The Semi-elliptic Conjecture: a Schanuel-style transcendence conjecture for the exponential of an extension of an elliptic curve by a torus

Bertolin (arXiv:2509.08700, 2025/2026) proposed the Semi-elliptic Conjecture, a Schanuel-type lower bound on the transcendence degree of the field generated by values of the exponential function, of the Weierstrass $\wp$ and $\zeta$ functions, and of Serre functions $f_{q}(p)$, taken at the points defining the semi-elliptic exponential of an extension $G$ of an elliptic curve $\mathcal{E}$ by a torus $\mathbb{G}_m^r$. The same paper proves that this conjecture is equivalent to the Grothendieck–André period conjecture for 1-motives whose abelian part is $\mathcal{E}$, and proves that period conjecture only in the case of an elliptic curve with complex multiplication (CM, i.e. an endomorphism ring larger than the expected $\mathbb{Z}$), algebraic invariants and torsion points (where it reduces to Chudnovsky's theorem). The problem is to prove or disprove the Semi-elliptic Conjecture in its stated full generality. A proof would settle the classical Schanuel conjecture, which the Semi-elliptic Conjecture implies, and would also settle the Lindemann-Weierstrass-style and $\sigma$-conjectures derived from it in the same paper; beyond the CM torsion case nothing is known.

Background

Schanuel's conjecture predicts that if $t_1,\dots,t_s$ are $\mathbb{Q}$-linearly independent complex numbers, then the field $\mathbb{Q}(t_1,\dots,t_s,\mathrm{e}^{t_1},\dots,\mathrm{e}^{t_s})$ has transcendence degree at least $s$ over $\mathbb{Q}$; it implies the Lindemann-Weierstrass theorem (the statement that the exponentials of $\mathbb{Q}$-linearly independent algebraic numbers are linearly independent over the field of algebraic numbers) and is the central open problem of transcendence theory. Bertolin showed in her thesis work (J. Number Theory 97 (2002)) that Schanuel's conjecture has a motivic origin: it is equivalent to the Grothendieck–André period conjecture (Conjecture 0.3 of the paper) applied to a 1-motive with no abelian part. A 1-motive is an algebraic data structure $M=[u:\mathbb{Z}\to G]$ consisting of a semi-abelian variety $G$ (an extension of an abelian variety by a torus $\mathbb{G}_m^r$, i.e. a product of $r$ copies of the multiplicative group $\mathbb{G}_m=\mathbb{C}^{\times}$) together with a point of $G$; its periods are the complex numbers obtained by pairing de Rham cohomology of $M$ with its singular cohomology. The Grothendieck–André period conjecture asserts that the transcendence degree of the field generated by the periods of a motive $M$ over a field of definition $K$ is at least the dimension of the motivic Galois group of $M$ (an algebraic group attached to $M$ that measures the expected algebraic relations among its periods; formally, the fundamental group of the tannakian category generated by $M$), with equality when $M$ is defined over $\overline{\mathbb{Q}}$.

Bertolin's paper extends this circle of ideas from the toric case to the semi-elliptic case. Let $\Omega=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ be a lattice in $\mathbb{C}$ with invariants $g_2,g_3$, let $\mathcal{E}=\mathbb{C}/\Omega$ be the corresponding elliptic curve (a one-dimensional abelian variety, i.e. a projective algebraic group), and let $\wp$, $\zeta$, $\sigma$ be the Weierstrass functions attached to $\Omega$: $\wp(z)$ is the doubly-periodic meromorphic function generating the function field of $\mathcal{E}$, $\zeta(z)$ is its quasi-periodic primitive with $\zeta'(z)=-\wp(z)$ (not the Riemann zeta function), and $\sigma(z)$ is the entire function with $\sigma'/\sigma=\zeta$. The Serre function is $f_{q}(p)=\sigma(p+q)/(\sigma(p)\sigma(q))\,\mathrm{e}^{-\zeta(q)p}$; it is the exponential of an elliptic integral of the third kind and appears as a period of extensions of $\mathcal{E}$ by $\mathbb{G}_m$. Such an extension $G$ of $\mathcal{E}$ by $\mathbb{G}_m^r$ is parametrized by points $Q_j$ of the dual elliptic curve $\mathcal{E}^*$ (the curve parametrizing degree-zero line bundles on $\mathcal{E}$, canonically isomorphic to $\mathcal{E}$ itself), and its exponential map $\exp_G$ is called the semi-elliptic exponential; the split case $G=\mathbb{G}_m\times\mathcal{E}$ was treated earlier by Bertolin and Waldschmidt (Hardy-Ramanujan J. 48 (2025)). Write $k=\operatorname{End}(\mathcal{E})\otimes\mathbb{Q}$, so $k=\mathbb{Q}$ unless $\mathcal{E}$ has complex multiplication (CM, i.e. a larger-than-expected endomorphism ring), in which case $k$ is an imaginary quadratic field.

