([\tau _j]_N\ne [\infty ]_N\) andFootnote 4
together with neighborhoods \(U_j\) of the \(x_j\), such that for a suitable neighborhood V of \(v_0\):
as a disjoint union of open sets, and such that the restricted functions
are bi-holomorphic.
For each \(j=1,\ldots ,n\) and \(v\in V\) our goal, achieved in Lemma 6.2(ii), is to determine expressions for \(q_j(v):=q^{2 \pi i \tau (j)}\), where \(\tau (j)\) is close to \(\tau _j\), such that
For \(q=e^{2 \pi i \tau }\) with \(\tau \in \mathbb {H}\), we have
Here, we assume that the first coefficient in this q-expansion of t is 1. Now, if
and
then
To fix a branch of the \(n\hbox {th}\) root, we choose the preimage \(\tau _n\) and recall that
Now, for each \(v\in \mathbb {C}\) close to \(v_0\), there is for each \(j\in \{1,\ldots , n\}\) a uniquely determined \(\tau (j) \in \mathbb {H}\) close to \(\tau _j\), such that
By choosing a neighborhood of \(\tau _n\), we fix a branch of the \(n\hbox {th}\) root of \(v\in \mathbb {C}\) close to \(v_0\):
where \(\tau (n)\) is close to \(\tau _n\) and determined as in (6.3).
In addition, let W be such that \(U(W(q))=W(U(q))=q\), and define
After this preparation, in view of (6.4) we can put things together as follows.
Lemma 6.2
In the given setting, for \(j=1,\ldots ,n\) and \(v\in \mathbb {C}\) close to \(v_0\), let
where \(\root n \of {v}\) is defined as in (6.4).
Then, for \(j=1,\ldots ,n\) and \(v\in \mathbb {C}\) close to \(v_0\):
where \([\tau (j)]_N=T_j(v)\) with \( \tau (j)\) as in (6.3), and
where the values \(q_j(v)\) are pairwise distinct for \(j=1,\ldots ,n\).
Proof
The values \(q_j(v)\), \(j=1,\ldots , n\), are defined by power series in \(q=q(\tau (n))\):
For a fixed v close to \(v_0\), these values are pairwise different for \(j=1,\ldots , n\), because
By (6.2)
This implies (i) and (ii). \(\square \)
Lemma 6.2 enables us to compute the polynomial \(D_t(1,b_1,\ldots ,b_{n-1})(v)\), because by part (i) with \(i=1,\ldots , n-1\) and \(j=1,\ldots , n\):
This means, each \((b_i^*\circ T_j)(v)\) can be represented as a Laurent series in powers of \(1/v^{1/n}\):
with coefficients \(\alpha _{i,j}, \beta _{i,j}\), etc., in \(\mathbb {C}\), and under the assumption that the first Laurent series coefficient \(\beta _{-{{\,\mathrm{pord}\,}}b_i}^{(i)}\) of each \(\widetilde{b_i}(q_j(v))\) is equal to 1. Owing to Lemma 5.2, \(D_t(1,b_1,\ldots ,b_{n-1})(v)\) as defined in (5.3) must be a polynomial in v. Consequently, we can compute it by taking suitable truncated versions of the expansions (6.5).
Remark 6.3
This is how we computed the order-reduction polynomials in (5.4) and (5.5).
7 Discriminant Polynomials
Important special cases of order-reduction polynomials are produced by order-complete module bases of \(\mathbb {C}[t,f]\) of the form as in Proposition 3.5.
Definition 7.1
Let \(t, f\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t\ge 1\), and \(\gcd (n,{{\,\mathrm{pord}\,}}f)=1\). Then
is called the discriminant polynomial for the order-complete basis \((1,f,\ldots ,f^{n-1})\) of the \(\mathbb {C}[t]\)-module:
The discriminant polynomial
factors as the square of a Vandermonde determinant. Now, invoking (6.5) with \(b_i=f\), and thus, \({{\,\mathrm{pord}\,}}b_i= {{\,\mathrm{pord}\,}}f\), gives
Hence
and thus
Summarizing, we have the following.
Lemma 7.2
Let \(t, f\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t\ge 1\), and \(\gcd (n,{{\,\mathrm{pord}\,}}f)=1\). Then, the degree of the discriminant polynomial \(D_t(f)(x)\in \mathbb {C}[x]\) is
8 Reduction Steps and Order-Reduction Polynomials
In Sect. 4, we described how order-complete bases can be transformed into integral bases of \(M^\infty (N)\) by a finite sequence of pole-order-reduction steps. In this section, we establish a link between pole-order-reduction steps and order-reduction polynomials.
To this end, we consider again our standard situation: let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\), let \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) be an order-complete basis of the \(\mathbb {C}[t]\)-module:
By Corollary 4.4, when \(M\ne M^\infty (N)\), there exist \(c_j\in \mathbb {C}\), not all zero, and \(\alpha \) in \(\mathbb {C}\), such that
In particular, there exists a \(k\in \{1,\ldots ,n-1\}\), such that
Proposition 8.1
With regard to order-reduction polynomials, this setting is reflected by
Proof
After filling the right side of (8.1) into the determinant definition (5.3) of \(D_t(1, b_1,\ldots ,b_{k-1}, h_\alpha , b_{k+1},\ldots , b_{n-1})(v)\) and noticing that \(t^*(T_j(v))=v\), \(j=1,\ldots ,n\), the proof is a straightforward consequence of determinant calculus. \(\square \)
In other words, a pole-order-reduction step associated with \(\alpha \in \mathbb {C}\):
from one order-complete basis to another corresponds to factoring the order-reduction polynomial as
Example 8.2
In the situation of Example 5.5
9 Local Puiseux Expansions
By considering local expansions at finitely many points \([\tau _j]_N\in X_0(N)\) for \(\tau _j \in {\hat{\mathbb {H}}}=\mathbb {H}\cup \mathbb {Q}\cup \{\infty \}\), in this section, we derive important ingredients for our proof of Theorem 12.2. To this end, we consider charts \(\varphi _{\tau _0}:U_0\rightarrow \mathbb {C}\) with \(\varphi _{\tau _0}([\tau ]_N):=\phi _{\tau _0}(\tau )\) defined in a standard way either by
if \(\tau _0\in \mathbb {H}\) is not an elliptic point, or by
if \(\tau _0\in \mathbb {H}\) is an elliptic point (cf. (9.5)), or according to (15.3) by
if \(\tau _0 =\frac{a}{c}=\gamma \infty \in \mathbb {Q}\cup \{\infty \}\).
Here, \(U_0 \subseteq X_0(N)\) is a neighborhood of \([\tau _0]_N\); furthermore, the periods \(h(\tau _0)\) equal either 2 or 3. We note explicitly that all these charts are centered at 0, that is,
Remark 9.1
The explanation why such charts have to be chosen can be found, for instance, in [5, Sect. 2.2 and Sect. 2.3]. Charts, being homeomorphisms between open subsets of Riemann surfaces and of \(\mathbb {C}\), are used to set up local series expansions. Charts of the kind as in (9.2) have to be taken when \([\tau _0]_N\) is an elliptic point, i.e., if
Throughout this section, again \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\). Now, we reconsider the setting in Sect. 5 by dropping the assumption that \(v_0\in \mathbb {C}\) is not a branch point of \(t^*\). This means, we allow \(\ell \le n\) pairwise distinct points \(x_j=[\tau _j]_N\in X_0(N)\) with \(\tau _j\in \mathbb {H}\cup \mathbb {Q}\), such that \([\tau _j]_N\ne [\infty ]_N\) andFootnote 5
There exists a neighborhood \(V_0\) of \(v_0\) and neighborhoods \(U_j\) of the \(x_j\), such that
as a disjoint union of open sets.
