The Relative Semi-abelian Schanuel Conjecture for extensions of abelian varieties by a torus (Bertolin–Philippon–Saha–Saha, Conjecture 1.3 = 4.7)

The Relative Semi-abelian Schanuel conjecture (Conjecture 1.3, restated as Conjecture 4.7, of Bertolin–Philippon–B. Saha–E. Saha, Journal of Algebra 596 (2022) 250–288) predicts a lower bound on the transcendence degree, over the period field of a semi-abelian variety $G$ that extends an abelian variety $A$ by a multiplicative torus, of the field generated by the periods of $G$, the coordinates of points of $G$, and their generalized semi-abelian logarithms; the predicted bound is $2\dim B_Q+\dim Z(1)$, where $B_Q$ and $Z(1)$ are explicit motivic invariants of the data. The conjecture is exactly the hypothesis of the paper's main theorem, which derives from it the algebraic independence (linear disjointness) of the fields of iterated semi-abelian exponential values and iterated generalized semi-abelian logarithms, a semi-abelian case of a question of Waldschmidt. All known results are conditional: the conjecture contains the Weak Abelian Schanuel conjecture and, in its purely toric degeneration (up to the known transcendence of $\pi$), the classical Schanuel conjecture, and no proof, refutation, or weakening of it is known. The problem is to prove the conjecture unconditionally in its stated generality, or to refute it by one explicit instance.

Background

Schanuel's conjecture, formulated in the 1960s, predicts that if complex numbers $x_1,\dots,x_n$ are linearly independent over the rationals, then the field $\mathbb{Q}(x_1,\dots,x_n,e^{x_1},\dots,e^{x_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$; here the transcendence degree of a field extension counts the maximal number of elements satisfying no polynomial relation with coefficients in the base field. It is the central open problem of transcendence number theory and implies, for example, the algebraic independence of logarithms of algebraic numbers. A far-reaching generalization is the Grothendieck period conjecture and its extension to mixed motives, the Generalized Period conjecture (attributed to Y. André and stated as Conjecture 1.2 of the source paper): for a motive $M$ over a subfield of $\mathbb{C}$, the transcendence degree of the field generated by the periods of $M$ — the integrals of algebraic differential forms over singular homology cycles — is at least the dimension of the motivic Galois group $\mathrm{Gal}^{mot}(M)$, the Tannakian automorphism group of the tensor category generated by $M$, whose dimension measures the extent to which period relations must have an algebraic (as opposed to accidental analytic) origin.

The natural test bed for this circle of conjectures beyond the classical exponential is the category of 1-motives, i.e. data $M=[u:\mathbb{Z}^n\to G]$ where $G$ is a semi-abelian variety — an algebraic group sitting in an exact sequence $0\to\mathbb{G}_m^s\to G\to A\to 0$ with $A$ an abelian variety (a projective connected commutative algebraic group) and $\mathbb{G}_m$ the multiplicative group (an algebraic torus). Their periods combine ordinary logarithms of algebraic numbers with abelian integrals of the first, second, and third kind. Bertolin (2002) proved that Schanuel's conjecture is equivalent to the Generalized Period conjecture applied to 1-motives with trivial abelian part. Cheng, Dietel, Herblot, Huang, Krieger, Marques, Mason, Mereb and Wilson (2009) showed that Schanuel's conjecture implies the algebraic independence of the values of the iterated exponential and the iterated logarithm, answering a question of M. Waldschmidt. Philippon, B. Saha and E. Saha (2020) developed the abelian analogue (1-motives with trivial toric part, via the Weak Abelian Schanuel conjecture) and proved the corresponding conditional statement for iterated abelian exponentials and generalized abelian logarithms.

The paper (Bertolin, Philippon, B. Saha, E. Saha, 2022) extends this program to general 1-motives, mixing toric and abelian parts. Extensions of an abelian variety $A$ defined over $\overline{\mathbb{Q}}$ by the torus $\mathbb{G}_m^s$ are classified by $s$ points $Q=(Q_1,\dots,Q_s)$ of the dual abelian variety $A^*=\mathrm{Pic}^0(A)$ (the moduli space of line bundles of degree zero on $A$, itself an abelian variety). The semi-abelian exponential $\exp_G:\mathrm{Lie}\,G_\mathbb{C}\to G(\mathbb{C})$, from the complexified tangent space at the origin, is a universal covering whose kernel is the lattice $H_1(G(\mathbb{C}),\mathbb{Z})$; in the elliptic case it is built from the Weierstrass $\wp$-function together with exponentials of third-kind theta functions. A semi-abelian logarithm $\log_G$ is an inverse of $\exp_G$ given by integration, and the generalized semi-abelian logarithm $\widetilde{\log}_G$ (Definition 2.4 of the paper) records, for a point $R\in G(\mathbb{C})$ above $P\in A(\mathbb{C})$, the abelian integrals of the first kind $\int_O^P\omega_i$, of the second kind $\int_O^P\eta_i$, and of the third kind $\int_O^R\xi^G_{Q_j}$; these numbers are precisely the periods of the 1-motive. The period matrix $\Omega_G$ collects the integrals over $H_1(G(\mathbb{C}),\mathbb{Z})$ of a basis of the first de Rham cohomology $H^1_{dR}(G)$ (the algebraic differential forms on $G$; pullbacks of the $\omega_i,\eta_i$ and the third-kind forms $\xi^G_{Q_j}$); it includes $2\pi i$ and, in the elliptic case, combinations $\eta_iq-\omega_i\zeta(q)$ of Weierstrass zeta-values.

