Pfaffian definitions of Weierstrass elliptic functions

Abstract

We give explicit definitions of the Weierstrass elliptic functions \(\wp \) and \(\zeta \) in terms of pfaffian functions, with complexity independent of the lattice involved. We also give such a definition for a modification of the Weierstrass function \(\sigma \). Our work has immediate applications to Diophantine geometry and we answer a question of Corvaja, Masser and Zannier on additive extensions of elliptic curves. We also point out further applications, also in connection with Pila–Wilkie type counting problems.

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Communicated by Ngaiming Mok.

Appendix

Appendix

Here we establish some limits on how far our results can be pushed. To begin, we show that no analogue of our results for \(\wp , \zeta \) and \(\varphi \) holds for \(\sigma \). Here it will be convenient to use some terminology from logic.

We assume that \(\lambda \in {\mathcal {F}}\) and recall the singular expansion of \(\omega _2\) (28) and the relations (10),(11). We set \(\omega = \omega _1 + \omega _2, \eta = \eta _1 + \eta _2\). We first note that as

$$\begin{aligned} \omega _2 + \omega _1 = i(\omega _1/\pi )\log (\lambda ) + \omega _1+u \end{aligned}$$

we have

$$\begin{aligned} \omega _2' + \omega _1' = i(\omega _1'/\pi )\log (\lambda ) +i\omega _1/(\lambda \pi )+ \omega _1'+u'. \end{aligned}$$

Using the expansion \(\omega _1 = \pi + O(\lambda )\) we get

$$\begin{aligned} 2\lambda (1-\lambda )(\omega _2' + \omega _1') = 2i +O(\lambda \log (\lambda )) \end{aligned}$$

and also using \(u = i4\log 2 + O(\lambda )\) we have

$$\begin{aligned} \omega _2 + \omega _1 = i \log (\lambda ) + \pi + i4\log 2 + O(\lambda \log (\lambda )). \end{aligned}$$

With the help of these expressions we compute

$$\begin{aligned} \omega \eta = -\frac{1}{3}\log (\lambda )^2 +\frac{2}{3}(i\pi - 4\log 2)\log (\lambda ) - 2\log (\lambda ) +O(1). \end{aligned}$$

If we express \(\log (\lambda )^2 = \log (|\lambda |) +2i\arg (\lambda )\log (|\lambda |) -\arg (\lambda )^2\) we can further compute

$$\begin{aligned} \mathfrak {I}(\omega \eta ) = \frac{2}{3}(\pi -\arg (\lambda ))\log |\lambda | +O(1). \end{aligned}$$

It follows from this that as \(\lambda \in {\mathcal {F}}\) approaches 0, the imaginary part of \(\omega \eta \) tends to infinity, for we have \(|\arg (\lambda )|\le \pi /2\) for such \(\lambda \).

Let \(\sigma _\lambda \) be the Weierstrass sigma function associated to the lattice spanned by \(\omega _1,\omega _2\). This function is odd and satisfies

$$\begin{aligned} \sigma _\lambda (z + \omega ) = -\sigma _\lambda \exp \left( \eta \left( z + \frac{\omega }{2}\right) \right) \end{aligned}$$

[16, Theorem 1, page 241]. Using these two properties we deduce that

$$\begin{aligned} \frac{\sigma _\lambda \left( \left( \frac{1}{2}+r\right) \omega \right) }{\sigma _\lambda \left( \left( \frac{1}{2}-r\right) \omega \right) } = \exp (r\eta \omega ). \end{aligned}$$