Now, if \(\ell < n\), not all of the restricted functions
are bi-holomorphic.
Summarizing this setting,
with multiplicities \(k_1, \ldots , k_\ell \), respectively, i.e., \(k_1+\cdots +k_{\ell }=n\).
Hence, if V is an open subset of \(V_0\) not containing \(v_0\), then for each \(j=1,\ldots ,\ell \), there exist pairwise disjoint open subsets \(U_{j,k}\subseteq U_j\), \(k=1,\ldots ,k_j\), such that
as a disjoint union, and for \(k=1,\ldots ,k_j\), the restricted functions
are bi-holomorphic.
For all \([\tau ]_N \in U_j\), \(j=1,\ldots ,\ell \), one has expansions
Again, using (6.1), one has
where
For \(j=1,\ldots , \ell \), let
Now, by inverting the relation (9.8) and using the Puiseux series, the situation of (9.6) is reflected as follows: for each \(v\in V\), there is for fixed (j, k), \(j=1,\ldots ,k_j\) and \(k\in \{1,\ldots , k_j\}\), a uniquely determined \(\tau =\tau (j,k)\in U_{j,k}\), such that
For such pairs \(\tau =\tau (j,k)\) and v, one has
As in Sect. 5, one works with a fixed branch of the \(k_j\hbox {th}\) root; moreover, we note that as a consequence of the definition of \(A_j(z)\), \(A_{j,1}\ne 0\) for all \(j=1,\ldots ,\ell \).
To connect to discriminant polynomials, let \(f\in M^\infty (N)\) be such that \(\gcd (n,{{\,\mathrm{pord}\,}}f)=1\). Moreover, without loss of generality, for \(j=1,\ldots , \ell \), we can assume that the neighborhoods \(U_{j}\) are chosen, such that the following expansions exist for all \([\tau ]_N \in U_{j}\):
Invoking (9.9), one obtains
Lemma 9.2
For \(v_0\in \mathbb {C}\) and \(j=1,\ldots , \ell \), suppose that open neighborhoods \(U_{j,k}\), \(k=1,\ldots ,k_j\) and V are chosen as above. Then, there exist series expansions with complex coefficients \(c_{j,p}\), such that for all \(v \in V:\)
Proof
Setting \([\tau ]_N := (t^*\mid U_{j,k})^{-1}(v)\in U_{j,k}\), the statement follows from applying (9.9) to (9.10):
\(\square \)
To adapt to the refined setting (9.6), we extend our \(T_j\)-notation to the additional restricted functions:
Finally, we use the information we obtained in terms of the local holomorphic Puiseux series expansion to represent the discriminant polynomial at \(v_0\in \mathbb {C}\). Namely, for all \(v \in V\):
The last equality is by (9.11); it gives rise to the following.
Proposition 9.3
Let \(t\in M^\infty (N)\) with \(n={{\,\mathrm{pord}\,}}t \ge 1\), let \(f\in M^\infty (N)\) be such that \(\gcd (n,{{\,\mathrm{pord}\,}}f)=1\). For \(v_0\in \mathbb {C}\), suppose that
with multiplicities \(k_1, \ldots , k_\ell \), respectively, i.e., \(k_1+\cdots +k_{\ell }=n.\)
If
and
then there exists a polynomial \(p(x)\in \mathbb {C}[x]\), such that for all \(v\in \mathbb {C}:\)
Proof
The statement follows from the last equality of the derivation preceding this proposition. Namely, under the condition (9.12), the second product on the right side of this equality is non-zero for \(v=v_0\). Condition (9.13) means that \(b_{j,1}\ne 0\) in (9.10), thus \(c_{j,1}\ne 0\) for \(j=1,\ldots ,\ell \) in (9.11). Consequently, from the first product in the expression under consideration, one can pull out \(v - v_0\) as follows:
Recalling that \(k_1+\cdots + k_l=n\) completes the proof for all \(v\in V\), where V is an open subset of a neighborhood \(V_0\) of \(v_0\), such that V does not contain \(v_0\). However, invoking the identity theorem from complex analysis, the statement extends to all \(v\in \mathbb {C}\). \(\square \)
Properties (9.12) and (9.13) are sufficiently important to deserve a
Definition 9.4
(separation property). Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\), let \(f\in M^\infty (N)\) be such that \(\gcd (n,{{\,\mathrm{pord}\,}}f)=1\). For \(v_0\in \mathbb {C}\), suppose that
We say that f has the separation property for \((t,v_0)\) if f satisfies (9.12) and (9.13).
Remark 9.5
In Sect. 14, we describe how to construct such an f having the separation property.
An immediate consequence of Proposition 9.3 is
Corollary 9.6
Let f have the separation property for \((t, \beta )\) with \(\beta \in \mathbb {C}\). Then
Another consequence of our analysis above is
Proposition 9.7
Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\), and let \(f\in M^\infty (N)\) be such that \(\gcd (n,{{\,\mathrm{pord}\,}}f)=1\). For \(v_0\in \mathbb {C}\), suppose that
For complex numbers \(a_0,\ldots , a_{n-1}\), not all zero, define a meromorphic function on \(\mathbb {H}\) by
If f has the separation property for \((t,v_0)\), thenFootnote 6
Proof
Suppose \(F^*\) is analytic on \(X_0(N)\backslash \{[\infty ]_N\}\). Then, assuming the setting as above, by (9.7), one has for all \(\tau \in U_1\), a series expansion
with \(a_{1,0}\ne 0\). Hence, owing to (9.4),
which implies a factorization
where the last equality is by (9.10) with \(b_{1,0}\ne 0\) owing to (9.13). As a consequence of (9.17), if \(k_1>1\):
and by iteration
Notice that if
is the zero polynomial (e.g., if \(k_1=n\)), the assumption that \((1,f,\ldots , f^{n-1})\) is an order-complete basis would imply that all \(a_j=0\), and the proof would stop with this contradiction.
