104 research outputs found
Horizontal variation of Tate--Shafarevich groups
Let be an elliptic curve over . Let be an odd prime and
an embedding. Let
be an imaginary quadratic field and the corresponding Hilbert class
field. For a class group character over , let be
the field generated by the image of and the prime
of above determined via . Under mild
hypotheses, we show that the number of class group characters such that
the -isotypic Tate--Shafarevich group of over is finite with
trivial -part increases with the absolute value of the
discriminant of
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 -ordinary cuspidal eigenforms whose
associated Galois representation becomes decomposable upon restriction to a
decomposition group at . It is expected that such -ordinary eigenforms
are precisely those with complex multiplication. In this paper, we study
Coleman-Greenberg's question using Galois deformation theory. In particular,
for -ordinary eigenforms which are congruent to one with complex
multiplication, we prove that the conjectured answer follows from the
-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 -nearly ordinary cohomology groups for
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
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