The paper defines the semi-abelian analogue of Schanuel's conjecture as the Generalized Period conjecture applied to $M=[u:\mathbb{Z}^n\to G]$ (Conjecture 4.5, which Proposition 4.6 shows is exactly that instance), and introduces the relative versions needed for its main theorem. Two motivic invariants control the expected bound: $B_Q=B\cap(A^n\times\{Q\})$, where $B$ is the smallest abelian subvariety of $A^n\times A^{*s}$ (up to isogeny, i.e. up to a surjective algebraic group homomorphism with finite kernel) containing the point $(P,Q)$, and $Z(1)$, an algebraic subtorus of $\mathbb{G}_m^{ns}$ of dimension $t\le ns$, containing the subtorus $Z'(1)$ attached to the image of the Lie-bracket data of the Poincaré biextension (the canonical $\mathbb{G}_m$-bundle over $A\times A^*$ whose fibre over a point $(P',Q')$ is the line $Q'\vert_{P'}$, i.e. the fibre at $P'$ of the degree-zero line bundle on $A$ represented by $Q'$) restricted to $B$, and smallest such that the quotient torus $(Z/Z')(1)$ contains the push-down $\pi(\widetilde{R})$ of the point $\widetilde{R}$ in the trivialized biextension torsor (Construction 3.4 of the paper). The Relative Semi-abelian conjecture (Conjecture 1.3 = 4.7) is the inequality stated in problem_statement. Writing $g=\dim A$: for $s=0$ it reduces to the Weak Abelian Schanuel conjecture (Remark 4.8), and for $g=0$ (in the equivalent Explicit form, Conjecture 4.10 with $n=1$, Remark 4.11) it reduces to the classical Schanuel conjecture up to the known transcendence of $\pi$. The main theorem (Theorem 1.6) deduces from Conjecture 4.7 that the tower of fields $\mathcal{E}=\bigcup_n\mathcal{E}_n$ obtained by iterating $\exp_G$ starting from $\overline{\mathbb{Q}}$ and the tower $\mathcal{L}=\bigcup_n\mathcal{L}_n$ obtained by iterating $\widetilde{\log}_G$ are algebraically independent (linearly disjoint) over $\mathbb{Q}$ — the semi-abelian answer to Waldschmidt's question. All these statements are conditional: no instance of Conjectures 4.5, 4.7, 4.9, 4.10, or 6.5 with genuinely mixed toric and abelian parts is known, and Proposition 4.12 shows the conjectures follow from the (unproven) Generalized Period conjecture.

Problem Statement

Prove, or disprove by an explicit counterexample, the Relative Semi-abelian conjecture (Conjecture 1.3, restated as Conjecture 4.7, of Bertolin–Philippon–B. Saha–E. Saha, Journal of Algebra 596 (2022)). Let $A$ be an abelian variety of dimension $g$ defined over $\overline{\mathbb{Q}}$, let $G$ be an extension of $A$ by the torus $\mathbb{G}_m^s$, corresponding to a point $Q=(Q_1,\dots,Q_s)\in A^*(\overline{\mathbb{Q}})^s$ under the classification of extensions by the dual abelian variety, and let $\Omega_G$ denote the matrix of periods of $G$ (the integrals over $H_1(G(\mathbb{C}),\mathbb{Z})$ of a basis of $H^1_{dR}(G)$). If $R=(R_1,\dots,R_n)$ is a point of $G(\mathbb{C})^n$ above a point $P=(P_1,\dots,P_n)\in A(\mathbb{C})^n$, then the conjecture asserts $\mathrm{tran.deg}_{\mathbb{Q}(\Omega_G)}\,\mathbb{Q}\big(\Omega_G,\,R,\,\widetilde{\log}_{G}^{\,n}(R)\big)\;\ge\;2\,\dim B_Q+\dim Z(1),$ where $\widetilde{\log}_{G}^{\,n}(R)=(\widetilde{\log}_G(R_1),\dots,\widetilde{\log}_G(R_n))$ is the tuple of generalized semi-abelian logarithms (integrals of the first, second, and third kind, Definition 2.4 of the paper), $B_Q=B\cap(A^n\times\{Q\})$ with $B$ the smallest abelian subvariety (modulo isogenies, i.e. up to a surjective algebraic group homomorphism with finite kernel) of $A^n\times A^{*s}$ containing the point $(P,Q)$, and $Z(1)$, of dimension $t\le ns$, is the subtorus of $\mathbb{G}_m^{ns}$ attached to the data by Construction 3.4(3) of the paper (the smallest such that $(Z/Z')(1)$ contains the push-down $\pi(\widetilde{R})$ of the trivialized Poincaré biextension torsor).

The problem asks for this inequality to be established unconditionally for all choices of $A$, $G$, $n\ge 1$, and $R$ as above, or to be refuted by one explicit instance satisfying the hypotheses for which the inequality provably fails. A derivation from the Grothendieck–André Generalized Period conjecture or from any other unproven hypothesis (such as Schanuel's conjecture) does not resolve the problem: Proposition 4.12 of the source paper already gives the former implication. Note the degenerate cases that any solution must in particular cover: $s=0$ recovers the Weak Abelian Schanuel conjecture, and the purely toric case $g=0$ (with $n=1$, in the equivalent Explicit form, Conjecture 4.10) recovers, up to the known transcendence of $\pi$, the classical Schanuel conjecture.

References

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