Using the same argument, one derives
etc., up to
If one of the polynomial factors in the role of B(x) above would be the zero polynomial, we are done. Otherwise, invoking condition (9.12) implies that
Recalling that not all the \(a_j\) are zero and \(k_1+\cdots +k_\ell =n\), we obtain a contradiction to the assumption that \(F^*\) is analytic on \(X_0(N)\backslash \{[\infty ]_N\}\). \(\square \)
Corollary 9.8
Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\), let \(f\in M^\infty (N)\) be such that \(\gcd (n,{{\,\mathrm{pord}\,}}f)=1\). For \(v_0\in \mathbb {C}\), suppose that
For complex numbers \(a_0,\ldots , a_{n-1}\), define a meromorphic function on \(\mathbb {H}\) by
If f has the separation property for \((t,v_0)\), then
Proof
If one of the \(a_j\) would be non-zero, Proposition 9.7 would imply a pole of \(F^*\) at some \([\tau _j]_N\ne [\infty ]_N\), \(\tau _j\in \mathbb {H}\cup \mathbb {Q}\). \(\square \)
10 Order Reduction and Discriminant Polynomials
In this section, we relate discriminant polynomials to order-reduction polynomials associated with integral bases. Throughout this section, let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t\ge 1\), let \((1,b_1,\ldots ,b_{n-1})\), \(b_j\in M^\infty (N)\), be an order-complete tuple forming an integral basis for \(M^\infty (N)\) over \(\mathbb {C}[t]\), that is,
Moreover, let \(f\in M^\infty (N)\) again be chosen, such that \(\gcd (n,{{\,\mathrm{pord}\,}}f)=1\). By Proposition 3.5, such an f gives rise to an order-complete basis \((1,f,\ldots ,f^{n-1})\) of the \(\mathbb {C}[t]\)-module:
By exemplifying the case for \(n=3\), we shall see how the discriminant polynomial
is related to the order-reduction polynomial:
By the identity theorem from complex analysis, it is sufficient to consider the situation for v from a neighborhood V of \(v_0\in V\). With the setting as in (5.3), one has
because owing to (10.1), there exist polynomials \(r_j^{(i)}(x)\in \mathbb {C}[x]\), such that
This implies
using
Taking determinants of both sides of the matrix equation squared, this gives for the general case:
Next, we consider the other direction. By Proposition 4.3, there exist polynomials \(q_j(x)\) and \(p_i(x)\) in \(\mathbb {C}[x]\), such that
where \(q_j(t)\) is either a constant or such that
As before, this can be expressed as a matrix equation. We display the case for \(n=3\):
where
Again, taking determinants of both sides of the matrix equation squared, for the general case, this gives another polynomial relation in v:
where \(s(x), q_1(x), \ldots , q_{n-1}(x)\) are polynomials in \(\mathbb {C}[x]\). It will be convenient to cancel out possible common factors and to write, as polynomials in \(\mathbb {C}[x]\):
such that
and
where \(q_j(x)\) are determined as in (10.3).
11 Order-Reduction Polynomials: Further Results
In this section, we continue the considerations made in the previous section. Again, \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\), and \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) is assumed to be an integral basis for \(M^\infty (N)\) over \(\mathbb {C}[t]\).
Lemma 11.1
Let \((1,\beta _1,\ldots ,\beta _{n-1})\) with \(\beta _j\in M^\infty (N)\) be an integral bases for \(M^\infty (N)\) over \(\mathbb {C}[t]\). Then, there exists a \(c\in \mathbb {C}\) such that
Proof
Applying the same kind of argument as used to derive (10.2), we obtain the polynomial relations:
and
This proves the statement. \(\square \)
Another application of the argument we used to derive (10.2) is the existence of some polynomial \(R(x)\in \mathbb {C}[x]\), such that
This, using (10.5), implies
Finally, as a consequence of (10.6), we obtain
We summarize the following.
Lemma 11.2
There is \(c\in \mathbb {C}\) such that
where for \(j=1,\ldots , n-1,\) the polynomials \(Q_j(x)\) divide the polynomials \(q_j(x)\) which are determined as in (10.3).
Lemma 11.3
Let \(Q_j(x)\) be the polynomials as in Lemma 11.2. Suppose f has the separation property for \((t, \beta )\) for some \(\beta \in \mathbb {C}\). Then
Proof
Suppose \(x-\beta \mid Q_j(x)\) for some \(j\in \{1,\ldots , n-1\}\). By Lemma 11.2, \(Q_j(x)\mid q_j(x)\) with \(q_j(x)\) as in relation (10.3). Hence, \(x-\beta \) divides \(q_j(x)\), and (10.3) can be rewritten as
As in the proof of Corollary 4.4, by division with remainder, there are polynomials \(p_l(x)\in \mathbb {C}[x]\) and \(a_l\in \mathbb {C}\), such that \(p_l^{(j)}(x)=(x-\beta ) p_l(x) + a_l\), \(l=0,\ldots ,n-1\). This means
Owing to the fact that f has the separation property for \((t,\beta )\), one has by Corollary 9.8:
Iterating this argument cancels out all powers of \(t-\beta \) and one arrives at a representation of \(b_j\) of the form:
with polynomials \(P_l(x)\) and Q(x), such that
Comparing this to the representation (10.3), which rewrites as
produces a contradiction to the uniqueness of the basis representation since in contrast to (11.5), \(x-\beta \) divides the denominator polynomial \(q_j(x)\). \(\square \)
Proposition 11.4
For any \(\beta \in \mathbb {C}\):
Proof
For the proof we choose f having the separation property for \((t,\beta )\).Footnote 7 For the “\(\Rightarrow \)” direction of the statement, suppose \(D_t(1,b_1,\ldots ,b_{n-1})(\beta )=0\). Then (11.4) implies \(D_t(f)(\beta )=0\) which, owing to Corollary 9.6, is true if and only if \(\beta \in {\mathrm {BranchPts}}(t)\). For the other direction, we use the reverse direction of this “if and only if” relation: \(\beta \in {\mathrm {BranchPts}}(t^*)\) implies \(x-\beta \mid D_t(f)(x)\). Next, we apply Lemma 11.3 to the equation (10.5) and obtain
which completes the proof. \(\square \)
From all this we obtain the complete factorization of order-reduction polynomials of integral bases. To state it, it is convenient to define
in order to keep the point \(\infty \) out, as the image of the only pole at \([\infty ]_N\).
Proposition 11.5
Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\) and \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) be an integral basis for \(M^\infty (N)\) over \(\mathbb {C}[t]\). Then, there exists a \(c\in \mathbb {C}\) and positive integers \(m_\beta \), such that
Moreover, for any \(\beta \in {\mathrm {BranchPts}}_\mathbb {C}(t^*)\), suppose that \(t^*(x)=\beta \) has
with multiplicities \(k_1^{(\beta )}, \ldots , k_{\ell (\beta )}^{(\beta )}\), respectively, i.e., \(k_1^{(\beta )}+\cdots +k_{\ell _{\beta }}^{(\beta )}=n.\)
Then
Proof
The factorization (11.7) is immediate from Proposition 11.4. To prove (11.8), let \(\beta \in {\mathrm {BranchPts}}_\mathbb {C}(t^*)\) be a branch point of the kind as stated. Choose f to have the separation property for \((t,\beta )\). Then (9.14) implies the existence of a polynomial \(p(x)\in \mathbb {C}[x]\), such that
According to (11.4), there exist polynomials \(Q_j(x)\), such that
and owing to Lemma 11.3,
Hence, \((x-\beta )^{n-\ell (\beta )}\) divides \(D_t(1,b_1,\ldots ,b_{n-1})(x)\) with the maximal power, which proves (11.8). \(\square \)
For the next consideration, we again have to use the charts as in (9.1), (9.2), and (9.3). In the setting of Proposition 11.5, one has
where the last line is by the fact that if \(t(\tau _0)\not \in {\mathrm {BranchPts}}(t^*)\), then
Here, we use the notion of multiplicity \({{\,\mathrm{mult}\,}}_x(f)\), also explained in Sect. 16, which stands for the multiplicity at the point \(x\in X\) of a meromorphic function f on a (compact) Riemann surface X. For \(x_0=[\tau _0]_N\in X_0(N)\), one has (e.g., [11, Lemma 4.7] and [5, Sect. 2.4]) with respect to our charts \(\phi _{\tau _0}(\tau )\) centered at 0:Footnote 8
Hence, we obtain Proposition 11.5.
