Universal (all-tuples) Schanuel-type conjecture for elliptic and quasi-elliptic functions: prove or refute the semi-elliptic conjecture and its sigma-refinement

Schanuel's conjecture predicts that $\mathbb{Q}$-linearly independent complex numbers together with their exponentials generate a field of transcendence degree at least the number of arguments; it has been open since the 1960s. Waldschmidt (2025) proved the analogous strong algebraic-independence statement for the identity function $z$, the Weierstrass functions $\wp(z)$, $\zeta(z)$, $\sigma(z)$, the exponentials $\mathrm{e}^{t_j z}$ and the Serre functions $f_u(z)$ attached to third-kind elliptic integrals, but only for almost all tuples, and posed the challenge of a universal version valid for all tuples: correct linear-independence hypotheses over the endomorphism field, optimal transcendence-degree lower bounds in the non-CM and CM cases, classification of the exceptional cases where the bound drops, the natural base field $\mathbb{Q}(g_2,g_3,\wp(u_i))$, and a precise connection with the Grothendieck-Andre period conjecture. The formulation half of that challenge has since been completed: Bertolin-Waldschmidt (2025) stated the split semi-elliptic conjecture (bound $s+2n$ with a drop-by-one exceptional case, proved equivalent to the Grothendieck-Andre period conjecture for the associated 1-motive), and Bertolin (2025-2026) stated the general non-split semi-elliptic conjecture involving the Serre functions, equivalent to the Grothendieck-Andre period conjecture for the associated 1-motive, together with a sigma-conjecture that follows from the Grothendieck-Andre period conjecture for the auto-dual 1-motive. The surviving open core, and the subject of this problem, is the unconditional proof or refutation of these universal all-tuples conjectures; since the semi-elliptic conjecture implies Schanuel's conjecture itself, the problem is at least as hard as one of the deepest open problems of transcendence theory.

Background

Schanuel's conjecture (formulated around 1960) predicts that if complex numbers $x_1,\dots,x_n$ are linearly independent over the rationals, then the field $\mathbb{Q}(x_1,\dots,x_n,\mathrm{e}^{x_1},\dots,\mathrm{e}^{x_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$. It implies nearly every known transcendence result about exponentials and logarithms and remains open. A trivially true 'almost all' companion is classical: for Lebesgue-almost all tuples the $2n$ numbers $x_i,\mathrm{e}^{x_i}$ are algebraically independent.

The source paper transports this picture to elliptic functions. Fix a lattice $\Omega=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ in $\mathbb{C}$ and the Weierstrass functions: $\wp(z)$, the doubly periodic meromorphic function whose graph $y^2=4x^3-g_2x-g_3$ is the elliptic curve $E$ (so $\wp$ is the first coordinate of the exponential map of $E$); $\zeta(z)$, the negative integral of $\wp$, quasi-periodic in the sense $\zeta(z+\omega_j)=\zeta(z)+\eta_j$ (the constants $\eta_j$ are the quasi-periods), which parametrizes the exponential of an extension of $E$ by the additive group; $\sigma(z)$, the entire function with $(\log\sigma)''=-\wp$, quasi-periodic of multiplicative type; and the Serre functions $f_u(z)=\mathrm{e}^{-\zeta(u)z}\sigma(z+u)/(\sigma(z)\sigma(u))$ for $u\notin\Omega$, which occur in the parametrization of the exponential of extensions of $E$ by the multiplicative group and are related to elliptic integrals of the third kind. The functions $\zeta$, $\sigma$ and $f_u$ are only quasi-periodic; they are the 'quasi-elliptic' functions of the title. If $E$ has complex multiplication (CM), its endomorphisms form an order in an imaginary quadratic field; the endomorphism field $k=\mathrm{End}(E)\otimes\mathbb{Q}$ equals $\mathbb{Q}$ in the non-CM case and $\mathbb{Q}(\tau)$ with $\tau=\omega_2/\omega_1$ in the CM case, and it is over $k$ (not merely over $\mathbb{Q}$) that independence of arguments must be measured in the elliptic setting.

