arXiv subjects

Cristiana Bertolin

Publications and source records attributed to Cristiana Bertolin.

At least 19 recordsLinked to original sources

arXiv2026

Algebraic independence of the exponential and Weierstrass $\wp$-functions

Let $Ω$ be a lattice in $\mathbb{C}$ with algebraic invariants and complex multiplication, let $\mathcal{E}$ be the elliptic curve associated with $Ω$, and let $\wp$ be the Weierstrass function relative to $Ω$. Set $ k:=\operatorname{End}(\mathcal{E}) \otimes_{\mathbb{Z}}\mathbb{Q}.$ We prove that if $t_1,\dots,t_s$ are $\mathbb{Q}$-linearly independent algebraic numbers and $p_1,\dots,p_n$ are $k$-linearly independent algebraic numbers, then the $s+n$ numbers \[ \mathrm{e}^{t_1},\dots,\mathrm{e}^{t_s}, \wp(p_1),\dots,\wp(p_n) \] are algebraically independent over $\overline{\mathbb{Q}}$. The proof uses the Tannakian description of the Lie algebra of the unipotent radical of the $1$-motive associated with these points.

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arXiv2026

Dimension of the motivic Galois group of a 1-motive

We compute the dimension of the motivic Galois group of a 1-motive M defined over the field of complex numbers, expressing it explicitly in terms of the rank of the multiplicative group generated by the points defining M. As an application, we obtain a new formulation of the Grothendieck--André periods Conjecture in the setting of 1-motives.

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arXiv2026

A conjecture in Schanuel style for 1-motives

Schanuel Conjecture contains all ``reasonable" statements that can be made on the values of the exponential function. In particular it implies the Lindemann-Weierstrass Theorem. In my Ph.D. I showed that Schanuel Conjecture has a geometrical origin: it is equivalent to the Grothendieck-André periods Conjecture applied to a 1-motive without abelian part. In this paper, we state a conjecture in Schanuel style, which will imply conjectures in Lindemann-Weierstrass style, for the semi-elliptic exponential function, that is for the exponential map of an extension G of an elliptic curve E by a multiplicative group. We propose the semi-elliptic Conjecture, which concerns the exponential function, the Weierstrass $\wp,$ $ζ$ functions and Serre functions. The case of a trivial extension has been treated in \cite{BW}, where we introduced the split semi-elliptic Conjecture. As in Schanuel's case, we expect that the semi-elliptic Conjecture contains all ``reasonable" statements that can be made on the values of the exponential function, of the Weierstrass $\wp$, $ζ$ functions and of Serre functions. We show that the semi-elliptic Conjecture has a geometrical origin (as Schanuel Conjecture): it is equivalent to the Grothendieck-André periods Conjecture applied to a 1-motive whose underlying abelian part is an elliptic curve. We prove the Grothendieck-André periods Conjecture for 1-motives defined by an elliptic curve with algebraic invariants and complex multiplication and by torsion points. We introduce the $σ$-Conjecture which involves the Weierstrass $\wp$, $ζ$ and $σ$ functions and we show that this conjecture is a consequence of the Grothendieck-André periods Conjecture applied to an adequate 1-motive.

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arXiv2026

1-motives and admissible variations of mixed Hodge structures

Let S be a connected scheme smooth and of finite type over the field of complex numbers. To every 1-motive over S, André associated the enriched Hodge realization given by a torsion-free, graded-polarizable and admissible variation of mixed Hodge structures of type (0,0), (-1,0), (0,-1), (-1,-1) over the associated complex analytic space. In this paper, we prove that every admissible variation of mixed Hodge structures of the above type arises, up to isogeny, from a 1-motive over S, thereby providing a positive answer to a question of André concerning the geometric origin of such variations. More precisely, we establish a Hodge-theoretic interpretation of sections of semi-abelian varieties by combining André's description of the abelian case with a new analysis of the toric part. As a consequence, we prove a relative analogue of Deligne's equivalence over the field of complex numbers. Namely, under suitable assumptions on S and on the lattices and the tori underlying 1-motives, the enriched Hodge realization functor induces an equivalence between the category of 1-motives over S and the category of torsion-free, graded-polarizable and admissible variations of mixed Hodge structures of type (0,0), (-1,0), (0,-1), (-1,-1). In general, the corresponding statement holds only up to isogeny. Finally, we introduce the global Mumford--Tate group of a 1-motive over S and show that its neutral connected component identifies with the Mumford-Tate group of the generic fiber.

