Biextensions of 1-motives by 1-motives

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.

Realizations of Biextensions

Let k be a field of characteristic 0 and M(k) is the category of 1-motives over k. We prove that Biext^1(M_1,M_2;M_3)=Hom_{MR(k)}(T(M_1)\otimes T(M_2), T(M_3))

Gerbes and abelian motives

Assuming the Hodge conjecture for abelian varieties of CM-type, one obtains a good category of abelian motives over the algebraic closure of a finite field and a reduction functor to it from the category of CM-motives. Consequentely, one obtains a morphism of gerbes of fibre functors with certain properties. We prove unconditionally that there exists a morphism of gerbes with these properties, and we classify them.

1-动机范畴中晶体类的塔类型定理

Crystalline Realizations of 1-Motives: Integral Structure and Classification via Linear Frobenius

本文在1-动机范围内证明了p进版本的塔猜想，通过Deligne 1-动机的全忠实性定理，将Barsotti-Tate晶体函子与滤过Dieudonné模的Hom群联系起来，从而得出Frobenius兼容的晶体类本质上是动机的，进而给出动机映射的线性代数描述。

Hirzebruch-Zagier cycles and twisted triple product Selmer groups

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $A$ be another elliptic curve over a real quadratic number field. We construct a $\mathbb{Q}$-motive of rank $8$, together with a distinguished class in the associated Bloch-Kato Selmer group, using Hirzebruch-Zagier cycles, that is, graphs of Hirzebruch-Zagier morphisms. We show that, under certain assumptions on $E$ and $A$, the non-vanishing of the central critical value of the (twisted) triple product $L$-function attached to $(E,A)$ implies that the dimension of the associated Bloch-Kato Selmer group of the motive is $0$; and the non-vani…