For \(r \in [0,\frac{1}{2})\) both \((\frac{1}{2}-r)\omega \) and \((\frac{1}{2}+r)\omega \) lie in the fundamental parallelogram spanned by \(\omega _1,\omega _2\). Thus the function

$$\begin{aligned} \psi _\lambda (r) = \mathfrak {I}\left( \frac{\sigma _\lambda \left( \left( \frac{1}{2}+r\right) \omega \right) }{\sigma _\lambda \left( \left( \frac{1}{2}-r\right) \omega \right) }\right) \end{aligned}$$

restricted to \([0,\frac{1}{2})\) is definable in the expansion of the real field by the restrictions of \(\mathfrak {R}\sigma _\lambda , \mathfrak {I}\sigma _\lambda \) to this fundamental parallelogram. However \(\psi _\lambda (r_0) = 0\) whenever \(\theta (r_0) = \frac{1}{\pi }(\mathfrak {I}(r_0\eta \omega ))\) is an integer. The function \(\theta \) is real and continuous on \([0,\frac{1}{2})\) and its image contains an interval of length

$$\begin{aligned} \left| \frac{\mathfrak {I}(\eta \omega )}{2\pi }\right| \end{aligned}$$

and so at least \(|\frac{\mathfrak {I}(\eta \omega )}{2\pi }|-1\) distinct integers. By our observations above, this latter expression tends to \(+\infty \) as \(\lambda \) approaches 0 and so the number of zeroes of \(\psi _\lambda \) is unbounded as \(\lambda \) tends to 0. Using this we have the following.

Proposition 26

Let L be the language of the real ordered field together with two binary functions f and g. Then there is a formula \(\theta (x)\) with the following property. For every positive integer n there exists \(\varepsilon >0\) such that if \(\lambda \in {\mathcal {F}}\) with \(|\lambda |<\varepsilon \) then, upon interpreting f and g as the real and imaginary parts of \(\sigma _\lambda \), the set defined by \(\theta \) has at least n connected components.

To prove this from the above, we simply let \(\theta (x)\) be a formula which expresses \(\psi _\lambda (x)=0\) in the structures mentioned in the proposition.

From the proposition, it follows immediately that there is no analogue of Peterzil and Starchenko’s well-known result on the two-variable \(\wp \)-function [23] for \(\sigma \) as a function of two variables. Similarly, it follows easily from the proposition (and Khovanskii’s theorem) that if \(B(\lambda )\) is a bound on the entries of the format of a representation of \(\sigma |_{\mathfrak {F}_\Omega }\) as a piecewise subpfaffian set, then \(B(\lambda )\) is unbounded as \(\lambda \) varies in \({\mathcal {F}}\). So there is no analogue for \(\sigma \) of our results for \(\wp ,\zeta \) and \(\varphi \).

Finally, we discuss the choice of the fundamental domain \(\mathfrak {F}_\Omega \). We have chosen \(\omega _1\) and \(\omega _2\) such that \(\omega _2/\omega _1\) lies in the standard fundamental domain in the upper half plane. Surprisingly (to us at least), this choice is important. In fact, if we change the fundamental domain of \(\Omega \), the format of the corresponding definition of \(\wp \) might go up. To see this, let

$$\begin{aligned} \mathfrak {F}_\Omega ^{(a,b,c,d)}= \{r_1(a\omega _1+b\omega _2) +r_2(c\omega _1+ d\omega _2); 0\le r_1,r_2<1, r_1^2+r_2^2\ne 0\}. \end{aligned}$$

for some integers abcd with \(ad-bc =1\). Let \(B'(a,b,c,d)\) be a bound on the entries of the format of a representation of \(\wp |_{\mathfrak {F}^{(a,b,c,d)}_\Omega }\) as a piecewise subpfaffian set.

Proposition 27

The number \(B'\) tends to infinity as \(\max \{|a|,|b|,|c|,|d|\}\) tends to infinity.

Proof

Suppose not. Then we can find an infinite sequence of distinct tuples for which \(B'\) is smaller than a fixed constant. Take a tuple (abcd) of that sequence and pick the entry that has modulus \(n =\max \{|a|,|b|,|c|,|d|\}\). Say it is a. By our assumption there is a representations of the curves \(C=\{ \wp (r\omega _2): r\in (0,1)\}\) and \(C_n=\{ \wp (r(a\omega _1+b\omega _2)) : r\in (0,1)\}\) as piecewise subpfaffian sets whose formats are bounded independently of n. And then there is a similar representation of \(C\cap C_n\). So by Khovanskii’s theorem the number of connected components of this set is again bounded independently of n. However, this set contains at least \((n-1)/2\) isolated points. If one of the other entries has modulus n we can make an analogous construction. Thus n is bounded along that sequence. This is a contradiction. \(\square \)

Motivated by a question that Corvaja and Zannier asked us, we now show how the results in this paper lead to an effective bound on the Betti map of a section of \(E_\lambda \) restricted to a small triangle in \({\mathbb {C}}\) with a vertex 0.