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arXiv2025

Variations on Schanuel's Conjecture for elliptic and quasi-elliptic functions I: the split case

It is expected that Schanuel's Conjecture contains all ``reasonable" statements that can be made on the values of {\em the exponential function}. In particular it implies the Lindemann-Weierstrass Theorem and the Conjecture on algebraic independence of logarithms of algebraic numbers. Our goal is to state conjectures {\em à la Schanuel}, which imply conjectures {\em à la Lindemann-Weierstrass}, for the exponential map of an extension $G$ of an elliptic curve ${\mathcal E}$ by the multiplicative group ${\mathbb G}_m$. In the present paper we assume that the extension is split, that is $G={\mathbb G}_m\times {\mathcal E}$. In a second paper in preparation we will deal with the non-split case, namely when the extension is not a product. Here we propose the {\em split semi-elliptic Conjecture}, which involves the exponential function and the Weierstrass $\wp$ and $ζ$ functions, related with integrals of the first and second kind. In the second paper, our {\em non-split semi-elliptic Conjecture} will also involve Serre's functions, related with integrals of the third kind. We expect that our conjectures contain all ``reasonable" statements that can be made on the values of these functions. In the present paper we highlight the geometric origin of the split semi-elliptic Conjecture: it is {\em equivalent to} the Grothendieck-André generalized period Conjecture applied to the 1-motive $M=[u:\mathbb{Z} \rightarrow {\mathbb G}_m^s \times {\mathcal E}^n ]$, which is the Elliptico-Toric Conjecture of the first author. We show that our split semi-elliptic Conjecture implies three theorems of Schneider on elliptic analogs of the Hermite-Lindemann and Gel'fond-Schneider's theorems, as well as a conjecture on the Weierstrass zeta function.

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arXiv2024

Mumford-Tate groups of 1-motives and Weil pairing

We show how the geometry of a 1-motive $M$ (that is existence of endomorphisms and relations between the points defining it) determines the dimension of its motivic Galois group ${\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}(M)$. Fixing periods matrices $Π_M$ and $Π_{M^*}$ associated respectively to a 1-motive $M$ and to its Cartier dual $M^*,$ we describe the action of the Mumford-Tate group of $M$ on these matrices. In the semi-elliptic case, according to the geometry of $M$ we classify polynomial relations between the periods of $M$ and we compute exhaustively the matrices representing the Mumford-Tate group of $M$. This representation brings new light on Grothendieck periods conjecture in the case of 1-motives.

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arXiv2023

The group law of Picard stacks via matrices

Let S be a site. We show that the 2-stack of strictly commutative Picard stacks over S is algebraic, i.e. it is 2-equivalent to the 2-stack of 2-algebras for an adequate algebraic 2-stack theory over S.

