A Divisor Formula and a Bound on the \(\mathbb {Q}\)-Gonality of the Modular Curve \(X_1(N)\)

Abstract

We give a formula for divisors of modular units on \(X_1(N)\) and use it to prove that the \(\mathbb {Q}\)-gonality of the modular curve \(X_1(N)\) is bounded above by \([{11N^2}/{840}]\), where \([\cdot ]\) denotes the nearest integer.

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Fig. 1

Notes

  1. 1.

    \({\dagger }\) These subsets are Galois orbits, see Sect. 4, or alternatively, Appendix A.1

References

  1. 1.

    Baaziz, H.: Equations for the modular curve \(X_1(N)\) and models of elliptic curves with torsion points. Math. Comp., 79(272):2371–2386, 2010. ISSN 0025-5718. https://doi.org/10.1090/S0025-5718-10-02332-X

  2. 2.

    Bourdon, A., Ejder, Ö., Liu, Y., Odumodu, F., Viray, B.: On the level of modular curves that give rise to isolated j-invariants. Advances in Mathematics 357, 106824 (2019)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Carlucci, P.: Cuspidal divisor class groups of non-split Cartan modular curves. Acta Arith., 187(4):301–327, (2019). ISSN 0065-1036. https://doi.org/10.4064/aa8516-6-2018

  4. 4.

    Chen, Y-H.: Cuspidal \({\mathbb{Q}}\)-rational torsion subgroup of \(J(\Gamma )\) of level \(P\). Taiwanese J. Math., 15(3):1305–1323, 2011. ISSN 1027-5487. https://doi.org/10.11650/twjm/1500406301

  5. 5.

    Csirik, J.A.: The Kernel of the Eistenstein Ideal. J. of Number Theory 92, 348–375 (2002)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Derickx, M., Etropolski, A., van Hoeij, M., Morrow, J.S., Zureick-Brown, D.: Sporadic cubic torsion. arXiv e-prints, art. arXiv:2007.13929, July (2020). To appear in Algebra & Number Theory

  7. 7.

    Derickx, M., van Hoeij, M.: Gonality of the modular curve \(X_1(N)\). J. Algebra, 417:52–71, (2014). ISSN 0021-8693. https://doi.org/10.1016/j.jalgebra.2014.06.026

  8. 8.

    Derickx, M., van Hoeij, M., Zeng, J.: Computing Galois representations and equations for modular curves \(X_H(l)\). arXiv e-prints, art. arXiv:1312.6819, Mar. (2014)

  9. 9.

    Diamond, F., Shurman, J.: A first course in modular forms, volume 228 of Graduate Texts in Mathematics. Springer-Verlag, New York, (2005). ISBN 0-387-23229-X

  10. 10.

    DLMF. NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.26 of 2020-03-15. URL http://dlmf.nist.gov/. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

  11. 11.

    Hurwitz, A., Courant, R.: Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Interscience Publishers Inc, New York (1944)

    Google Scholar 

  12. 12.

    Kubert, D.S., Lang, S.: Modular units, volume 244 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science]. Springer-Verlag, New York-Berlin, (1981). ISBN 0-387-90517-0

  13. 13.

    Mazur, B.: Some comments on elliptic curves over general number fields and Brill-Noether modular varieties. 2014. URL http://people.math.harvard.edu/~mazur/papers/For.Maine.0718.2014(3).pdf

  14. 14.

    Najman, F.: Torsion of rational elliptic curves over cubic fields and sporadic points on \(X_1(n)\). Math. Res. Lett., 23(1):245–272, 2016. ISSN 1073-2780. URL https://doi.org/10.4310/MRL.2016.v23.n1.a12

  15. 15.

    Serre, J.-P.: Local fields, volume 67 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1979. ISBN 0-387-90424-7. Translated from the French by Marvin Jay Greenberg

  16. 16.

    Silverman, J.H.: The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics. Springer, Dordrecht, second edition, ISBN 978-0-387-09493-9. (2009) URL https://doi.org/10.1007/978-0-387-09494-6

  17. 17.

    Smith, H.: Ramification in the Division Fields of Elliptic Curves and an Application to Sporadic Points on Modular Curves. arXiv e-prints, art. arXiv:1810.04809, Oct. (2018)

  18. 18.