Waldschmidt's Corollary 2.3 proves the strong Schanuel property for almost all tuples: for almost all $n$-tuples $(z_i)$, the $n(4+r+s)$ numbers $z_i,\wp(z_i),\zeta(z_i),\sigma(z_i),\mathrm{e}^{t_jz_i},f_{u_k}(z_i)$ are algebraically independent over $K(g_2,g_3)$ for any finitely generated base field $K$ (with a CM variant $n(4+r+2s)$ involving $f_{u_k}(\alpha_l z_i)$ for endomorphisms $\alpha_l$). The paper then poses the challenge addressed here: state, and prove, a Schanuel-style conjecture holding for ALL tuples. The source lists the constraints: the lower bound cannot exceed $n(3+r+s)$ in the non-CM case and $n(3+r+2s)$ in the CM case (in the paper's notation), and even that is sometimes too optimistic because of an 'exceptional case' where the degree drops; independence of the arguments must be required over $k$; the finitely generated field $K$ should be replaced by the natural base field $\mathbb{Q}(g_2,g_3,\wp(u_1),\dots,\wp(u_s))$; and the conjecture should connect with the Grothendieck-Andre period conjecture.

That last connection runs through 1-motives. A 1-motive (Deligne) is a two-term complex $[u:\mathbb{Z}\to G]$ with $G$ a semi-abelian variety (an extension of an abelian variety by a torus); for $G=\mathbb{G}_m^s\times E^n$ one obtains the split semi-elliptic exponential whose coordinates are precisely the functions $\mathrm{e}^{t}$, $\wp$, $\zeta$ above, while non-split extensions involve the Serre functions. The Grothendieck-Andre period conjecture (GAC) predicts that the dimension of the motivic Galois group of a motive (the fundamental group of the category of motives that the object generates) equals the transcendence degree of the field generated by its periods. For 1-motives this yields exactly the predicted lower bounds of Schanuel type: Bertolin (J. Number Theory 2002) stated the Elliptico-Toric conjecture, the restriction of the period conjecture to 1-motives whose abelian part is a power of an elliptic curve, from this principle, and Bertolin-Philippon-B. Saha-E. Saha (J. Algebra 2022) stated semi-abelian analogues of Schanuel's conjecture (their Conjecture 4.2 is an elliptic Schanuel conjecture). Under GAC the challenge's formulation is therefore not free: the correct all-tuples statement is dictated, and the task is to prove it.

Problem Statement

The problem posed by the source is the universal, all-tuples version of Schanuel's conjecture for elliptic and quasi-elliptic functions, with optimal lower bounds, correct independence hypotheses over the endomorphism field $k=\mathrm{End}(E)\otimes\mathbb{Q}$, classified exceptional cases, the natural base field $\mathbb{Q}(g_2,g_3,\wp(u_i))$, and a precise connection with the Grothendieck-Andre period conjecture for 1-motives. The formulation half of this challenge has been completed in the derived literature; the remaining problem, stated in its now-precise form, is to prove or refute the following universal conjectures, which are jointly the target (each is a facet of the single objective of establishing the exact algebraic-independence structure of values of the elliptic and quasi-elliptic functions at arbitrary complex arguments):

(i) Split Semi-Elliptic Conjecture (Bertolin-Waldschmidt, Conjecture 2.1). For $\Omega$ a lattice in $\mathbb{C}$, $t_1,\dots,t_s$ $\mathbb{Q}$-linearly independent complex numbers and $p_1,\dots,p_n$ $k$-linearly independent elements of $\mathbb{C}\setminus\Omega$, the transcendence degree over $\mathbb{Q}$ of the field $\mathbb{Q}(t_1,\dots,t_s,\mathrm{e}^{t_1},\dots,\mathrm{e}^{t_s},g_2,g_3,p_1,\dots,p_n,\wp(p_1),\dots,\wp(p_n),\zeta(p_1),\dots,\zeta(p_n))$ is at least $s+2n$, except in the exceptional case $2\pi i\mathbb{Q}\subset\sum_l\mathbb{Q}t_l$ and $\Omega\subset\sum_i kp_i$, in which case it is at least $s+2n-1$; this bound is optimal.