Corollary 11.6
Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\) and \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) be an integral basis for \(M^\infty (N)\) over \(\mathbb {C}[t]\). Then
Next, recall from Sect. 16 the definition of \({\mathrm {Deg}}(f)\), the degree of a meromorphic function f on a compact Riemann surface X:
Choosing \(v:=\infty \), we have \({\mathrm {Deg}}(t^*)=n\). Let
Recall the Riemann–Hurwitz formula [11, Thmorem 4.16]Footnote 9 for a non-constant holomorphic map \(F:X\rightarrow Y\) between compact Riemann surfaces:
Now, we apply this to our setting where \(X:=X_0(N)\) and \(F:=t^*:X_0(N)\rightarrow {\hat{\mathbb {C}}}\). Owing to \(g({\hat{\mathbb {C}}})=0\), together with (11.9) and Corollary 11.6, this gives
We summarize in
Corollary 11.7
Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\) and \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) be an integral basis for \(M^\infty (N)\) over \(\mathbb {C}[t]\). Then
12 Proof of the Weierstraß Gap Theorem
In this section, we prove the gap theorem for modular functions in \(M^\infty (N)\).
Definition 12.1
(Gaps in modular function algebras) Let M be a subalgebra of \(M^\infty (N)\), the modular functions for \(\Gamma _0(N)\) which are holomorphic in \(\mathbb {H}\) and with a pole at \(\infty \). A positive integer n is called a gap in M if there is no \(f\in M\) with \({{\,\mathrm{pord}\,}}f =n\). We also define the gap number \(g_M\) as the total number of gaps in M, that is,
In this section, we prove the gap theorem in the following version.
Theorem 12.2
(Weierstraß gap theorem for \(X_0(N)\)) Let \(g:=g(X_0(N))\) be the genus of \(X_0(N)\). If \(g\ge 1\), then \(M^\infty (N)\) has exactly g gaps \(n_j\) with
If \(g=0\), then \(M^\infty (N)\) has no gaps i.e., there exists an \(h\in M^\infty (N)\), such that \({{\,\mathrm{pord}\,}}h=1\).Footnote 10
To prepare for the proof, we determine the gap number \(g_{\mathbb {C}[t,f]}\), where \(t, f\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t\ge 2\), \(l:={{\,\mathrm{pord}\,}}f \ge 2\), and \(\gcd (l,n)=1\). To construct such functions with relatively prime pole orders is straightforward; see, for instance, Example 2.3. By Proposition 3.5, we know that
where \((1,f,f^2,\ldots , f^{n-1})\) is an order-complete module basis. Hence, there are \(l_j\in \mathbb {Z}_{\ge 0}\), \(j=1,2, \ldots , n-1,\) such that
Thus, inspecting each of the residue classes modulo n for \(j\in \{1,\ldots ,n-1\}\) makes clear that one cannot find any function of pole order
in \(\mathbb {C}[t,f]\). Hence, for fixed j, \(l_j\) pole orders are missing; summing j from 1 to \(n-1\) gives the total number of missing pole orders of functions in \(\mathbb {C}[t,f]\):
where the last equality is by [8, (3.32)]. We summarize in
Lemma 12.3
Let \(t, f\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t\ge 1\), \(l:={{\,\mathrm{pord}\,}}f \ge 1\), and \(\gcd (l,n)=1\). Then, the total number of missing pole orders of functions in \(\mathbb {C}[t,f]\) is
Proof
If \(n=1\) or \(\ell =1\) then \(\mathbb {C}[t,f]=M^\infty (N)\); i.e., there is no gap. The case both n and \(\ell \) greater or equal to 2 was treated above. \(\square \)
Proof of Theorem 12.2.
Recalling Definition 4.5, each pole-order-reduction step associated to some \(\alpha \in \mathbb {C}\),
between order-complete bases
- (i)
by Corollary 4.4 (4.4) reduces the total number of gaps by exactly one.
- (ii)
by Proposition 8.1 (8.3) reduces the degree of the order-reduction polynomial by exactly two.
The gap theorem now will be proved by successively applying pole-order-reduction steps to the order-complete basis \((1,f,\ldots , f^{n-1})\) of \(\mathbb {C}[t,f]\), where t and f are chosen from \(M^\infty (N)\), such that \(n:={{\,\mathrm{pord}\,}}t\ge 2\), \(l:={{\,\mathrm{pord}\,}}f \ge 2\), and \(\gcd (l,n)=1.\) Such a pair (t, f) can be easily constructed, see, for instance, Example 2.3.
Suppose that after r reduction steps, we arrive at the integral basis \((1,b_1,\ldots ,b_{n-1})\) of \(M^\infty (N)\). Defining
the reduction observation (ii) made above gives
furthermore, reduction observation (i) implies for the gap numbers:
Combining (12.2) and (12.3) gives the desired Weierstraß estimate for the total number of gaps in \(M^\infty (N)\) in terms of the genus g:
Hence, we proved that \(M^\infty (N)\) has exactly g gaps. If \(g=0\) there are no gaps; i.e., in this case, after relabelling indices,
Hence, \(M^\infty (N)=\mathbb {C}[h]\) for \(h:=b_1\).
To prove the remaining part of the gap theorem, namely, the bound (12.1) for the gaps \(\{n_1=1,n_2,\ldots ,n_g\}\) where \(g\ge 1\), we will use a general combinatorial argument. Notice that \(n_1=1\), because otherwise there would be no gap, which, as we proved, is only possible if \(g=0\).
To prepare for the combinatorial argument, recall that after choosing t and f from \(M^\infty (N)\) as above, by applying pole-order-reduction steps, we arrived, after relabelling indices, at an integral basis \((1,b_1,\ldots , b_{n-1})\) for \(M^\infty (N)\) where \({{\,\mathrm{pord}\,}}b_j \equiv j \pmod n\), \(j=1,\ldots ,n-1\). Defining
this basis gives rise to the additive submonoid
of \((\mathbb {Z}_{\ge 0},+)\) which describes the gap set of \(M^\infty (N)\):
Let \(m+1\) be the smallest non-gap of \(M^\infty (N)\); \(m\ge 1\) owing to \(n_1=1\).
To prove the desired bound (12.1) for the gap sizes \(n_j\), we change the representation of the monoid S with respect to \(m+1\). Namely, it is easy to see that there exist positive integers \(s_1,\ldots , s_m \in \mathbb {Z}_{>0}\), such that \(s_j\equiv j \pmod {m+1}\) for all \(j\in \{1,\ldots ,m\}\) and
In Sect. 13, we denote the number of gaps in a monoid S by \(\gamma (S)\). Hence, in the given context, \(g=\gamma (S)\). Recall that \(m+1\) is chosen to be the smallest non-gap of \(M^\infty (N)\). Therefore, we choose a monoid representation with respect to \(m+1\). Concretely, in this case, there are \(k_j\in \mathbb {Z}_{>0}\), such that
Now, Lemma 13.1 implies
Since \(j+(m+1)(k_j-1)\) are the largest non-gaps in each residue class modulo \(m+1\), this proves the bound given in (12.1), and the proof of the Weierstraß gap Theorem 12.2 is completed. \(\square \)
13 A Gap Property of Monoids
Let \(m\in \mathbb {Z}_{>0}\) and \(s_1,\ldots , s_m \in \mathbb {Z}_{>0}\), such that \(s_j\equiv j \pmod {m+1}\) for all \(j\in \{1,\ldots ,m\}\). We consider the additive submonoid:
of \((\mathbb {Z}_{\ge 0},+)\). A positive integer \(\ell \not \in S\) is called a gap of S.Footnote 11 Let \(\gamma (S)\) be the total number of gaps of S. Relating to our proof setting in Sect. 12, we choose this representation of S under the assumption that \(m+1\) is the smallest non-gap of S. By the definition of S, there are positive integers \(k_j\), such that
An easy count gives
Lemma 13.1
(Monoid gap lemma). Under these assumptions, one has for all \(j=1, \ldots ,m\):
In other words, the largest possible gap is bounded by \(2 \gamma (S)-1\). Before proving this statement, we prove two elementary observations.