In order to keep the discussion brief we restrict our attention to triangles contained in \({\mathcal {F}}\) but this could be extended without difficulty.

Let \(Q \in {\mathbb {C}}[X,Y]\ne 0\) and choose \(\Delta \) an open triangle in the set \({\mathcal {F}}\) with 0 as one of its vertices such that there is an analytic solution \(\xi \) to

$$\begin{aligned} Q(\xi ,\lambda )=0 \end{aligned}$$

on \(\Delta \) and each such solution satisfies \(\xi (\lambda ) \ne 0,1,\lambda \). Now we fix such an analytic solution \(\xi \). If \(\xi \) is non-constant we pick \(\lambda _0 \in \Delta \) such that \(\xi (\lambda )\in X_\lambda \) for \(\lambda \) in a neighbourhood of \(\lambda _0\) and define the Betti coordinates for \(\xi (\lambda )\) as usual. If \(\xi \) is constant we also define the Betti coordinates with z defined as the continuation of z from the north if \(\xi \) lies on the boundary of \(X_\lambda \). We continue the Betti-coordinates from this neighbourhood analytically to \(\Delta \).

Proposition 28

The Betti map of \(\xi \) on \(\Delta \) is bounded effectively by a constant that depends only on the degree of Q.

Proof

Let \(\lambda _1\in \Delta \) and \(l:[0,1]\rightarrow \Delta \) be a straight line joining \(\lambda _0\) and \(\lambda _1\). The set

$$\begin{aligned} \{t \in [0,1]; Q(\xi ,l(t))=0 \text { and } \xi \in (-\infty ,0]\cup [1,\infty )\cup \{\text {image of } l\}\} \end{aligned}$$

has a finite number of connected components \(N(\lambda _1)\) and this number is bounded solely and effectively by the degree of Q.

The map \(L =(l,\xi \circ l)\) defines a path in the space \(S= \Delta \times {\mathbb {C}}{\setminus }((\Delta \times \{0\}) \cup (\Delta \times \{1\})\cup \{(\lambda ,\lambda ); \lambda \in \Delta \})\) and we can choose a path from \((\lambda _1,\xi (\lambda _1))\) to \(p_0=(\lambda _0,\xi (\lambda _0))\) lying entirely in the fibred product \(\Delta \times _\lambda X_\lambda \). We can compose those two paths to get a loop \(\gamma \in \pi _1(p_0,S)\).

The fundamental group \(F=\pi _1(p_0,S)\) is generated by the three loops \(\gamma _1, \gamma _2\) and \( \gamma _3\) around \(\Delta \times \{0\}, \Delta \times \{1\}\) and \( \{(\lambda ,\lambda ); \lambda \in \Delta \}\) respectively. These are chosen such that, say, the compositum with \(\xi \) of the first two are small loops around 0 and 1 respectively while the compositum of the third with \(\xi -\lambda \) is a small loop around 0. There is a group homomorphism \(\rho :F \rightarrow S_2\ltimes {\mathbb {Z}}^2\) where the group law on \(S_2\ltimes {\mathbb {Z}}^2\) is defined by \((x_1,y_1)\cdot (x_2,y_2) =(x_1x_2, x_1y_1 +y_2)\) (where we identified \(S_2\) with \(\{\pm 1\}\)). From (7) and (8) we can deduce that it is given by

$$\begin{aligned} \rho (\gamma _1) =(-1,(0,1)), \rho (\gamma _2) = (-1,(1,0)), \rho (\gamma _3) = (-1,(1,1)) \end{aligned}$$

and the action of F on the Betti-coordinates (given by analytic continuation) can be expressed by \((a,b):(b_1,b_2) \rightarrow a(b_1,b_2) +b\).