    Streng, M.: Generators of the group of modular units for \(\Gamma ^1(N)\) over \({\mathbb{Q}}\). arXiv e-prints, art. arXiv:1503.08127, Mar (2015)

  19. 19.

    Sutherland, A.: Torsion subgroups of elliptic curves over number fields. (2012a). URL https://math.mit.edu/~drew/MazursTheoremSubsequentResults.pdf

  20. 20.

    Sutherland, A., van Hoeij, M.: Defining equations for \(X_1(N)\). 2014. URL https://math.mit.edu/~drew/X1_optcurves.html

  21. 21.

    Sutherland, A.V.: Constructing elliptic curves over finite fields with prescribed torsion. Mathematics of Computation, 81(278):1131–1147, (2012b). ISSN 00255718, 10886842. http://www.jstor.org/stable/23267989

  22. 22.

    Sutherland, A.V., Zywina, D.: Modular curves of prime-power level with infinitely many rational points. Algebra Number Theory, 11(5):1199–1229, (2017). ISSN 1937-0652. URL https://doi.org/10.2140/ant.2017.11.1199

  23. 23.

    Takagi, T.: Cuspidal class number formula for the modular curves \(X_1(p)\). J. Algebra, 151(2):348–374, (1992). ISSN 0021-8693. URL https://doi.org/10.1016/0021-8693(92)90119-7

  24. 24.

    van Hoeij, M.: Low Degree Places on the Modular Curve X1(N). (2013). URL https://www.math.fsu.edu/~hoeij/files/X1N/LowDegreePlaces

  25. 25.

    van Hoeij, M.: Minformula (in file: cusp divisors program). (2013). URL www.math.fsu.edu/~hoeij/files/X1N

  26. 26.

    van Hoeij, M., Smith, H.: Computations for gonality bound. April 28, (2020). URL https://doi.org/10.5281/zenodo.4396300

  27. 27.

    Yang, Y.: Modular units and cuspidal divisor class groups of \(X_1(N)\). J. Algebra, 322(2):514–553, (2009). ISSN 0021-8693. URL https://doi.org/10.1016/j.jalgebra.2009.04.012

  28. 28.

    Yang, Y., Yu, J.-D.: Structure of the cuspidal rational torsion subgroup of \(J_1(p^n)\). J. Lond. Math. Soc. (2), 82(1):203–228, (2010). ISSN 0024-6107. URL https://doi.org/10.1112/jlms/jdq013

  29. 29.

    Yoo, H.: The rational cuspidal divisor class group of \(X_0(N)\). arXiv e-prints, art. arXiv:1908.06411, Aug (2019)

  30. 30.

    Yu, J.: A cuspidal class number formula for the modular curves \(X_{1}(N)\). Math. Ann., 252(3):197–216, (1980). ISSN 0025-5831. URL https://doi.org/10.1007/BF01420083

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Acknowledgements

The first author was supported by NSF grants 1618657 and 2007959. The authors would like to extend a very appreciative thanks to the anonymous referees for their careful reading and detailed comments. The second author would also like to thank Katherine E. Stange for the many helpful conversations about modular units.

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Appendices

A. A second proof of the MinFormula

A.1 Cusps: a modular interpretation

Take the congruence subgroup

$$\begin{aligned} \Gamma _1(N)=\left\{ \begin{bmatrix} a &{} b\\ c &{} d \end{bmatrix}\in {\text {SL}}_2(\mathbb {Z}) \ \bigg | \ \begin{bmatrix} a &{} b\\ c &{} d \end{bmatrix}\equiv \begin{bmatrix} 1 &{} *\\ 0 &{} 1 \end{bmatrix} \bmod N \right\} \end{aligned}$$

where \(*\) indicates the entry is unspecified. The extended complex upper half plane is

$$\begin{aligned} \overline{{\mathcal H}}={\mathcal H}\cup \mathbb {Q}\cup \{\infty \}, \end{aligned}$$

where \({\mathcal H}\) is the usual complex upper half plane. The groups \(\Gamma _1(N) \subseteq {\text {SL}}_2(\mathbb {Z})\) act on the extended complex upper half plane \(\overline{{\mathcal H}}\) by fractional linear transformations. The quotient is the modular curve \(X_1(N)\).