Lemma 13.2
If i and \(\ell \) in \(\mathbb {Z}_{>0}\) are such that \(i + \ell = j\) for \(j\in \{1,\ldots , m\}\), then
Proof
The inequality is by \(s_i+s_\ell \in S\) with \(s_i+s_\ell \equiv j \pmod {m+1}\), and \(s_j\in S\) is minimal with this property. \(\square \)
Lemma 13.3
If i and \(\ell \) in \(\mathbb {Z}_{>0}\) are such that \(i + \ell = j+m+1\) for \(j\in \{1,\ldots , m\}\), then \(k_i + k_\ell +1\ge k_j\).
Proof
The inequality is by \(s_i+s_\ell \in S\) with \(s_i+s_\ell \equiv j \pmod {m+1}\), and \(s_j=j+(m+1)k_j\in S\) is minimal with this property. \(\square \)
Proof of Lemma 13.1
By (13.1), the statement to prove is equivalent to
By Lemma 13.2,
Summing the left and right sides, respectively; of these, \(j-1\) inequalities gives
By Lemma 13.3,
Summing the left and right sides, respectively; of these, \(m-j\) inequalities gives
Combining the two inequalities, we obtain that
which is (13.3). \(\square \)
14 Functions with Separation Property
The setting which we use throughout this section is: \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 2\) and \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) is an integral basis for \(M^\infty (N)\) over \(\mathbb {C}[t]\). Because of \({{\,\mathrm{pord}\,}}t = n\), for any fixed \(\alpha \in \mathbb {C}\), we have thatFootnote 12
We note that, as above, owing to \(t\in M^\infty (N)\), \(x_j=[\tau _j]_N\in \Gamma _0(N)\) with \(\tau _j\in \mathbb {H}\cup \mathbb {Q}\) are such that \([\tau _j]_N\ne [\infty ]_N\).
In Definition 9.4, we defined the separation property of f for \((t,v_0)\) with \(v_0=\alpha \) as in (14.1). At various places, we required f to have this property, for instance, in Proposition 11.5. In this section, we prove the existence of such f. In addition, here, we have to use the charts as in (9.1), (9.2), and (9.3).
Lemma 14.1
Given the setting of this section with \(\ell \ge 2\), let \(\tau , \tau '\in \{\tau _1,\ldots ,\tau _\ell \}\) be such that \(\tau \ne \tau '\). Then
Proof
We are free to relabel the indices of the preimages of \(\alpha \). Hence, it is sufficient to prove the statement for \(\tau _1:=\tau \) and \(\tau _2:=\tau '\). Suppose
As in (9.7), for \(j=1,\ldots ,\ell \) and suitable neighborhoods \(U_j\), one has local expansions for \(\tau \in U_j\):
Moreover, as in (9.10), for \(j=1,\ldots , \ell \), we can assume that the neighborhoods \(U_{j}\) are chosen, such that the following expansions exist for all \(\tau \in U_{j}\):
Taking \(a_j\in \mathbb {C}\) the quotient
defines a modular function \(g\in M(N)\). Now, \(g\in M^{\infty }(N)\) if and only if all the zeros \(\tau _j\) of \(t(\tau _j)-\alpha =0\) cancel out. Indeed, assuming (\(\star \)), one can determine \(a_j\in \mathbb {C}\), not all zero, such that this cancellation happens. Namely, using the local expansions the cancellation condition translates into a system of linear equations, where j runs form 1 to \(\ell \):Footnote 13
This gives in total \(k_1+\cdots +k_\ell =n\) equations. However, owing to (\(\star \)), two of these equations are the same. This means, we are left with \(n-1\) equations in n unknowns \(a_0,\ldots , a_{n-1}\). This implies that there exists a solution to the system with the \(a_j\) not all 0 . This produces a contradiction: since \(g\in M^\infty (N)\), there are polynomials \(p_j(x)\in \mathbb {C}[x]\), such that
Combining this with (14.2), the uniqueness of the basis representation gives
Consequently, all \(p_j(x)\) and all \(a_j\) must be zero. Hence, (\(\star \)) leads to a contradiction and the lemma is proved. \(\square \)
Lemma 14.2
Given the setting of this section with \(\ell \ge 2\), let
be a subset of \(m\in \{2,\ldots ,\ell \}\) pairwise distinct preimages of \(\alpha \). Then, there exist \(\alpha _i\in \mathbb {C}\), such that for \(f=\alpha _1 b_1+\cdots +\alpha _{n-1} b_{n-1}\):
Proof
We proceed by induction on m. If \(m=2\), then by the previous lemma, there exists an \(i\in \{1,\ldots ,n-1\}\), such that \(b_i(\tau _{i_1})\ne b_i(\tau _{i_2})\), and we choose \(f:=b_i\). Suppose \(m\ge 2\). Because of index relabeling, we can choose \(S_m:=\{\tau _1,\ldots ,\tau _m\}\), and the induction hypothesis gives an \(F\in M^\infty (N)\) as a \(\mathbb {C}\)-linear combination of \(b_j\), such that the values \(F(\tau _1),\ldots ,F(\tau _{m})\) are pairwise distinct. If \(F(\tau _{m+1})\ne F(\tau _i)\) for all \(i=1,\ldots ,m\), the induction step is done. Otherwise, \(F(\tau _{m+1})=F(\tau _r)\) for some \(r\in \{1,\ldots ,m\}\). By the previous lemma, there is some \(k\in \{1,\ldots ,n-1\}\), such that \(b_k(\tau _{m+1})\ne b_k(\tau _r)\), and we can choose a non-zero \(c\in \mathbb {C}\), such that
By inspection, one verifies for \(F_r:=F+cb_k\) that \(F_r(\tau _{m+1})\ne F_r(\tau _r)\) and also that the values
Suppose \(F_r(\tau _{m+1})=F_r(\tau _s)\) for some \(s\in \{1,\ldots ,m\}\backslash \{r\}\). If there is no such s, we are done with \(f:=F_r\). Otherwise, by the previous lemma, there is some \(l\in \{1,\ldots ,n-1\}\), such that \(b_l(\tau _{m+1})\ne b_l(\tau _s)\), and we can choose a non-zero \(d\in \mathbb {C}\), such that
Now, we set \(F_{r,s}:=F_r+db_l\), and see that,
together with pairwise distinct values \(F_{r,s}(\tau _1),\ldots ,F_{r,s}(\tau _{m})\). Iterating this argument exhausts all possibilities and the induction step is proved. \(\square \)
Under the assumptions as in (14.1), to have the separation property for \((t,\alpha )\), f additionally has to satisfy the conditions (9.13) which rewritten as order conditions are
Remark 14.3
The order in (14.4) has to be interpreted in view of (9.10) and in the sense of the \(\phi \)-order defined in Definition 15.3. This deviates slightly from the standard notation used in the theory of the Riemann surfaces, where (14.4) would be stated in the format:
This notation suppresses the explicit mentioning of the chart. We also use this notation, for instance, in cases like (2.4), where the chart is clear from the context, or in Lemma 16.1 when citing from Riemann surface theory.
