Universal (all-tuples) Schanuel conjecture for elliptic and quasi-elliptic functions: prove the semi-elliptic Schanuel conjectures, equivalently the Grothendieck–André period conjecture for 1-motives with elliptic part

Waldschmidt (2025) proved that a strong Schanuel property holds for almost all tuples of evaluation points for the Weierstrass functions $\wp$ , $\zeta$ , $\sigma$ , exponentials, and Serre functions (the functions attached to elliptic integrals of the third kind), and posed the challenge of formulating and proving a universal analogue valid for all tuples, with optimal transcendence-degree lower bounds, correct linear-independence hypotheses over the endomorphism field, and a classification of exceptional drops. Follow-up work has completed the formulation half and supplied the motivic (Grothendieck–André period) explanation in the direction the paper announced: Bertolin and Waldschmidt stated the split semi-elliptic Conjecture, and Bertolin the non-split semi-elliptic Conjecture — each equivalent to the Grothendieck–André generalized period conjecture for a suitable 1-motive — together with the $\sigma$ -Conjecture, which is a consequence of that period conjecture. The surviving open core is the proof: these conjectures are unproved in every nontrivial case (any elliptic curve without complex multiplication, or any non-torsion evaluation points), and they contain Schanuel's conjecture itself as the case with no elliptic factor. Only the special case of elliptic curves with algebraic invariants, complex multiplication, and torsion points is proved.

Background

Schanuel's conjecture (1960) predicts that if $x_1,\dots,x_n$ are complex numbers linearly independent over the rationals $\mathbb{Q}$ , then at least $n$ of the $2n$ numbers $x_1,\dots,x_n,e^{x_1},\dots,e^{x_n}$ are algebraically independent over $\mathbb{Q}$ (no nonzero polynomial with rational coefficients vanishes on them); equivalently the transcendence degree of the field they generate — the maximal number of algebraically independent elements — is at least $n$ . It is one of the main open problems of transcendence theory: apart from the Lindemann–Weierstrass theorem (the case of algebraic $x_i$ ), essentially no case is known. A stronger statement holds for almost all tuples (with respect to Lebesgue measure): the $2n$ numbers are algebraically independent. Waldschmidt's paper 'Schanuel Property for Elliptic and Quasi--Elliptic Functions' (arXiv:2504.14041, published in the Hardy–Ramanujan Journal 48) transfers this 'almost all' phenomenon from the exponential to the functions attached to an elliptic curve. Fix a lattice $\Omega$ in the complex plane and the associated Weierstrass functions: the elliptic function $\wp(z)$ (a doubly meromorphic function parametrizing the elliptic curve $E=\mathbb{C}/\Omega$ ), the quasi-periodic Weierstrass zeta function $\zeta(z)$ (its primitive, occurring in the parametrization of extensions of $E$ by the additive group), the Weierstrass sigma function $\sigma(z)$ (an entire function with $\sigma'/\sigma=\zeta$ , related to third-kind data), and the Serre functions $f_u(z)$ attached to