Following [9, Chapter 3.8] and similar to Sect. 2.2, we represent cusps of \(X_1(N)/\overline{\mathbb {Q}}\) with pairs of order N vectors

$$\begin{aligned} \pm \begin{bmatrix} a \\ c \\ \end{bmatrix}\in \left( \mathbb {Z}/N\mathbb {Z}\right) ^2. \end{aligned}$$

The Galois action on the cusps can be represented with matrices of the form

$$\begin{aligned}\pm \begin{bmatrix} y &{} z\\ 0 &{} 1 \end{bmatrix}\in {\text {GL}}_2(\mathbb {Z}/N\mathbb {Z}) \end{aligned}$$

on the order N vectors in \(\left( \mathbb {Z}/N\mathbb {Z}\right) ^2\), see [9, Sections 7.6 and 7.7]. Two vectors

$$\begin{aligned}\begin{bmatrix} a' \\ c' \\ \end{bmatrix} \ \ \ \text { and } \ \ \ \begin{bmatrix} a \\ c \\ \end{bmatrix} \end{aligned}$$

represent the same cusp when

$$\begin{aligned}\begin{bmatrix} a' \\ c' \\ \end{bmatrix} = \pm \begin{bmatrix} a + jc \\ c \\ \end{bmatrix}\end{aligned}$$

for some \(j \in \mathbb {Z}\). Two cusps represented this way are in the same Galois orbit if and only if \(c = \pm c'\). Hence Each Galois orbit is uniquely determined by \(\pm c\), in other words, by an element of the Cartan  \(C(N):=\left( \mathbb {Z}/ N\mathbb {Z}\right) /\pm \), which is identified with \(\{0,\ldots ,\lfloor N/2 \rfloor \}\). We will denote such orbit by \(C_c(N)\). Let \(\mathbf{n} _c(N), \mathbf{e} _c(N), \mathbf{f} _c(N)\) be as in Sect. 2.2. There are \(\mathbf{f} _c(N)\) cusps in \(C_c(N)\), each of which is represented by \(\mathbf{e} _c\) pairs of vectors in \((\mathbb {Z}/N\mathbb {Z})^2\), for a total of \(\mathbf{n} _c = \mathbf{e} _c\,\mathbf{f} _c\) pairs.

The width of a cusp [9, pp. 59 and 60] is defined as follows. Let \(A\in {\text {SL}}_2(\mathbb {Z})\) be such that \(A\cdot \left[ {\begin{matrix} a \\ c \end{matrix}}\right] =\infty =\left[ {\begin{matrix} 1\\ 0 \end{matrix}}\right] \). The width \(\mathbf{e }_{\scriptscriptstyle {\left[ {\begin{array}{c} a\\ c \end{array}} \right] }}(N)\) is the smallest positive integer for which

$$\begin{aligned} A\begin{bmatrix} 1 &{} \mathbf{e }_{\scriptscriptstyle {\left[ {\begin{array}{c} a\\ c \end{array}} \right] }}(N)\\ 0 &{} 1 \end{bmatrix}A^{-1}\in \Gamma _1(N). \end{aligned}$$

A computation shows that this is \({N}/{\gcd (c,N)}\). Thus the width \(\mathbf{e }_{{\scriptscriptstyle {\left[ {\begin{array}{c} a\\ c \end{array}} \right] }}}(N)\) is \({N}/{\gcd (c,N)}\), which equals the number \(\mathbf{e} _c(N)\) from Sects. 2.2 and 4 with one exception, namely \(C_2(4)\). The cusp corresponding to \(\left[ {\begin{matrix} 1\\ 2 \end{matrix}}\right] \) on \(X_1(4)\) is the lone cusp in the orbit \(C_2(4)\). It is the only irregular cusp for any modular curve \(X_1(N)\), \(X_0(N)\), or X(N) [9, p. 75]. It has width 2, but it has ‘order’ 1. Throughout this paper \(\mathbf{e} _c(N)\) denotes the width, except for the case \(\mathbf{e} _2(4)\) where it denotes the ‘order’ which is 1.

Siegel functions

We would like to define a class of functions on the complex upper half plane \({\mathcal H}\).