Lemma 14.4
Given the setting of this section with \(\ell \ge 1\), assume that \(k_i={{\,\mathrm{ord}\,}}_{\phi _{\tau _i}(\tau )}(t(\tau )-\alpha )>1\) for some \(i\in \{1,\ldots ,n-1\}\). Then
Proof
Let us assume that
As in (9.7), for \(j=1,\ldots ,\ell \) and suitable neighborhoods \(U_j\), one has local expansions for \(\tau \in U_j\):
Now, we proceed with the proof exactly as above. Namely, as in (9.10), for \(j=1,\ldots , \ell \), we can assume that the neighborhoods \(U_{j}\) are chosen such that the following expansions exist for all \(\tau \in U_{j}\) and \(k\in \{1,\ldots ,n-1\}\):
Now, we apply the same strategy as in the proof of Lemma 14.1 and determine \(a_j\in \mathbb {C}\), not all zero, such that
This leads us to consider the same system of \(k_1+\cdots +k_\ell =n\) linear equations. This time, owing to (\(\star \star \)), we have \(d_1^{(k,i)}=0\) for all \(k=1,\ldots ,n-1\), and the equation containing the \(d_1^{(k,i)}\) is always satisfied and can be removed. This means, we are left with \(n-1\) equations in n unknowns \(a_0,\ldots , a_{n-1}\).Footnote 14 This implies that there exists a solution to the system with the \(a_j\) not all 0, which produces a contradiction as in the proof of Lemma 14.2. \(\square \)
Lemma 14.5
Again, we assume the setting of this section with \(\ell \ge 1\). Then, there exist \(\alpha _i\in \mathbb {C}\), such that for \(f:=\alpha _0t +\alpha _1 b_1+\cdots +\alpha _{n-1}b_{n-1}\):
Proof
For the proof, it is convenient to introduce an auxiliary function:
where j is chosen according to Lemma 14.4, namely, such that \({{\,\mathrm{ord}\,}}_{\phi _{\tau _i}(\tau )}(b_j(\tau )-b_j(\tau _i))=1\). We will show: given \(F=a_0 t + a_1 b_1 +\cdots + a_{n-1} b_{n-1}\) with \(a_j\in \mathbb {C}\), such that
then there is an \(f=\alpha _0 t + \alpha _1 b_1 +\cdots + \alpha _{n-1} b_{n-1}\) with \(\alpha _j\in \mathbb {C}\) which satisfies (14.6) and
In other words, we prove Lemma 14.5 by induction on k.
The base case \(k=1\) corresponds to the induction step from \(k=0\) to \(k=1\). The existence of F such that (14.8) holds is by Lemma 14.2. If, in addition,
we take \(f:=F\), and the base case \(k=1\) is done. If \({{\,\mathrm{ord}\,}}_{\phi _{\tau _1}(\tau )}(F(\tau )-F(\tau _1))>1\), define
where the quotient is taken for all \(i,j\in \{1,\ldots ,\ell \}\) for which the denominator is non-zero. Now, it is a straightforward verification that for this f the condition (14.6) holds and also that
This settles the base case \(k=1\).
For the induction step \(k\rightarrow k+1\), we assume that we have F of required form, such that (14.8) and (14.9) hold. If, in addition,
we are done. Otherwise, define
where the quotient is taken for all \(i,j\in \{1,\ldots ,\ell \}\) for which the denominator is non-zero. Now, we additionally require that
for \(a_{1,j}\) and \(b_{1,j}\) coming from the expansions:
Again, it is straightforward to verify that for such a choice f has the properties (14.6) and (14.10). The extra requirement (14.11) is needed to guarantee the first k instances of the latter condition. This completes the proof of the induction step and also the proof of Lemma 14.5. \(\square \)
Summarizing, using an integral basis \((1,b_1,\ldots , b_{n-1})\) for \(M^\infty (N)\) over \(\mathbb {C}[t]\), we constructed an f which proves
Corollary 14.6
For every \(\alpha \in {\mathbb {C}}\) , there is an \(f\in M^{\infty }(N)\), such that f has the separation property for \((t,\alpha )\).
15 Appendix: Modular Functions–Basic Notions
To make this article as much self-contained as possible, in this section, we recall most of the facts we need about modular functions.
The modular group \({\mathrm {SL}}_2({\mathbb {Z}})=\{\left( {\begin{matrix} a&{}b\\ c&{}d \end{matrix}} \right) \in \mathbb {Z}^{2\times 2}:a d - b c =1\}\) acts on the upper half \(\mathbb {H}\) of the complex plane by \(\left( {\begin{matrix} a&{}b\\ c&{}d \end{matrix}} \right) \tau := \frac{a \tau +b}{c \tau +d}\); this action is inherited by the congruence subgroups:
where throughout this paper, N is a fixed positive integer. Note that \(\Gamma _0(1)={\mathrm {SL}}_2({\mathbb {Z}})\). These subgroups have a finite index in \({\mathrm {SL}}_2({\mathbb {Z}})\):
see the standard literature on modular forms like [1] or [5]. Particularly related to our context are [9] and [12].
The action of \({\mathrm {SL}}_2({\mathbb {Z}})\) on \(\mathbb {H}\) extends to an action on meromorphic functions \(f:\mathbb {H}\rightarrow {\hat{\mathbb {C}}}:=\mathbb {C}\cup \{\infty \}\). A meromorphic function \(f:{\mathbb {H}}\rightarrow {\hat{\mathbb {C}}}\) is called a meromorphic modular function for\(\Gamma _0(N)\), if (i) for all \(\left( {\begin{matrix} a&{}b\\ c&{}d \end{matrix}} \right) \in \Gamma _0(N)\):
and (ii) for any \(\gamma =\left( {\begin{matrix} a&{}b\\ c&{}d \end{matrix}} \right) \in {\mathrm {SL}}_2({\mathbb {Z}})\), there exists an \(M=M(\gamma )\in {\mathbb {Z}}\) together with a Fourier expansion:
where \(q=q(\tau ):=e^{2\pi i\tau }\) and \(w_N(c):= N/\gcd (c^2,N)\). By M(N), we denote the set of meromorphic modular functions for \(\Gamma _0(N)\).
