Horizontal variation of Tate--Shafarevich groups

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and $\iota: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field. For a class group character $\chi$ over $K$, let $\mathbb{Q}(\chi)$ be the field generated by the image of $\chi$ and $\mathfrak{p}_{\chi}$ the prime of $\mathbb{Q}(\chi)$ above $p$ determined via $\iota_p$. Under mild hypotheses, we show that the number of class group characters $\chi$ such that the $\chi$-isotypic Tate--Shafarevich group of $E$ over $H_{K}$ is finite with trivial $\mathfrak{p}_{\chi}$-part increases with the absolute value of the discriminant of $K$

Class groups and local indecomposability for non-CM forms

In the late 1990's, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those $p$-ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at $p$. It is expected that such $p$-ordinary eigenforms are precisely those with complex multiplication. In this paper, we study Coleman-Greenberg's question using Galois deformation theory. In particular, for $p$-ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the $p$-indivisibility of a certain class group.Comment: 40 pages, with a 11-page appendix by Haruzo Hida. v3: improvements to exposition, minor correction

BIG IMAGE OF GALOIS REPRESENTATIONS AND CONGRUENCE IDEALS

2. Galois representations associated to Siegel modular forms 2 3. Fullness of the image for Galois representations in GSp(4) 3 3.1. Irreducibility and open image

Control theorems of pp-nearly ordinary cohomology groups for SL(n)\text {SL} (n)

ABSTRACT. — In this paper, we prove control theorems for the p-adic nearly ordinary cohomology groups for SL(n) over an aribitrary number field, generalizing the result already obtained for SL(2). The result should have various implications in the study of p-adic cohomological modulat forms on GL(n). In particular, in a subsequent paper, we will study p-adic analytic families of cuch Hecke eigenforms. RÉSUMÉ. — Dans cet article, on démontre le théorème de contrôle pour les groupes de cohomologie quasi-ordinaire p-adique de SL(n) sur un corps de nombre arbitraire en généralisant le résultat déjà connu pour SL(2). Le résultat doit avoir des implications variées dans la théorie des formes modulaires p-adiques cohomologiques sur GL(n). En particulier, on étudiera des familles p-adiques analytiques des formes propres de Hecke dans un prochain article