Definition 2

Let \((a_1,a_2) \in \mathbb {Q}^2 - \mathbb {Z}^2\). For \(\tau \in {\mathcal H}\), define the Siegel function associated to \((a_1,a_2)\), denoted \(g_{(a_1,a_2)}\), by the product

$$\begin{aligned}&g_{(a_1,a_2)}(\tau ):=-q^{\frac{1}{2}\mathbb {B}_2(a_1)}e^{2\pi i\frac{1}{2}(a_2(a_1-1))}(1-e^{2\pi ia_2}q^{a_1})\prod _{n=1}^\infty (1-e^{2\pi ia_2}q^{n+a_1})(1-e^{-2\pi i a_2}q^{n-a_1}), \end{aligned}$$

where \(q=e^{2\pi i \tau }\), and \(\mathbb {B}_2(x)=x^2-x+\frac{1}{6}\) is the second Bernoulli polynomial.

One can check that adding an integral vector to \((a_1,a_2)\) does not change the order of \(g_{(a_1,a_2)}\), so we can interpret \((a_1,a_2)\) as a non-zero element of \(\left( \mathbb {Q}/ \mathbb {Z}\right) ^2\).

We are interested in the divisors of Siegel functions. From the q-expansion, we see that

$$\begin{aligned} {\text {ord}}_\infty g_{(a_1,a_2)}= \mathbf{e }_\infty \cdot \frac{1}{2}\,\mathbb {B}_2(a_1). \end{aligned}$$

Recall, \(\infty \) denotes the standard prime at infinity given by the equivalence class of \(\left[ {\begin{matrix} 1\\ 0 \end{matrix}}\right] \) under the action of \(\Gamma _1(N)\) and \(\mathbf{e }_\infty = 1\) is its width. Consider another cusp of the modular curve \(X_1(N)\) that corresponds to the orbit of \(\left[ {\begin{matrix} a\\ c \end{matrix}}\right] \). Let \(A\in {\text {SL}}_2(\mathbb {Z})\) be a matrix such that

$$\begin{aligned} A\cdot \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}=\begin{bmatrix} a \\ c \\ \end{bmatrix}. \end{aligned}$$

When \(g_{(a_1,a_2)}\) is a function on \(X_1(N)\), the order of \(g_{(a_1,a_2)}\) at the cusp corresponding to \(\left[ {\begin{matrix} a\\ c \end{matrix}}\right] \) is

$$\begin{aligned} {\text {ord}}_{{\scriptscriptstyle {\left[ {\begin{array}{c} a\\ c \end{array}} \right] }}}\left( g_{(a_1,a_2)}\right) = \mathbf{e }_c \cdot \frac{1}{2} \,\mathbb {B}_2\left( \,\left\{ \, \left[ (a_1,a_2)\cdot A\right] _1 \, \right\} \,\right) , \end{aligned}$$
(33)

where \(\{\cdot \}=\cdot -\lfloor \cdot \rfloor \) denotes the fractional part  and \([\cdot ]_1\) denotes the first entry of the vector. The paper [22] has a concise description of the above for an arbitrary modular curve, but [12, Chapter 2] has a more thorough exposition for X(N); specifically, see the boxed equation on page 40. The reader should note that in [12], Kubert and Lang are considering the \(q^\frac{1}{N}\) expansion. In the remainder of this paper, we will consider Siegel functions of the form \(g_{(0,a)}\), with a a nonzero element of \(\mathbb {Q}/\mathbb {Z}\) of order dividing N. Following [18], we write

$$\begin{aligned} H_{k} := g_{\left( 0,\frac{k}{N}\right) }, \ \ \ \mathrm{with} \ \ k \in \mathbb {Z}- N \mathbb {Z}. \end{aligned}$$

Caution In [18], Streng considers the modular curve \(X^1(N)\), while we have \(X_1(N)\). The isomorphism \(\Gamma ^1(N)\setminus {\mathcal H}\rightarrow \Gamma _1(N)\setminus {\mathcal H}\) is given by \(\begin{bmatrix} 0 &{} -1\\ 1 &{} 0 \end{bmatrix}.\) This isomorphism sends \(g_{(a,0)}\) to \(g_{(0,-a)}\); however, \(g_{(0,-a)}=-g_{(0,a)}\).