By (ii) with \(\left( {\begin{matrix} a&{}b\\ c&{}d \end{matrix}} \right) =\left( {\begin{matrix} 1&{}0\\ 0&{}1 \end{matrix}} \right) \), any \(f\in M(N)\) admits a Laurent series expansion in powers of q with finite principal part, that is
Hence, in view of \(\lim _{{\mathrm {Im}}(\tau )\rightarrow \infty } q(\tau ) = 0\), one can extend f to \({\mathbb {H}}\cup \{\infty \}\) by defining \(f(\infty ):=\infty \), if \(M>0\), and \(f(\infty ):= f_0\), otherwise. Subsequently, a Laurent expansion of f as in (15.2) will be also called q-expansion offat infinity.Footnote 15
Given \(\gamma =\left( {\begin{matrix} a&{}b\\ c&{}d \end{matrix}} \right) \in {\mathrm {SL}}_2({\mathbb {Z}})\) and \(f\in M(N)\), consider the Laurent series expansion of \(f(\gamma \tau )\) in powers of \(q^{1/w_N(c)}\),
In view of \(\gamma \infty =\lim _{{\mathrm {Im}}(\tau )\rightarrow \infty } \gamma \tau = a/c\), we say that (15.3) is a q-expansion offata / c. Understanding that \(a/0=\infty \), this also covers the definition of q-expansions at \(\infty \). Concerning the uniqueness of such expansions, let \(\gamma '\in {\mathrm {SL}}_2({\mathbb {Z}})\) be such that \(\gamma ' \infty = \gamma \infty = a/c\), then the q-expansion of \(f(\gamma ' \tau )\) differs from that of \(f(\gamma \tau )\) only by a root-of-unity factor in the coefficients, namely, we have then \(\gamma ' = \gamma \left( {\begin{matrix} \pm 1&{}m\\ 0&{}\pm 1 \end{matrix}} \right) \) for some \(m \in \mathbb {Z}\), which implies
As a consequence, one can extend f from \(\mathbb {H}\) to \({\hat{\mathbb {H}}}:=\mathbb {H}\cup \{\infty \}\cup \mathbb {Q}\) by defining \(f(a/c):=\lim _{{\mathrm {Im}}(\tau )\rightarrow \infty } f(\gamma \tau )\), where \(\gamma \in {\mathrm {SL}}_2({\mathbb {Z}})\) is chosen, such that \(\gamma \infty = a/c\). Another consequence is that the q-expansions of f at \(\infty \) are uniquely determined owing to
Next, notice that the action of \({\mathrm {SL}}_2({\mathbb {Z}})\), and thus of \(\Gamma _0(N)\), extends in an obvious way to an action on \({\hat{\mathbb {H}}}\). The orbits of the \(\Gamma _0(N)\) action are denoted by
In cases where N is clear from the context, one also writes \([\tau ]\) instead of \([\tau ]_N\). The set of all such orbits is denoted by
The \(\Gamma _0(N)\) action maps \({\mathbb {Q}}\cup \{\infty \}\) to itself, and owing to (15.1), each \(\Gamma _0(N)\) produces only finitely many orbits \([\tau ]_N\) with \(\tau \in {\mathbb {Q}}\cup \{\infty \}\); such orbits are called cusps of \(X_0(N)\). One has, for example,
Lemma 15.1
For any prime \(\ell \)
- (1)
\(X_0(\ell ) \text { has two cusps}: [\infty ]_\ell \, \text {and }\ [0]_\ell \);
- (2)
\(X_0(\ell ^2) \ \text { has }\ell +1\text { cusps}: [\infty ]_{\ell ^2},[0]_{\ell ^2}, \text {and }\ [k/\ell ]_{\ell ^2},\, \, k=1,\ldots , \ell -1\).
Proof
This fact can be found in many sources; a detailed description of how to construct a set of representatives for the cusps of \(\Gamma _0(N)\), for instance, is given in [16, Lemma 5.3]. \(\square \)
Suppose that the domain of \(f\in M(N)\) is extended from \(\mathbb {H}\) to \({\hat{\mathbb {H}}}\) as described above, i.e., \(f:\mathbb {H}\rightarrow {\hat{\mathbb {C}}}\) is extended to \(f:{\hat{\mathbb {H}}}\rightarrow {\hat{\mathbb {C}}}\), where we keep the same name for the extended function. Then, using this extension gives rise to a function \(f^*:X_0(N)\rightarrow {\hat{\mathbb {C}}}\), which is defined as follows:
The fact that \(f^*\) is well-defined follows from our previous discussion. We say that \(f^*\) is induced by f.
As described in detail in [5], \(X_0(N)\) can be equipped with the structure of a compact Riemann surface. This analytic structure turns the induced functions \(f^*\) into meromorphic functions on \(X_0(N)\). The following classical lemma [11, Theorem 1.37], a Riemann surface version of Liouville’s theorem, is crucial for zero recognition of modular functions.
Lemma 15.2
Let X be a compact Riemann surface. Suppose that \(g:X \rightarrow \mathbb {C}\) is a holomorphic function on all of X. Then, g is a constant function.
Being meromorphic, modular functions form fields. A classic example is that \(M(N)=\mathbb {C}(j(\tau ), j(N\tau ))\), e.g., [5, Proposition 7.5.1], where j is the modular invariant (the Klein j function). The subset
which is important for our context and, obviously, is not a field but a \(\mathbb {C}\)-algebra. In this case, owing to the definition of induced functions \(f^*\), all possible poles of \(f^*\) can be spotted by checking whether \(f^*([a/c])=f(a/c)=\infty \) for \(a/c\in \mathbb {Q}\cup \{\infty \}\). Because of (15.1), \(\mathbb {Q}\cup \{\infty \}\) splits only into a finite number of cusps:
Hence, knowing all the cusps \([a_j/c_j]\) reduces the task of finding all possible poles to the inspection of q-expansions of f at \(a_j/c_j\); i.e., of q-expansions of \(f(\gamma _j \tau )\) as in (15.3) with \(\gamma _j\in {\mathrm {SL}}_2({\mathbb {Z}})\) such that \(\gamma _j \infty = a_j/c_j\). We call these expansions also local q-expansions of \(f^*\) at the cusps \([a_j/c_j]_N\); \(w_N(c_j)\) is called the width of the cusp \([a_j/c_j]_N\). It is straightforward to show that it is independent of the choice of the representative \(a_j/c_j\) of the cusp \([a_j/c_j]_N\), and that \(w_N(c_j)=N/\gcd (c_j^2,N)\) for relatively prime \(a_j\) and \(c_j\). Note that \([\infty ]_N= [1/0]_N\).
Definition 15.3
(Order and\(\phi \)-order). Let \(f=\sum _{n=m}^\infty a_n q^n\) with \(m\in \mathbb {Z}\), such that \(a_m\ne 0\). Then we define the order of f as
More generally, if \(\phi =\sum _{n= 1}^\infty b_n q^{n/w}\) for some fixed \(w\in \mathbb {Z}_{>0},\) and \(F=f\circ \phi :=\sum _{n=m}^\infty a_n \phi ^n\), then we define the \(\phi \)-order of f as
(e.g., if \(m={{\,\mathrm{ord}\,}}f=-1\) and \(\phi =q^2\), then \({{\,\mathrm{ord}\,}}_\phi F=-1\), but \({{\,\mathrm{ord}\,}}F=-2\); if \(m={{\,\mathrm{ord}\,}}f=-2\) and \(\phi =q^{1/2}\), then \({{\,\mathrm{ord}\,}}_\phi F=-2\), but \({{\,\mathrm{ord}\,}}F=-1\).) In addition, more generally, we extend this definition of \(\phi \)-order to the case, where \(\phi :=\phi _{\tau _0}(\tau )\) is one of the charts as in (9.1), (9.2), and (9.3).
The order \({{\,\mathrm{ord}\,}}_{[a/c]_\ell } f^*\) of \(f^*\) at a cusp \([a/c]_\ell \) is defined to be the \(q^{1/w_\ell (c)}\)-order of a local q-expansion of \(f^*\) at \([a/c]_N\), that is,
It is straightforward to verify that \({{\,\mathrm{ord}\,}}_{[a/c]}f^*\) is well-defined. For a concrete example, see Example 2.1.
16 Appendix: Meromorphic Functions on Riemann Surfaces–Basic Notions
To make this article as much self-contained as possible, in this second appendix section, we recall most of the facts that we need about meromorphic functions on Riemann surfaces. For the terminology, we basically follow [7]; other classic texts are [6] and [11].