parameters $u$ , which arise in the parametrization of extensions of $E$ by the multiplicative group $\mathbb{G}_m$ (elliptic integrals of the third kind). An elliptic curve has complex multiplication (CM) when its endomorphism ring is larger than $\mathbb{Z}$ ; in that case the field $k=\operatorname{End}(E)\otimes\mathbb{Q}$ is an imaginary quadratic field, while $k=\mathbb{Q}$ otherwise. Waldschmidt proves (Corollary 2.3) that, assuming the two periods $\omega_1,\omega_2$ (a basis of $\Omega$ ) and the parameters $u_1,\dots,u_s$ are $\mathbb{Q}$ -linearly independent, for almost all $n$ -tuples $(z_1,\dots,z_n)$ the $n(4+r+s)$ numbers $z_i,\wp(z_i),\zeta(z_i),\sigma(z_i),e^{t_i z_i},f_{u_j}(z_i)$ (with $t_1,\dots,t_r$ $\mathbb{Q}$ -linearly independent; $n(4+r+2s)$ numbers in the CM case with extra values at endomorphism multiples) are algebraically independent over $K(g_2,g_3)$ , where $g_2,g_3$ are the lattice invariants and $K$ is any finitely generated extension of $\mathbb{Q}$ . He then poses the central challenge: state and prove a version of this that holds for all tuples, not merely almost all. When the evaluation points are algebraic one cannot predict a transcendence degree better than $n(3+r+s)$ (resp. $n(3+r+2s)$ in the CM case), since $\zeta$ and $\wp$ satisfy additive relations; such a bound is sometimes still too optimistic in exceptional geometric situations; the hypotheses must require linear independence of the points over the endomorphism field $k$ rather than over $\mathbb{Q}$ ; and the base field should be the natural one $\mathbb{Q}(g_2,g_3,\wp(u_1),\dots)$ . The challenge is coupled to motives: a 1-motive is an algebraic object $M=[u:\mathbb{Z}\to G]$ mixing a torus, an abelian variety (here the elliptic curve), and a lattice map, and the Grothendieck–André generalized period conjecture predicts the transcendence degree of the period field of $M$ in terms of the dimension of its motivic Galois group (the algebraic group measuring the motivic relations among periods). Bertolin's earlier work showed that Schanuel's conjecture itself is equivalent to this period conjecture for 1-motives without abelian part, so an elliptic analogue is expected to be exactly a period conjecture for 1-motives with elliptic abelian part. What is unconditionally known is confined to classical elliptic transcendence theorems (elliptic analogues, due to Schneider, of the Hermite–Lindemann and Gel'fond–Schneider theorems — the classical transcendence theorems for exponentials of, and logarithms at, algebraic points — discussed in the follow-up paper arXiv:2504.14048) and to the CM case: for elliptic curves with algebraic invariants and complex multiplication, the transcendence degree of the field of periods and quasi-periods (the quasi-periods being $\eta_i=\zeta(\omega_i)$ for the period basis $\omega_1,\omega_2$ of $\Omega$ ) equals $2$ , a classical theorem quoted in arXiv:2509.08700, which extends to its Proposition 5.4 covering 1-motives defined by such curves and torsion points. No algebraic-independence statement of this strength is known for any elliptic curve without complex multiplication.