Bertolin's paper proposes the Semi-elliptic Conjecture (Conjecture 0.2), a lower bound on the transcendence degree over $\mathbb{Q}$ of the field generated by $t_l,\mathrm{e}^{t_l}$, the invariants $g_2,g_3$, the points $q_j,p_i$, the values $\wp(q_j),\zeta(q_j),\wp(p_i),\zeta(p_i)$ and the Serre values $f_{q_j}(p_i)$, and proves (her Theorem 0.4) that it is equivalent to the Grothendieck–André period conjecture applied to the 1-motive $M=[u:\mathbb{Z}\to G^n]$ built from the semi-elliptic exponential. She also derives from it Lindemann-Weierstrass-style algebraic independence conjectures (Conjectures 0.6 and 0.7 of v3), and $\sigma$-Conjectures (Conjectures 0.8 and 0.9 of v3) involving additionally the $\sigma$-values, showing these follow from the period conjecture applied to a suitable auto-dual 1-motive. Unconditionally, the paper proves the period conjecture only for 1-motives defined by a CM elliptic curve with algebraic invariants and torsion points (Proposition 5.4 of v3, numbered Theorem 7.3 in v1), where the statement reduces to a classical theorem of G. V. Chudnovsky (recorded as formula (5.9) of the paper): for a CM elliptic curve with algebraic invariants, $\operatorname{trdeg}_{\mathbb{Q}}\mathbb{Q}(\omega_1,\omega_2,\eta_1,\eta_2)=2$, where $\eta_i=\eta(\omega_i)$ are the quasi-periods, defined by $\zeta(z+\omega_i)=\zeta(z)+\eta_i$ (equivalently $\eta_i=2\zeta(\omega_i/2)$).

Problem Statement

Prove or disprove, in the full generality in which it is stated, the Semi-elliptic Conjecture (Conjecture 0.2 of C. Bertolin, arXiv:2509.08700) concerning the values of the exponential function, of the Weierstrass $\wp$, $\zeta$ functions and of the Serre functions $f_q(p)$ attached to a lattice $\Omega\subset\mathbb{C}$.

Its linearly independent special case (Conjecture 0.5 of the same paper) reads: let $\Omega$ be a lattice with invariants $g_2,g_3$ and let $k=\operatorname{End}(\mathbb{C}/\Omega)\otimes\mathbb{Q}$. Suppose $t_1,\dots,t_s$ are $\mathbb{Q}$-linearly independent complex numbers, and $q_1,\dots,q_r,p_1,\dots,p_n$ are $k$-linearly independent complex numbers in $\mathbb{C}\smallsetminus\Omega$. Choose subsets $I\subseteq\{1,\dots,n\}$ and $J\subseteq\{1,\dots,r\}$ such that the classes of $p_i,q_j$ for $i\in I$ and $j\in J$ form a $k$-basis of the space generated by the classes of $p_1,\dots,p_n,q_1,\dots,q_r$ in $\mathbb{C}/(\Omega\otimes_{\mathbb{Z}}\mathbb{Q})$. Then

$$\operatorname{trdeg}_{\mathbb{Q}}\,\mathbb{Q}\bigl(t_l,\,\mathrm{e}^{t_l},\,g_2,g_3,\,q_j,\,p_i,\,\wp(q_j),\,\zeta(q_j),\,\wp(p_i),\,\zeta(p_i),\,f_{q_j}(p_i)\ (i\in I,\,j\in J)\bigr)\ \ge\ s+2(r+n)+|I||J|,$$

except in the case $2\pi\mathrm{i}\,\mathbb{Q}\subset\sum_l\mathbb{Q}t_l$ and $\Omega\subset\sum_i k\,p_i+\sum_j k\,q_j$, where the bound is lowered by one.