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arXiv2022

Semi-abelian analogues of Schanuel Conjecture and applications

In this article we study Semi-abelian analogues of Schanuel conjecture. As showed by the first author, Schanuel Conjecture is equivalent to the Generalized Period Conjecture applied to 1-motives without abelian part. Extending her methods, the second, the third and the fourth authors have introduced the Abelian analogue of Schanuel Conjecture as the Generalized Period Conjecture applied to 1-motives without toric part. As a first result of this paper, we define the Semi-abelian analogue of Schanuel Conjecture as the Generalized Period Conjecture applied to 1-motives. C. Cheng et al. proved that Schanuel conjecture implies the algebraic independence of the values of the iterated exponential and the values of the iterated logarithm, answering a question of M. Waldschmidt. The second, the third and the fourth authors have investigated a similar question in the setup of abelian varieties: the Weak Abelian Schanuel conjecture implies the algebraic independence of the values of the iterated abelian exponential and the values of an iterated generalized abelian logarithm. The main result of this paper is that a Relative Semi-abelian conjecture implies the algebraic independence of the values of the iterated semi-abelian exponential and the values of an iterated generalized semi-abelian logarithm.

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arXiv2021

Brauer groups of 1-motives

Over a normal base scheme, we prove the generalized Theorem of the Cube for 1-motives and that a torsion class of the group H^2_ét(M,G_m)$ of a 1-motive M, whose pull-back via the unit section is zero, comes from an Azumaya algebra. In particular, we deduce that over an algebraically closed field of characteristic zero, all classes of H^2_ét(M,G_m) come from Azumaya algebras.

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arXiv2020

A note on divisorial correspondences of extensions of abelian schemes by tori

Let S be a locally noetherian scheme and consider two extensions G_1 and G_2 of abelian S-schemes by S-tori. In this note we prove that the fppf-sheaf Corr _S(G_1,G_2) of divisorial correspondences between G_1 and G_2 is representable. Moreover, using divisorial correspondences, we show that line bundles on an extension G of an abelian scheme by a torus define group homomorphisms between G and Pic_{ G/S}.

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arXiv2019

Third kind elliptic integrals and 1-motives

In our PH.D. thesis we have showed that the Generalized Grothendieck's Conjecture of Periods applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental conjecture involving elliptic integrals of the first and second kind, and logarithms of complex numbers. In this paper we investigate the Generalized Grothendieck's Conjecture of Periods in the case of 1-motives whose underlying semi-abelian variety is a non trivial extension of a product of elliptic curves by a torus. This will imply the introduction of elliptic integrals of the third kind for the computation of the period matrix of M and therefore the Generalized Grothendieck's Conjecture of Periods applied to M will be equivalent to a transcendental conjecture involving elliptic integrals of the first, second and third kind.

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arXiv2018

Morphisms of 1-motives defined by line bundles

Let $S$ be a normal base scheme. The aim of this paper is to study the line bundles on 1-motives defined over $S$. We first compute a dévissage of the Picard group of a 1-motive $M$ according to the weight filtration of $M$. This dévissage allows us to associate, to each line bundle $L$ on $M$, a linear morphism $φ_{L}: M \rightarrow M^*$ from $M$ to its Cartier dual. This yields a group homomorphism $Φ: Pic(M) / Pic(S) \to Hom(M,M^*)$. We also prove the Theorem of the Cube for 1-motives, which furnishes another construction of the group homomorphism $Φ: Pic(M) / Pic(S) \to Hom(M,M^*)$. Finally we prove that these two independent constructions of linear morphisms $M \to M^*$ using line bundles on $M$ coincide. However, the first construction, involving the dévissage of $Pic(M)$, is more explicit and geometric and it furnishes the motivic origin of some linear morphisms between 1-motives. The second construction, involving the Theorem of the Cube, is more abstract but perhaps also more enlightening.

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arXiv2018

Higher-dimensional study of extensions via torsors

Let S be a site. First we define the 3-category of torsors under a Picard S-2-stack and we compute its homotopy groups. Using calculus of fractions we define also a pure algebraic analogue of the 3-category of torsors under a Picard S-2-stack. Then we describe extensions of Picard S-2-stacks as torsors endowed with a group law on the fibers. As a consequence of such a description, we show that any Picard S-2-stack admits a canonical free partial left resolution that we compute explicitly. Moreover we get an explicit right resolution of the 3-category of extensions of Picard S-2-stacks in terms of 3-categories of torsors. Using the homological interpretation of Picard S-2-stacks, we rewrite this three categorical dimensions higher right resolution in the derived category of abelian sheaves on S.

