42 research outputs found
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.Comment: paper with an appendix of Michel Waldschmidt and a letter of Yves
Andr\'
Biextensions of 1-motives by 1-motives
Let S be a scheme. In this paper, we define the notion of biextensions of
1-motives by 1-motives. If M(S) denotes the Tannakian category generated by
1-motives over S (in a geometrical sense), we define geometrically the
morphisms of M(S) from the tensor product of two 1-motives M_1 and M_2 to
another 1-motive M_3, to be the isomorphism classes of biextensions of
(M_1,M_2) by M_3. Generalizing this definition we obtain, modulo isogeny, the
geometrical notion of morphism of M(S) from a finite tensor product of
1-motives to another 1-motive.Comment: 15 page
Extensions of Picard stacks and their homological interpretation
Let S be a site. We introduce the notion of extensions of strictly
commutative Picard S-stacks. We define the pull-back, the push-down, and the
sum of such extensions and we compute their homological interpretation: if P
and Q are two strictly commutative Picard S-stacks, the equivalence classes of
extensions of P by Q are parametrized by the cohomology group Ext^1([P],[Q]),
where [P] and [Q] are the complex associated to P and Q respectively.Comment: more reference
Multilinear morphisms between 1-motives
Let S be an arbitrary scheme. We define biextensions of 1-motives by
1-motives which we see as the geometrical origin of morphisms from the tensor
product of two 1-motives to a third one. If S is the spectrum of a field of
characteristic 0, we check that these biextensions define morphisms from the
tensor product of the realizations of two 1-motives to the realization of a
third 1-motive. Generalizing we obtain the geometrical notion of morphisms from
a finite tensor product of 1-motives to another 1-motive.Comment: new introduction
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
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}.Comment: new versio
Gerbes and Brauer groups over stacks
The aim of this paper is to develop the theory of Brauer groups for stacks,
which are not necessarily algebraic, using gerbes as foundamental tools.
As an application, we focus our attention on Brauer theory for mixed motives:
in particular, over a normal base scheme, we prove the generalized Theorem of
the Cube for 1-motives and that a torsion class of the H^2_et(M,G_m) of a
1-motive M, whose pull-back via the unit section is zero, comes from an Azumaya
algebra. Over an algebraically closed field, all classes of H^2_et(M,G_m) come
from Azumaya algebras.Comment: We add a section about 2-descent theory for stack