(ii) Semi-Elliptic Conjecture (Bertolin, Conjecture 0.2, with its special case Conjecture 0.5). The corresponding all-tuples statement for the exponential of an arbitrary (non-split) extension of $E^n$ by a torus, involving the Serre functions $f_{q_j}(p_i)=\mathrm{e}^{-\zeta(q_j)p_i}\sigma(p_i+q_j)/(\sigma(p_i)\sigma(q_j))$, with the correction terms for Cartier-dual pairs and the exceptional drop-by-one clause exactly as stated in that work (a representative special case: for $t_i$ $\mathbb{Q}$-linearly independent and $q_j,p_i$ $k$-linearly independent in $\mathbb{C}\setminus\Omega$, the field generated by the $2s+2+3r+3n+|I||J|$ numbers $t_i,\mathrm{e}^{t_i},g_2,g_3,q_j,p_i,\wp(q_j),\zeta(q_j),\wp(p_i),\zeta(p_i),f_{q_j}(p_i)$ for $(i,j)\in I\times J$ has transcendence degree at least $s+2(r+n)+|I||J|$ outside the exceptional case, where $I,J$ index a $k$-basis of the classes of the $p_i,q_j$ modulo $\Omega\otimes_{\mathbb{Z}}\mathbb{Q}$).

(iii) Sigma-Conjecture (Bertolin, Conjecture 0.8). If $p_1,\dots,p_n$ are $k$-linearly independent elements of $\mathbb{C}\setminus\Omega$, the transcendence degree over $\mathbb{Q}$ of the field $\mathbb{Q}(g_2,g_3,p_1,\dots,p_n,\wp(p_1),\dots,\wp(p_n),\zeta(p_1),\dots,\zeta(p_n),\sigma(p_1),\dots,\sigma(p_n))$ is at least $2n+\dim_k\langle p_i\rangle+\dim_{\mathbb{Q}}\langle\zeta(p_i)p_i\rangle$, where $\langle p_i\rangle$ is the sub-$k$-vector space of $\mathbb{C}/(\Omega\otimes_{\mathbb{Z}}\mathbb{Q})$ generated by the classes of the $p_i$, and $\langle\zeta(p_i)p_i\rangle$ is the sub-$\mathbb{Q}$-vector space of $\mathbb{C}/2\pi i\mathbb{Q}$ generated by the classes of the products $\zeta(p_i)p_i$ if the $k$-span of the $p_i$ meets $\Omega\otimes_{\mathbb{Z}}\mathbb{Q}$ non-trivially, and is trivial otherwise (the exact statement, including the precise condition on the torsion of the classes of the $p_i$, is Conjecture 0.8 of the reference).

Of the three statements, (i) is the split special case of the general statement (ii) (obtained when the extension of $E^n$ by the torus is split), so a proof of (ii) also settles (i); hence a complete proof must establish (ii) together with (iii), while a single explicit counterexample refuting any one of (i), (ii), (iii) resolves the whole task. Prove these conjectures unconditionally for all tuples of complex arguments (not merely for almost all tuples, algebraic arguments, torsion points, or CM curves with torsion points), or refute any of them by an explicit tuple satisfying the stated hypotheses for which the predicted lower bound fails outside the exceptional clause. By the proved connections (Bertolin-Waldschmidt Theorem 4.3: (i) is equivalent to the Grothendieck-Andre period conjecture for the 1-motive $[u:\mathbb{Z}\to\mathbb{G}_m^s\times E^n]$; Bertolin Theorem 0.4: (ii) is equivalent to the Grothendieck-Andre period conjecture for the associated 1-motive; Bertolin Theorem 7.4: the Grothendieck-Andre period conjecture for the auto-dual 1-motive implies (iii)), an admissible route is a proof of the Grothendieck-Andre period conjecture for 1-motives whose abelian part is a power of an elliptic curve, and conversely such a period-conjecture proof for the associated 1-motives would follow from (ii); any proof must in any case handle the non-CM and CM cases, the exceptional case, and the natural base field exactly as in the stated conjectures.

References

1. Michel Waldschmidt, Schanuel Property for Elliptic and Quasi--Elliptic Functions, arXiv:2504.14041 (2025), published in the Hardy-Ramanujan Journal, vol. 48, DOI 10.46298/hrj.2026.15601, https://arxiv.org/abs/2504.14041
2. Cristiana Bertolin and Michel Waldschmidt, Variations on Schanuel's Conjecture for elliptic and quasi-elliptic functions I: the split case, arXiv:2504.14048 (2025), published in the Hardy-Ramanujan Journal, vol. 48, DOI 10.46298/hrj.2026.15602, https://arxiv.org/abs/2504.14048
3. Cristiana Bertolin, A conjecture in Schanuel style for 1-motives, arXiv:2509.08700 (2025, revised through 2026), https://arxiv.org/abs/2509.08700
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