Now by an elementary geometric argument the word-length of \(\gamma \) as a word in \(\gamma _1,\gamma _2, \gamma _3\) can be bounded from above by \(N(\lambda _1)\) which as remarked above is bounded independently of \(\lambda _1\). It follows that if \(\rho (\gamma ) =(x,y)\) then y is bounded independently of \(\lambda _1\). Since by Proposition 4 the Betti-coordinates on \(X_\lambda \) are bounded effectively so are the Betti coordinates of \(\xi \) on \(\Delta \). \(\square \)

We note that although there is some choice involved in \(\Delta \), the bound obtained is independent of the choices made. To obtain a statement about a general open triangle with vertex 0, contained in \({\mathbb {C}}{\setminus }\{0,1\}\) we cut it into several simple regions. This construction can also be carried out in an effective manner.

Finally, again in connection with correspondence with Corvaja and Zannier, we discuss the definability of Betti maps (of sections of \(E_\lambda \)), viewed now as functions of \(\lambda \). To this end we fix \(U\subseteq {\mathbb {C}}{\setminus }\{0,1\}\), an open set, definable (by which we shall always mean definable in \({\mathbb {R}}_{\text {an,exp}}\)). We suppose that U is simply connected. For instance U could be a sector of the unit circle. On U we take some choice of period maps \(\omega _1\) and \(\omega _2\) (we need not make the particular choice made elsewhere in the paper, but we do number them such that the quotient below takes values in the upper half plane). We now write \(\lambda \) for the usual \(\lambda \)-function on the upper half plane, and so we will write t for the variable in U. The quotient \(\omega _2/\omega _1\) is a branch of the inverse of \(\lambda \). By a theorem of Peterzil and Starchenko [23], \(\lambda \) is definable on its usual fundamental domain and on the image of this domain under finitely many elements in \(\Gamma (2)\) (the elements needed will depend on U and on the choices of the periods). As the inverse of a definable function, the quotient above is definable. It follows that the derivative of this quotient is also definable (see, for example, Chapter 7 of van den Dries’s book [9]). Computing, we find that

$$\begin{aligned} \left( \frac{\omega _2}{\omega _1}\right) '(t)= \frac{c}{t(1-t)\omega _1(t)^2}, \end{aligned}$$

for some absolute \(c \ne 0\). So \(\omega _2\) is too.

To get the definability of the elliptic logarithms, we use the definability, also due to Peterzil and Starchenko [23], of the map \(\wp \), as a function of both z and \(\tau \), on the domain \(\{ (\tau , z): \tau \in {\mathcal {S}} \text { and } z\in \mathfrak {F}_{\Omega _\tau } \}\). Here \({\mathcal {S}}\) is the usual fundamental domain in the upper half plane, and \(\Omega _\tau \) is the lattice generated by 1 and \(\tau \). This definability clearly extends to the domain with \({\mathcal {S}}\) replaced by the union of fundamental domains for the \(\lambda \) function that we used above.

Suppose that \(\xi \) is an algebraic function of t, and that we have fixed a definite well-defined branch on U. We now write z for some branch on U of the elliptic logarthim determined by \(\xi \). This satisfies

$$\begin{aligned} \wp (\lambda ^{-1}(t), z(t))= \xi (t) -\frac{1}{3}(t+1), \end{aligned}$$

with the inverse of \(\lambda \) that we gave above. From this follows the definability of z on U. And once we have the periods and the logarithm we get the definability of the Betti maps on U, defined as usual by (5) and (6) but with the periods and logarithm as above. It then follows from general facts on definability, which can again be found for instance in Chapter 7 of van den Dries’s book [9], that the differential of the Betti map (considered in more general setting in [7]) is also definable.

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Jones, G., Schmidt, H. Pfaffian definitions of Weierstrass elliptic functions. Math. Ann. (2020). https://doi.org/10.1007/s00208-019-01948-8

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