The unweighted order of \(H_k\) at \(c\in C(N)\) is

$$\begin{aligned} \mathrm{uord}_c\left( H_k\right) = \frac{1}{2}\,\mathbb {B}_2\left( \left\{ c\cdot \frac{k}{N}\right\} \right) . \end{aligned}$$
(34)

Note that \(x \mapsto \mathbb {B}_2(\{x\})\) is a continuous function even though \(x \mapsto \{x\}\) is not.

Generators of the modular units

Describing the modular units on a given modular curve has long been a subject of interest. A significant motivation of Kubert and Lang’s text [12] is to describe the units of X(N) over \(\mathbb {Q}(\zeta _N)\). They show that, with the exception of some 2-torsion elements when N is even, the units are generated by the Siegel functions described above.

Motivated by [6, Conjecture 1], Streng [18] has used similar methods to describe all modular units on \(X^1(N)\) over \(\mathbb {Q}\). Before stating the result we introduce some of the relevant objects. We start with Tate normal form.

Lemma 3

([18, Lemma 2.1]) If E is an elliptic curve over a field K of characteristic 0 (such as the elliptic curves in Equations (14) and (22)) and P is a point on E of order greater than 3 with \(x(P) \in K\), then the pair \((E,\pm P)\) is isomorphic to a unique pair of the form

$$\begin{aligned} E_T:Y^2+(1-C)XY-BY=X^3-BX^2, \ \ \ \ \ P=(0,0), \end{aligned}$$
(35)

where \(B,C\in K\) and the discriminant

$$\begin{aligned} D=B^3(16B^2+(1-20C-8C^2)B+C(C-1)^3)\ne 0. \end{aligned}$$

Further, each pair \(B,C\in K\) with \(D\ne 0\) satisfying (35) yields an elliptic curve and with a distinguished point P of order greater than 3.

This form \(E_T\) is called Tate normal form. Let \(K = \mathbb {Q}(j)\) and E be as in Equation (22) and let \(K_0 = K(x_0)\) where \(x_0\) is transcendental over K. Let

$$\begin{aligned} P = \left( x_0, \sqrt{x_0^3 - 3 j_0 x_0 - 2 j_0}\right) . \end{aligned}$$

Sending P to (0, 0) and E to Tate normal form with affine linear transformations results in expressions \(B, C \in K_0\) (computation at [26]). Identify \(x_0\) with x so that \(K_0\) becomes \(\mathbb {Q}(x,j)\). Then \(B,C \in \mathbb {Q}(x,j)\) are \(B = -F_3\) and \(C = -F_4\). Due to the uniqueness of the Tate normal form, it should also be possible to write xj in terms of BC, and a computation [26] confirms that. Thus \(\mathbb {Q}(B,C) = \mathbb {Q}(x,j)\).

A computation [26] shows \(F_2 = B^4/D\). Conjecture 1 in [6], proved by Streng [18], says that for \(N > 2\), the modular units in \(\mathbb {Q}(X_1(N))\) modulo \(\mathbb {Q}^*\) are freely generated by \(F_2,\ldots ,F_{\lfloor N/2 \rfloor + 1}\).

Considering the Tate normal form over \(\mathbb {Z}[B,C]\), we can look at the \(k^{\text {th}}\) division polynomial \(\psi _{k,E_T}(x,y)\in \mathbb {Z}[B,C][x,y]\). As in [18, Example 2.2], evaluating \(\psi _{k,E_T}\) at (0, 0) gives:

$$\begin{aligned}&P_1:=\psi _{1,E_T}(0,0)=1, \quad \quad P_2:=\psi _{2,E_T}(0,0)=-B, \quad \quad P_3:=\psi _{3,E_T}(0,0)=-B^3,\\&P_4:=\psi _{4,E_T}(0,0)=CB^5, \quad \quad P_5:=\psi _{5,E_T}(0,0)=-(C-B)B^8, \\&P_6:=\psi _{6,E_T}(0,0)=-B^{12}(C^2-B+C), \quad \quad P_7:=\psi _{7,E_T}(0,0)=B^{16}(C^3-B^2+BC) . \end{aligned}$$

