scholarly journals Congruences with Eisenstein series and -invariants

2019 ◽  
Vol 155 (5) ◽  
pp. 863-901 ◽  
Author(s):  
Joël Bellaïche ◽  
Robert Pollack
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Eisenstein Series ◽  
Galois Representations ◽  
Selmer Groups ◽  
Main Conjecture ◽  
The Family ◽  
Eisenstein Ideal ◽  
Hida Families

We study the variation of $\unicode[STIX]{x1D707}$-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the $p$-adic zeta function. This lower bound forces these $\unicode[STIX]{x1D707}$-invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When $U_{p}-1$ generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the $p$-adic $L$-function is simply a power of $p$ up to a unit (i.e. $\unicode[STIX]{x1D706}=0$). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

scholarly journals The geometry of Hida families II: -adic -modules and -adic Hodge theory

2018 ◽  
Vol 154 (4) ◽  
pp. 719-760
Author(s):  
Bryden Cais
Keyword(s):  
Galois Representations ◽  
Hodge Theory ◽  
Geometric Proof ◽  
De Rham Cohomology ◽  
Etale Cohomology ◽  
The Family ◽  
Étale Cohomology ◽  
De Rham ◽  
Hida Families ◽  
And Control

We construct the $\unicode[STIX]{x1D6EC}$-adic crystalline and Dieudonné analogues of Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $\unicode[STIX]{x1D6EC}$-adic comparison isomorphisms between these cohomologies and the $\unicode[STIX]{x1D6EC}$-adic de Rham cohomology studied in Cais [The geometry of Hida families I:$\unicode[STIX]{x1D6EC}$-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s $\unicode[STIX]{x1D6EC}$-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules attached to Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology by Dee [$\unicode[STIX]{x1D6F7}$–$\unicode[STIX]{x1D6E4}$modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable $\unicode[STIX]{x1D6EC}$-adic duality theorems for each of the cohomologies we construct.

scholarly journals On the Eisenstein ideal for imaginary quadratic fields

2009 ◽  
Vol 145 (03) ◽  
pp. 603-632 ◽  
Author(s):  
Tobias Berger
Keyword(s):  
Lower Bound ◽  
Automorphic Forms ◽  
Quadratic Field ◽  
Galois Representations ◽  
Hecke Algebra ◽  
Quadratic Fields ◽  
Selmer Group ◽  
Imaginary Quadratic Fields ◽  
Eisenstein Ideal ◽  
Eisenstein Cohomology

AbstractFor certain algebraic Hecke charactersχof an imaginary quadratic fieldFwe define an Eisenstein ideal in ap-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the specialL-valueL(0,χ). We further prove that its index is bounded from above by thep-valuation of the order of the Selmer group of thep-adic Galois character associated toχ−1. This uses the work of R. Tayloret al. on attaching Galois representations to cuspforms of GL2/F. Together these results imply a lower bound for the size of the Selmer group in terms ofL(0,χ), coinciding with the value given by the Bloch–Kato conjecture.

scholarly journals Deformations of reducible Galois representations to Hida-families

2021 ◽  
pp. 1-24
Author(s):  
Anwesh Ray
Keyword(s):  
Deformation Theory ◽  
Galois Representations ◽  
Congruence Class ◽  
Representation Formula ◽  
Theoretic Approach ◽  
Hida Families ◽  
Global Deformation

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that [Formula: see text] lifts to a Hida line for which the weights range over a congruence class modulo-[Formula: see text]. The advantage of the purely Galois theoretic approach is that it allows us to construct [Formula: see text]-adic families of Galois representations lifting the actual representation [Formula: see text], and not just the semisimplification.

scholarly journals Euler characteristics and their congruences in the positive rank setting

2020 ◽  
pp. 1-18
Author(s):  
Anwesh Ray ◽  
R. Sujatha
Keyword(s):  
Elliptic Curves ◽  
Euler Characteristic ◽  
Selmer Groups ◽  
Euler Characteristics ◽  
Rank Zero ◽  
Homology Groups ◽  
Main Conjecture ◽  
Congruence Relations ◽  
Higher Rank ◽  
Positive Rank

Abstract The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

scholarly journals Analysis and combinatorics of partition zeta functions

2020 ◽  
pp. 1-10
Author(s):  
Robert Schneider ◽  
Andrew V. Sills
Keyword(s):  
Zeta Function ◽  
Explicit Formula ◽  
Zeta Functions ◽  
Complete Information ◽  
Combinatorial Proof ◽  
Fixed Length ◽  
The Family ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Partial Fraction Decomposition

We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon’s partial fraction decomposition of the generating function for partitions of fixed length.

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