Lemma 15.2 states the fundamental fact that any analytic function on a compact Riemann surface is constant. In Example 2.1, we have seen that \(z_5^*\) has its only zero of order 1 at \([\infty ]_5\) and its only pole at \([0]_5\) with multiplicity 1, i.e., \(z_5^*\) has order \(-1\) at \([\infty ]_5\).Footnote 16 This is also in accordance with Lemma 16.1, a corollary of another fundamental fact which says that meromorphic functions on compact Riemann surfaces have exactly as many zeroes as poles (counting multiplicities); see, for instance, [11, Proposition 4.12]:
Lemma 16.1
Let g be a non-constant meromorphic function on a compact Riemann surface X. Then
Here, \({{\,\mathrm{ord}\,}}_{x_0} g\) is defined as follows. Suppose \(g(x)= \sum _{n=m}^\infty c_n (\varphi (x)-\varphi (x_0))^n\), \(c_m\ne 0\), is the local Laurent expansion of g at \(x_0\) using the local coordinate chart \(\varphi :U_0\rightarrow \mathbb {C}\) which homeomorphically maps a neighborhood \(U_0\) of \(x_0\in X\) to an open set \(V_0\subseteq \mathbb {C}\). Then, \({{\,\mathrm{ord}\,}}_{x_0} g :=m\).
Let \({\mathcal {M}}(S)\) denote the field of meromorphic functions \(f:S\rightarrow {\hat{\mathbb {C}}}\) on a Riemann surface S.Footnote 17 Let \(f\in {\mathcal {M}}(S)\) be non-constant: then for every neighborhood U of \(x\in S\), there exist neighborhoods \(U_x\subseteq U\) of x and V of f(x), such that the set \(f^{-1}(v)\cap U_x\) contains exactly k elements for every \(v\in V\backslash \{f(x)\}\). This number k is called the multiplicity of f at x; notation: \(k={{\,\mathrm{mult}\,}}_x(f)\).Footnote 18 If S is compact, \(f\in {\mathcal {M}}(S)\) is surjective and each \(v \in {\hat{\mathbb {C}}}\) has the same number of preimages, say n, counting multiplicities; i.e., \(n=\sum _{x\in f^{-1}(v)}{{\,\mathrm{mult}\,}}_x(f)\), see, e.g., [7, Theorem 4.24]. This number n is called the degree of f; notation: \(n={\mathrm {Deg}}(f)\). One of the consequences is that non-constant functions on compact Riemann surfaces have as many (finitely many) zeros as poles counting multiplicities; this is Lemma 16.1.
\({\mathrm {RamiPts}}(f):= \{x\in S: {{\,\mathrm{mult}\,}}_x(f)\ge 2\}\) denotes the set of ramification points of f; \({\mathrm {BranchPts}}(f):= f({\mathrm {RamiPts}}(f))\subseteq {\hat{\mathbb {C}}}\) denotes the set of branch points of f. Ramification points, and also branch points, of a function f form sets having no accumulation point. Hence, for functions on compact Riemann surfaces, these sets have finitely many elements.
17 Conclusion
In this article, we present the first proof of the Weierstraß gap theorem (for modular functions) without using the Riemann–Roch theorem. The main ingredient in our proof is the concept of order-reduction polynomials which corresponds to the discriminant of a field extension of \(\mathbb {Q}\) in the setting of algebraic number theory, see, for instance, [10, III, §3]. In the field case, the structure of this discriminant is related to the ramification index [10, III, §2, Proposition 8, and III, §3, Proposition 14]. Analogously, in Proposition 11.5, we give a factorization of the order-reduction polynomial which in direct fashion relates to the branch points of the modular function t. This relation allows us to connect the degree of this polynomial to the genus of \(X_0(N)\). This observation is crucial for our proof of the Weierstraß gap theorem.
In addition, our approach gives new algebraic and algorithmic insight based on module presentations of modular function algebras, in particular, the usage of integral bases. For example, our proof also gives a method to compute the order-reduction polynomial by using the Puiseux series expansions at infinity. Another new feature concerns the gap bound: the main task of our proof is to show that there are exactly g gaps for any modular function algebra. The proof that the corresponding pole orders are bounded by \(2g-1\), with the help of an elementary combinatorial argument turns out to be an immediate consequence of our approach. Another by-product of our framework is a natural explanation of the genus \(g=0\) case as a consequence of the reduction to an integral basis.
Summarizing, our setting generalizes ideas from algebraic number theory, but still stays close to “first principles.” Hence, we feel that our approach has potential for further extensions and applications. For example, we are planning to exploit the algorithmic content of our approach for computer algebra applications, for instance, for the effective computation of suitable module bases for modular function algebras.
Notes
A \(\mathbb {C}\)-algebra is a commutative ring with 1 which is also a vector space over \(\mathbb {C}\).
Notice that \({{\,\mathrm{pord}\,}}1/z_{11}=5\).
The existence of such a \(v_0\) is owing to Corollary 4.4.
Notice that, in particular, \(\tau \ne \infty \).
Note that, in particular, \(\tau _j\ne \infty \).
Note that \(\tau _j\in \mathbb {H}\cup \mathbb {Q}\) are such that \([\tau _j]_N\ne [\infty ]_N\). Hence, (9.16) implies that \(F\not \in M^\infty (N)\).
How to construct such f is described in Sect. 14.
I.e., \(\phi _{\tau _0}(\tau _0)=0\).
Actually the special case we need, \(Y={\hat{\mathbb {C}}}\), was given by Riemann; e.g., [2].
This means, in this case, one has \(M^\infty (N)=\mathbb {C}[h]\).
The largest gap is called the Frobenius number of S.
Recall that \(t^*([\tau ]_n)=t(\tau )\).
Notice that the charts \(\phi _{\tau _j}(\tau )\) are centered at 0; i.e., \(\phi _{\tau _j}(\tau _j)=0\). Consequently, for fixed j, the numerator in (14.2) has to be of the form: \(\phi _{\tau _j}(\tau )^{k_j}(c_0+c_1\phi _{\tau _j}(\tau )+ \cdots ).\)
Notice that for this argument to work, we invoke \(k_i>1\).
This expansion and also those for \(f(\gamma \tau )\) are required to converge for all \(\tau \in \mathbb {H}\) with \({\mathrm {Im}}(\tau )\) sufficiently large.
Notice that we also say that \(z_5^*\) has pole order 1 at \([\infty ]_5\).
In this context, \({\hat{\mathbb {C}}}:=\mathbb {C}\cup \{\infty \}\) is understood to be a compact Riemann surface isomorphic to the Riemann sphere.
If x is a pole of \(f:\)\({{\,\mathrm{mult}\,}}_x f =-{{\,\mathrm{ord}\,}}_x f\); otherwise, \({{\,\mathrm{mult}\,}}_x f = {{\,\mathrm{ord}\,}}_x (f - f(x))\).
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Acknowledgements
Open access funding provided by Johannes Kepler University Linz. Both authors were supported by grant SFB F50-06 of the Austrian Science Fund (FWF). In November 2018, while working on parts of this paper, the first named author enjoyed the overwhelming hospitality of Bill Chen and his team at the Center for Applied Mathematics, Tianjin University. Finally, we thank Krishnaswami Alladi and an anonymous referee for constructive remarks which helped to improve the presentation of this paper.
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Dedicated to our good friend George Andrews at the occasion of his 80th birthday
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Paule, P., Radu, CS. A Proof of the Weierstraß Gap Theorem not Using the Riemann–Roch Formula. Ann. Comb. 23, 963–1007 (2019). https://doi.org/10.1007/s00026-019-00459-2
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DOI: https://doi.org/10.1007/s00026-019-00459-2