A computation shows \(P_k = (q_3 / q_2^3)^{k^2 - 1} Q_k\) for \(k < 5\). This must then be true for all k since both sequences \(P_k\) and \(Q_k\) satisfy the recurrence relations (15) and (16), which are preserved under scaling (defined in Sect. 3). From Eq. (19),

$$\begin{aligned} P_k = \left( \frac{q_3}{q_2^3}\right) ^{k^2 - 1} Q_k = \left( \frac{q_3}{q_2^{8/3}}\right) ^{k^2 - 1} \tilde{Q}_k = \left( \tilde{q}_3\right) ^{k^2 - 1} \prod _{d | k} \tilde{q}_d = F_3^{\lfloor k^2/3 \rfloor } \prod _{3 < d | k} F_d. \end{aligned}$$
(36)

In particular, the multiplicative group \(\left\langle D,-B,P_4,\ldots ,P_k \right\rangle \) equals \(\left\langle F_2,\ldots ,F_k\right\rangle \). Streng defined \(F_k\) for \(k>3\) to be \(P_k\) but where all factors \(P_j\) with \(j<k\) have been removed; Eq. (36) makes this precise.

Since \(\psi _{k,E_T}(P)=0\) if and only if P has order dividing k, we see \(F_k(P)=0\) if and only if P has exact order k. As mentioned in Sect. 3, the polynomial \(F_N\) is a model for the modular curve \(X_1(N)\) for \(N>3\). The Tate normal form (35) is only defined for \(N>3\), so [6] used xj coordinates to construct \(F_2\) and \(F_3\). Rewritten in terms of BC they are \(F_2=B^4D^{-1}\) and \(F_3=-B\). We can now state the main result of [18]

Theorem 3

[18, Theorem 1.1], [6, Conjecture 1] The modular units of \(X^1(N)\) are given by \(\mathbb {Q}^*\) times the free abelian group on \(B,D,F_4,F_5,\dots , F_{\lfloor N/2\rfloor +1}\), or equivalently, \(F_2,\ldots ,F_{\lfloor N/2\rfloor +1}\).

Streng gives \(P_k\) explicitly in terms of Siegel functions.

Lemma 4

[18, Lemma 3.3] For all \(k\in \mathbb {Z}- N\mathbb {Z}\)

$$\begin{aligned} P_k=\left( \frac{H_1^2H_3}{H_2^3}\right) ^{k^2-1}\frac{H_k}{H_1} \ \ \mathrm{and} \ \ D=\left( \frac{H_1^2H_3}{H_2^3}\right) ^{12} H_1^{12}. \end{aligned}$$

Defining \(\tilde{H}_k := H_k / H_1^{k^2}\), we get

$$\begin{aligned} P_k = \left( \frac{\tilde{H}_3}{\tilde{H}_2^3}\right) ^{k^2-1} \tilde{H}_k, \ \ F_3 = P_2 = \frac{\tilde{H}_3^3}{\tilde{H}_2^8}, \ \ \text {and} \ \ F_2 = \frac{P_2^4}{D} = \tilde{H}_2^4 . \end{aligned}$$
(37)

Setting \(t = c/N\), Eq. (34) gives

$$\begin{aligned} \mathrm{uord}_c\left( \tilde{H}_k\right) = \mathrm{uord}_c\left( H_k \right) - k^2 \mathrm{uord}_c\left( H_1\right) = \frac{1}{2} \left( \mathbb {B}_2\left( \left\{ kt \right\} \right) - k^2 \mathbb {B}_2\left( \left\{ t \right\} \right) \right) . \end{aligned}$$
(38)

We say that a function \(f: [0,1/2] \rightarrow \mathbb {R}\) is k-piecewise linear if it is continuous and \(f''(t) = 0\) for all \(t \not \in \frac{1}{k} \mathbb {Z}\). Two k-piecewise linear functions coincide if and only if they have: the same initial value f(0), the same initial slope \(f'(0^+)\), and the same change in slope at each \(t \in \frac{1}{k} \mathbb {Z}\). These three conditions hold for the right-hand sides of (38) and (39) and thus:

$$\begin{aligned} \mathrm{uord}_c( \tilde{H}_k ) = \frac{1}{2} \left( (k^2-k)t - \frac{1}{6}(k^2-1) \right) + k \sum _{0< i < k/2} \left( \min \left( t, \frac{i}{k}\right) - t\right) . \end{aligned}$$
(39)

Applying (39) to \(F_2\) and \(F_3\) in (37) produces the unweighted order functions \(v_2(t)\) and \(v_3(t)\) in Theorem 1.