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arXiv2013

Extensions of Picard 2-Stacks and the cohomology groups Ext^i of length 3 complexes

The aim of this paper is to define and study the 3-category of extensions of Picard 2-stacks over a site S and to furnish a geometrical description of the cohomology groups Ext^i of length 3 complexes of abelian sheaves. More precisely, our main Theorem furnishes (1) a parametrization of the equivalence classes of objects, 1-arrows, 2-arrows, and 3-arrows of the 3-category of extensions of Picard 2-stacks by the cohomology groups Ext^i, and (2) a geometrical description of the cohomology groups Ext^i of length 3 complexes of abelian sheaves via extensions of Picard 2-stacks. To this end, we use the triequivalence between the 3-category of Picard 2-stacks and the tricategory T^[-2,0](S) of length 3 complexes of abelian sheaves over S introduced by the second author in arXiv:0906.2393, and we define the notion of extension in this tricategory T^[-2,0](S), getting a pure algebraic analogue of the 3-category of extensions of Picard 2-stacks. The calculus of fractions that we use to define extensions in the tricategory T^[-2,0](S) plays a central role in the proof of our Main Theorem.

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arXiv2012

Homological interpretation of extensions and biextensions of 1-motives

Let k be a separably closed field. Let K_i=[A_i \to B_i] (for i=1,2,3) be three 1-motives defined over k. We define the geometrical notions of extension of K_1 by K_3 and of biextension of (K_1,K_2) by K_3. We then compute the homological interpretation of these new geometrical notions: namely, the group Biext^0(K_1,K_2;K_3) of automorphisms of any biextension of (K_1,K_2) by K_3 is canonically isomorphic to the cohomology group Ext^0(K_1 \otimes K_2,K_3), and the group Biext^1(K_1,K_2;K_3) of isomorphism classes of biextensions of (K_1,K_2) by K_3 is canonically isomorphic to the cohomology group Ext^1(K_1 \otimes K_2,K_3).

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arXiv2011

Biextensions of Picard stacks and their homological interpretation

Let S be a site. We introduce the 2-category of biextensions of strictly commutative Picard S-stacks. We define the pull-back, the push-down, and the sum of such biextensions and we compute their homological interpretation: if P,Q and G are strictly commutative Picard S-stacks, the equivalence classes of biextensions of (P,Q) by G are parametrized by the cohomology group Ext^1([P] {\otimes} [Q] ,[G]), the isomorphism classes of arrows from such a biextension to itself are parametrized by the cohomology group Ext^0([P]{\otimes} [Q] ,[G]) and the automorphisms of an arrow from such a biextension to itself are parametrized by the cohomology group Ext^{-1}([P]{\otimes}[Q] ,[G]), where [P],[Q] and [G] are the complex associated to P,Q and G respectively.

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arXiv2010

Some tensor products

We define the tensor product of 1-motives with motives of weight 0 and we construct explicitely the 1-motive underlying the quotient M_1 \otimes M_2 / W_{-3}(M_1 \otimes M_2).

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arXiv2010

Extensions and biextensions of locally constant group schemes, tori and abelian schemes

Let S be a scheme. We compute explicitly the group of homomorphisms, the S-sheaf of homomorphisms, the group of extensions, and the S-sheaf of extensions involving locally constant S-group schemes, abelian S-schemes, and S-tori. Using the obtained results, we study the categories of biextensions involving these geometrical objets. In particular, we prove that if G_i (for i=1,2,3) is an extension of an abelian S-scheme A_i by an S-torus T_i, the category of biextensions of (G_1,G_2) by G_3 is equivalent to the category of biextensions of the underlying abelian S-schemes (A_1,A_2) by the underlying S-torus T_3.

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