Problem Statement

Prove or refute the universal (all-tuples) Schanuel-style conjectures for elliptic and quasi-elliptic functions posed at the end of Section 2 of arXiv:2504.14041 and made precise in the follow-up literature. In its cleanest precise form (the Split Semi-Elliptic Conjecture of Bertolin–Waldschmidt, arXiv:2504.14048, Conjecture 2.1): let $\Omega$ be a lattice in $\mathbb{C}$ with invariants $g_2,g_3$ , let $k$ be the field of endomorphisms of the elliptic curve $\mathbb{C}/\Omega$ (so $k=\mathbb{Q}$ without complex multiplication, and an imaginary quadratic field with it), let $t_1,\dots,t_s$ be $\mathbb{Q}$ -linearly independent complex numbers and $p_1,\dots,p_n$ be $k$ -linearly independent complex numbers in $\mathbb{C}\setminus\Omega$ ; then the transcendence degree over $\mathbb{Q}$ of the field $\mathbb{Q}\bigl(t_1,\dots,t_s,e^{t_1},\dots,e^{t_s},g_2,g_3,p_1,\dots,p_n,\wp(p_1),\dots,\wp(p_n),\zeta(p_1),\dots,\zeta(p_n)\bigr)$ (where $\wp$ and $\zeta$ are the Weierstrass $\wp$ -function, the doubly meromorphic function parametrizing the curve $\mathbb{C}/\Omega$ , and its quasi-periodic primitive, the Weierstrass zeta function) is at least $s+2n$ , except in the exceptional case where $2\pi i\mathbb{Q}\subset\sum_j\mathbb{Q}t_j$ and $\Omega\subset\sum_i kp_i$ , in which case it is at least $s+2n-1$ ; these bounds are optimal. Here $s$ is the source's notation for the number of exponents $t_1,\dots,t_s$ ; it is unrelated to the letter $s$ used in the Background above, where $s$ counts the Serre-function parameters $u_1,\dots,u_s$ and the exponents are counted by $r$ . Prove this conjecture for all lattices and all tuples satisfying the stated hypotheses (equivalently, by the equivalence theorems of arXiv:2504.14048 and arXiv:2509.08700, prove the Grothendieck–André generalized period conjecture for all 1-motives whose abelian part is an elliptic curve), together with its companions covering the remaining quasi-elliptic data: the non-split semi-elliptic Conjecture involving the Serre functions (third-kind data) and the $\sigma$ -Conjecture involving the Weierstrass $\sigma$ -function. By the equivalence theorems of arXiv:2504.14048 and arXiv:2509.08700, the split semi-elliptic Conjecture and its non-split companion are each equivalent to the Grothendieck–André generalized period conjecture for a suitable 1-motive whose abelian part is an elliptic curve, and the $\sigma$ -Conjecture is a consequence of that period conjecture. A resolution must go beyond the cases known today — the classical elliptic transcendence theorems, the CM period transcendence-degree equality, and the CM-with-torsion-points case of Proposition 5.4 of arXiv:2509.08700 — since those do not touch non-CM curves or non-torsion points; note that the case $n=0$ with no elliptic curve (dropping $g_2,g_3$ ) is exactly Schanuel's conjecture, so a full proof would in particular settle it. A refutation would consist of an explicit tuple satisfying the stated independence hypotheses for which the claimed transcendence-degree lower bound fails, with a complete proof of the required algebraic dependence; since each of the three named conjectures is a consequence of the generalized period conjecture, one fully proved counterexample defeating the bound of any one of them — equivalently, refuting the generalized period conjecture — is by itself a resolution.

References

C. Bertolin and M. Waldschmidt, 'Variations on Schanuel's Conjecture for elliptic and quasi-elliptic functions I: the split case', Hardy–Ramanujan Journal 48 (2025), Part A, pp. 10–29. DOI: 10.46298/hrj.2026.15602 (arXiv:2504.14048). URL: https://arxiv.org/abs/2504.14048
M. Waldschmidt, 'Schanuel Property for Elliptic and Quasi--Elliptic Functions', Hardy–Ramanujan Journal 48 (2025). DOI: 10.46298/hrj.2026.15601 (arXiv:2504.14041, submitted 18 April 2025). URL: https://arxiv.org/abs/2504.14041
C. Bertolin, 'A conjecture in Schanuel style for 1-motives', arXiv:2509.08700 (10 September 2025). DOI: 10.48550/arXiv.2509.08700. URL: https://arxiv.org/abs/2509.08700
C. Bertolin, 'Dimension of the motivic Galois group of a 1-motive', arXiv:2604.24128 (27 April 2026). DOI: 10.48550/arXiv.2604.24128. URL: https://arxiv.org/abs/2604.24128
C. Bertolin, P. Philippon, B. Saha and E. Saha, 'Semi-abelian analogues of Schanuel conjecture and applications', Journal of Algebra 596 (2022), pp. 250–288. DOI: 10.1016/j.jalgebra.2021.12.040. URL: https://doi.org/10.1016/j.jalgebra.2021.12.040
B. Bakker and J. Tsimerman, 'Functional Transcendence of Periods and the Geometric André–Grothendieck Period Conjecture', arXiv:2208.05182 (10 August 2022). DOI: 10.48550/arXiv.2208.05182. URL: https://arxiv.org/abs/2208.05182
J. Noguchi, 'Analytic Ax–Schanuel for semi-abelian varieties and Nevanlinna theory', Journal of the Mathematical Society of Japan 76 (2024). DOI: 10.2969/jmsj/89588958. URL: https://doi.org/10.2969/jmsj/89588958