To verify \(v_k(t)\) for the remaining \(k > 3\), let \(\tilde{v}_k(t)\) be the unweighted order function of \(\tilde{q}_k\), i.e. \(\tilde{v}_k(t)\) is the right hand side of Eq. (29) without the factor \(s_k\). So \(\tilde{v}_2(t) = 0\), \(\tilde{v}_3(t) = \frac{1}{3} v_3(t)\) and \(\tilde{v}_k(t) = v_k(t)\) for \(k>3\). The unweighted order function for \(\tilde{Q}_k = \prod _{d | k} \tilde{q}_d\) according to Theorem 1 and Remark 1 is

$$\begin{aligned} \sum _{d | k} \tilde{v}_d(t) = \sum _{d|k} \sum _{0<j<d/2} \mathbf{n} _{j}(d) m_{{j}/d}(t) = k \sum _{0<i<k/2} m_{{i}/k}(t), \end{aligned}$$
(40)

where \(m_a(t) = \min (t,a) - 4 a(1-a)t\). We also used that k is the sum of \(\mathbf{n} _{j}(d)\), taken over all \(0<j<d/2\) with d|k and \(j/d = i/k\).

Applying (39) to \(\tilde{Q}_k = \tilde{H}_k / \tilde{H}_2^{(k^2-1)/3}\) gives the same result. To see this, note that \(m_{i/k}(t)\), which contains \(\min (t, i/k)\), appears in (40) with the same coefficient k as the coefficient of \(\min (t, i/k)\) in (39). The terms \(\frac{1}{6}(k^2-1)\) from (39) cancel out for \(\tilde{H}_k / \tilde{H}_2^{(k^2-1)/3}\) but then the remaining terms \((\cdots ) t\) in (39),(40) must also match by the integral argument from Remark 1. This confirms \(\tilde{v}_k(t)\) and thus \(v_k(t)\) for the remaining k. This gives a second proof for most (except the case N|k, see Lemma 4) of the reformulation of Theorem 1 given in Remark 1.

B. Proof of primitivity

Proposition 1

The \(k{\text {th}}\) division polynomial of the Tate normal form, \(P_k\), is primitive in \(\mathbb {Z}[B,C]\).

Proof

Order the monomials lexicographically with the following rule

$$\begin{aligned} B^{n_1}C^{n_2}<B^{m_1}C^{m_2} \ \ \mathrm{when} \ \ n_1<m_1 \ \mathrm{or} \ (n_1=m_1 \ \mathrm{and} \ n_2<m_2). \end{aligned}$$

If \(R\in \mathbb {Q}[B,C]\), let M(R) denote the smallest monomial of R. For example, if \(R=3B^2C^5+B^3C\), then \(M(R)=3B^2C^5\). A key property is \(M(R_1R_2)=M(R_1)M(R_2)\).

Let \(c_k\) denote \(\lceil k/3\rceil \). It is enough to prove that

$$\begin{aligned} M(P_k)=(-1)^{c_k}(-B)^{\lfloor k^2/3\rfloor } C^{c_k(c_k-1)/2} \end{aligned}$$
(41)

since it shows that \(P_k\) has at least one coefficient equal to \(\pm 1\).

We will prove (41) by induction. First, a direct verification shows that (41) holds for \(k=1, 2, 3,4\). Suppose now that k is even, and write \(l=\frac{k}{2}\). Recall the recursion relation (16)

$$\begin{aligned} P_k=\frac{P_l}{P_2}\left( P_{l+2}P_{l-1}^2-P_{l-2}P_{l+1}^2\right) . \end{aligned}$$

the smallest monomial of the first summand \(P_{l+2}P_{l-1}^2\) is

$$\begin{aligned} (-1)^{c_{l+2}+2c_{l-1}}(-B)^{\lfloor (l+2)^2/3\rfloor +2\lfloor (l-1)^2/3\rfloor } C^{c_{l+2}(c_{l+2}-1)/2+c_{l-1}(c_{l-1}-1)}. \end{aligned}$$