The full Semi-elliptic Conjecture 0.2 additionally admits points that are dependent over $k$ in three layers ($n\le n'\le n''$, $r\le r'\le r''$) and correspondingly corrects the bound: a term $\dim_{\mathbb{Q}}\langle\beta_{i,j}+\beta_{i,j}^{t}\rangle$ counting homomorphisms of the smallest abelian subvariety containing the points, reduced by $\tfrac{1}{2}|\mathcal{D}_{n',r'}|$ to count each Cartier-dual pair of 1-motives only once, and a term $(n''-n')(r''-r')$ contributed by the parameters $t_{ij}$ together with the values $t_{ij}$ and $\mathrm{e}^{t_{ij}}f_{q_j}(p_i)$; the full statement, including the exceptional cases, is Conjecture 0.2 of the cited paper, while the display above states only its linearly independent special case, Conjecture 0.5. By Theorem 0.4 of the paper (proved there), an equivalent target is the Grothendieck–André period conjecture for all 1-motives of the form $M=[u:\mathbb{Z}\to G^n]$ whose abelian part is an elliptic curve and whose extension $G$ of it by $\mathbb{G}_m^r$ is parametrized by points $Q_j$ of the dual curve. The single configuration settled unconditionally is the CM elliptic curve with algebraic invariants and torsion points (Proposition 5.4 of the paper, via Chudnovsky's theorem); a resolution of the conjecture in either direction, over arbitrary lattices and arbitrary configurations of the defining points, is sought. A proof would in particular settle the classical Schanuel conjecture, the derived Lindemann-Weierstrass-style conjectures (Conjectures 0.6 and 0.7) and the $\sigma$-Conjectures (Conjectures 0.8 and 0.9), since the paper proves unconditionally that the core conjecture implies them.

References

1. C. Bertolin, “A conjecture in Schanuel style for 1-motives”, arXiv:2509.08700 (v1 September 2025, v2 November 2025, v3 June 2026), DOI 10.48550/arXiv.2509.08700, https://arxiv.org/abs/2509.08700
2. 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, https://doi.org/10.46298/hrj.2026.15602
3. C. Bertolin, “Périodes de 1-motifs et transcendance”, Journal of Number Theory 97 (2002), no. 2, pp. 204-221, DOI 10.1016/S0022-314X(02)00002-1, https://doi.org/10.1016/S0022-314X(02)00002-1
4. C. Bertolin, “Third kind elliptic integrals and 1-motives”, Journal of Pure and Applied Algebra 224 (2020), no. 10, article 106396, DOI 10.1016/j.jpaa.2020.106396, https://doi.org/10.1016/j.jpaa.2020.106396
5. 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, https://doi.org/10.1016/j.jalgebra.2021.12.040
6. M. Waldschmidt, “Schanuel Property for Elliptic and Quasi--Elliptic Functions”, Hardy-Ramanujan Journal 48 (2025), DOI 10.46298/hrj.2026.15601, https://doi.org/10.46298/hrj.2026.15601
7. C. Bertolin, “Dimension of the motivic Galois group of a 1-motive”, arXiv:2604.24128 (v1 April 2026), DOI 10.48550/arXiv.2604.24128, https://arxiv.org/abs/2604.24128
8. A. Huber and G. Wüstholz, “Transcendence and Linear Relations of 1-Periods”, Cambridge Tracts in Mathematics 227, Cambridge University Press, 2022, ISBN 978-1-316-51993-6, DOI 10.1017/9781009019729, https://doi.org/10.1017/9781009019729
9. B. Bakker and J. Tsimerman, “Functional Transcendence of Periods and the Geometric André–Grothendieck Period Conjecture”, arXiv:2208.05182 (2022), DOI 10.48550/arXiv.2208.05182, https://arxiv.org/abs/2208.05182
10. B. Zilber, “Model theory of special subvarieties and Schanuel-type conjectures”, Annals of Pure and Applied Logic 167 (2016), no. 10, pp. 1000-1028, DOI 10.1016/j.apal.2015.02.002, https://doi.org/10.1016/j.apal.2015.02.002
11. M. Waldschmidt, “Nombres transcendants et fonctions sigma de Weierstrass”, C. R. Math. Rep. Acad. Sci. Canada 1 (1978/79), no. 2, pp. 111-114, zbMATH 3623681, https://zbmath.org/3623681