On Selmer Groups of Adjoint Modular Galois Representations

Number Theory ◽  
1996 ◽  
pp. 89-132 ◽  
Author(s):  
Haruzo Hida
Keyword(s):  
Galois Representations ◽  
Selmer Groups

scholarly journals Fine Selmer groups of congruent Galois representations

2018 ◽  
Vol 187 ◽  
pp. 66-91 ◽  
Author(s):  
Meng Fai Lim ◽  
Ramdorai Sujatha
Keyword(s):  
Galois Representations ◽  
Selmer Groups

scholarly journals Akashi series, characteristic elements and congruence of Galois representations

2016 ◽  
Vol 12 (03) ◽  
pp. 593-613
Author(s):  
Meng Fai Lim
Keyword(s):  
Galois Representations ◽  
The Other ◽  
Selmer Groups ◽  
Euler Characteristics

In this paper, we compare the Akashi series of the Pontryagin dual of the Selmer groups of two Galois representations over a strongly admissible [Formula: see text]-adic Lie extension. Namely, we show that whenever the two Galois representations in question are congruent to each other, the Akashi series of one is a unit if and only if the Akashi series of the other is also a unit. We will also obtain a similar result for the Euler characteristics of the Selmer groups and the characteristic elements attached to the Selmer groups.

scholarly journals Irreducibility of limits of Galois representations of Saito–Kurokawa type

2021 ◽  
Vol 7 (3) ◽  
Author(s):  
Tobias Berger ◽  
Krzysztof Klosin
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Galois Representations ◽  
Siegel Modular Forms ◽  
Selmer Groups ◽  
Two Dimensional ◽  
Elliptic Modular Form

AbstractWe prove (under certain assumptions) the irreducibility of the limit $$\sigma _2$$ σ 2 of a sequence of irreducible essentially self-dual Galois representations $$\sigma _k: G_{{\mathbf {Q}}} \rightarrow {{\,\mathrm{GL}\,}}_4(\overline{{\mathbf {Q}}}_p)$$ σ k : G Q → GL 4 ( Q ¯ p ) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to $$1 \oplus \rho \oplus \chi $$ 1 ⊕ ρ ⊕ χ with $$\rho $$ ρ irreducible, two-dimensional of determinant $$\chi $$ χ , where $$\chi $$ χ is the mod p cyclotomic character. More precisely, we assume that $$\sigma _k$$ σ k are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as $$k\rightarrow 2$$ k → 2 appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to $$\rho $$ ρ ) which we assume are non-zero.

Compatibility of arithmetic and algebraic local constants (the case )

2015 ◽  
Vol 151 (9) ◽  
pp. 1626-1646 ◽  
Author(s):  
Jan Nekovář
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Selmer Groups ◽  
Real Multiplication ◽  
Special Cases ◽  
Congruent Modulo ◽  
Multiplication Theorem ◽  
Parity Conjecture

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$ are compatible with the usual local constants at all primes not dividing $p$ and in two special cases also at primes dividing $p$. We deduce new cases of the $p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

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.

EULER CHARACTERISTIC OF Λ-ADIC FORMS OVER KUMMER EXTENSIONS

2014 ◽  
Vol 10 (02) ◽  
pp. 401-419 ◽  
Author(s):  
SUDHANSHU SHEKHAR
Keyword(s):  
Euler Characteristic ◽  
Modular Forms ◽  
Galois Representations ◽  
Rational Numbers ◽  
Selmer Groups

In this paper we compute the Euler characteristic of the Selmer groups associated to modular forms over certain Kummer extensions of the field of rational numbers. We also discuss the Euler characteristic of Λ-adic deformations of Galois representations associated to modular forms.

scholarly journals Adjoint modular Galois representations and their Selmer groups

1997 ◽  
Vol 94 (21) ◽  
pp. 11121-11124 ◽  
Author(s):  
H. Hida ◽  
J. Tilouine ◽  
E. Urban
Keyword(s):  
Galois Representations ◽  
Selmer Groups
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