For the second summand \(-P_{l-2}P_{l+1}^2\) it is

$$\begin{aligned} (-1)^{c_{l-2}+2c_{l+1}}(-B)^{\lfloor (l-2)^2/3\rfloor +2\lfloor (l+1)^2/3\rfloor } C^{c_{l-2}(c_{l-2}-1)/2+c_{l+1}(c_{l+1}-1)}. \end{aligned}$$

When \(l\equiv 1 \bmod 3\), the second summand has the smallest monomial, and when \(l\equiv 2 \bmod 3\), the first summand has the smallest monomial. When \(3\mid l\), we have to consider the exponent of C. In this case, the first summand is the smallest.

In each case, verifying Eq. (41) is straightforward. For example, when \(l\equiv 0 \bmod 3\) we have

$$\begin{aligned} \lfloor l^2/3\rfloor +\lfloor (l+2)^2/3\rfloor +2\lfloor (l-1)^2/3\rfloor =\frac{4l^2+3}{3}=\lfloor k^2/3\rfloor +1, \end{aligned}$$

and

$$\begin{aligned} c_l(c_l-1)/2+c_{l+2}(c_{l+2}-1)/2+c_{l-1}(c_{l-1}-1)=\frac{l}{3}\left( \frac{2l-3}{3}\right) =\frac{c_k}{2}(c_k-1). \end{aligned}$$

Now suppose k is odd and write \(k=2l+1\). Recall the recursion relation (15)

$$\begin{aligned} P_k=P_{l+2}P_{l}^3-P_{l-1}P_{l+1}^3. \end{aligned}$$

For the first summand, the smallest monomial is

$$\begin{aligned} (-1)^{c_{l+2}+3c_{l}}(-B)^{\lfloor (l+2)^2/3\rfloor +3\lfloor l^2/3\rfloor } C^{c_{l+2}(c_{l+2}-1)/2+3c_{l}(c_{l}-1)/2}, \end{aligned}$$

and for the second summand it is

$$\begin{aligned} (-1)^{c_{l-1}+3c_{l+1}}(-B)^{\lfloor (l-1)^2/3\rfloor +3\lfloor (l+1)^2/3\rfloor } C^{c_{l-1}(c_{l-1}-1)/2+3c_{l+1}(c_{l+1}-1)/2}. \end{aligned}$$

When \(l\equiv 0\bmod 3\), the second summand has the smaller monomial; when \(l\equiv 1 \bmod 3\), considering the exponent of C shows the second summand has the smaller monomial; and when \(l\equiv 2 \bmod 3\), the first summand has the smaller monomial.

Verifying Eq. (41) is again straightforward for each case. For example, when \(l\equiv 1\bmod 3\)

$$\begin{aligned} \lfloor (l-1)^2/3\rfloor +3\lfloor (l+1)^2/3\rfloor =\frac{4l^2+4l+1}{3}=\lfloor k^2/3\rfloor , \end{aligned}$$

and

$$\begin{aligned} c_{l-1}(c_{l-1}-1)/2+3c_{l+1}(c_{l+1}-1)/2=\frac{4l^2-2l-2}{18}=\frac{c_k}{2}(c_k-1). \end{aligned}$$

Repeating these computations for the remaining cases proves the proposition.

Recall that \(-B = F_3 = \tilde{q}_3^3\) and \(-C = F_4\). From (36) we find that \(\tilde{Q}_{k \setminus 3}\) from Sect. 3 is \(P_k/(-B)^{\lfloor k^2/3 \rfloor }\) which is primitive in \(\mathbb {Z}[B,C]=\mathbb {Z}[F_3,F_4]\) by Eq. (41).

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van Hoeij, M., Smith, H. A Divisor Formula and a Bound on the \(\mathbb {Q}\)-Gonality of the Modular Curve \(X_1(N)\). Res. number theory 7, 22 (2021). https://doi.org/10.1007/s40993-021-00243-3

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Keywords

  • Modular curves
  • Gonality
  • modular units
  • Siegel functions
  • Torsion points on elliptic curves

Mathematics Subject Classification

  • Primary 11G16
  • Secondary 14H52
  • 11G05
  • 14G35
  • 11F03