Let $J_0\left(𝔫\right)$ be the Jacobian variety of the Drinfeld modular curve $X_0\left(𝔫\right)$ over ${𝔽}_q\left(t\right)$, where $𝔫$ is an ideal in ${𝔽}_q\left[t\right]$. Let $0\to B\to J_0\left(𝔫\right)\to A\to 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0\left(𝔫\right)$ , is stable under the action of the Hecke algebra $𝕋\subset$ End $\left(J_0\left(𝔫\right)\right)$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0\to {\Phi }_{B,\infty }\to {\Phi }_{J,\infty }\to {\Phi }_{A,\infty }\to 0$ of the Néron models of these abelian varieties over ${𝔽}_q\left[\left[\frac{1}{t}\right]\right]$. This criterion is always satisfied when either $A$ or $B$ is one-dimensional. Moreover, we prove that...

We obtain coniugacy classes (with respect to a $𝔰l_2$ action) in the space of Hodge cycles in the middle cohomology of an Abelian variety of quaternionic type. The existence of such a class simplifies the study of the Hodge conjecture.

We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection $C\left(G\right)/C_{E^c}\left(G\right)\hookrightarrow L{²}_E\left(G\right)$ is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.

We study the structure of the filtered Gauss-Manin system associated to a holomorphic function with an isolated singularity, and get a basis of the Brieskorn lattice ${\Omega }_{X,0}^{n+1}/df\wedge d{\Omega }_{X,0}^{n+1}$ over $ℂ\left\{\right\{{\partial }_t^{-1}\left\}\right\}$ such that the action of $t$ is expressed by $$tv=A_0+A_1{\partial }_t^{-1}v$$ for two matrices $A_0,A_1$ with $A_1$ semi-simple, where $v={}^t(v_1...v_{\mu })$ is the basis. As an application, we calculate the $b$-function of $f$ in the case of two variables.

If $K/k$ is a finite Galois extension of number fields with Galois group $G$, then the kernel of the capitulation map $C{l}_k\to C{l}_K$ of ideal class groups is isomorphic to the kernel $X\left(H\right)$ of the transfer map $H/H^{\text{'}}\to A,$ where $H=\text{Gal}(\tilde{K}/k),A=\text{Gal}(\tilde{K}/K)$ and $\tilde{K}$ is the Hilbert class field of $K$. H. Suzuki proved that when $G$ is abelian, $\left|G\right|$ divides $\left|X\right(H\left)\right|$. We call a finite abelian group $X$ a transfer kernel for $G$ if $X\cong X\left(H\right)$ for some group extension $A\hookrightarrow H\twoheadrightarrow G$. After characterizing transfer kernels in terms of integral representations of $G$, we show that $X$ is a transfer...

Sia $X$ una varietà abeliana e $L$ un fibrato in rette ampio di tipo $\left(2,2d_2,\mathrm{\dots },2d_g\right)$ su $X$; sia ${\phi }_L$ l'applicazione associata a $L$. In questo lavoro si dimostra il seguente fatto: se $d_i\le 2$ per qualsiasi $i$, $L$ non è mai normalmente generato (quindi, se ${\phi }_L$ è un embedding, ${\phi }_L\left(X\right)$ non è proiettivamente normale); negli altri casi invece $L$ è normalmente generato per $\left(X,c_1\left(L\right)\right)$ generico nello spazio dei moduli delle varietà abeliane polarizzate di tipo $\left(2,2d_2,\mathrm{\dots },2d_g\right)$.

Let $n$ be a positive integer and $A^{\prime }$ a complete characteristic zero discrete valuation ring with maximal ideal $𝔪$, absolute ramification index $e\<p-1$ and perfect residue field $k$ of characteristic $p\>2$. In this paper we classify smooth finite dimensional formal $p$- groups over $A_n^{\prime }=A^{\prime }/{𝔪}^n{A}^{\prime }$, groups on which the “multiplication by $p$” morphism is faithfully flat, in particular $p$-divisible groups. As applications, we prove that $p$-divisible groups over $k$, and the morphisms between them, lift canonically to $A^{\prime }/p{A}^{\prime }$, and...