scholarly journals Selmer groups of symmetric powers of ordinary modular Galois representations

2021 ◽  
Vol 143 (1) ◽  
pp. 125-173
Author(s):  
Xiaoyu Zhang
Keyword(s):  
Galois Representations ◽  
Selmer Groups ◽  
Symmetric Powers

scholarly journals SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS

2018 ◽  
Vol 19 (5) ◽  
pp. 1521-1572
Author(s):  
Haruzo Hida ◽  
Jacques Tilouine
Keyword(s):  
Power Series ◽  
Galois Representation ◽  
Symmetric Power ◽  
Selmer Groups ◽  
Abelian Surfaces ◽  
Symmetric Powers ◽  
Branching Laws ◽  
Hida Family

We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over $\mathbb{Q}$ of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use $R=T$ theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from $\text{GSp}(4)$ to $U(4)$. In that case we obtain a corollary for abelian surfaces.

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 Galois representations attached to moments of Kloosterman sums and conjectures of Evans

2014 ◽  
Vol 151 (1) ◽  
pp. 68-120 ◽  
Author(s):  
Zhiwei Yun ◽  
Christelle Vincent
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Kloosterman Sums ◽  
Symmetric Power ◽  
Reductive Groups ◽  
Local Systems ◽  
Symmetric Powers ◽  
Exterior Powers ◽  
Power Moments ◽  
Tensor Powers

AbstractKloosterman sums for a finite field $\mathbb{F}_{p}$ arise as Frobenius trace functions of certain local systems defined over $\mathbb{G}_{m,\mathbb{F}_{p}}$. The moments of Kloosterman sums calculate the Frobenius traces on the cohomology of tensor powers (or symmetric powers, exterior powers, etc.) of these local systems. We show that when $p$ ranges over all primes, the moments of the corresponding Kloosterman sums for $\mathbb{F}_{p}$ arise as Frobenius traces on a continuous $\ell$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ that comes from geometry. We also give bounds on the ramification of these Galois representations. All of this is done in the generality of Kloosterman sheaves attached to reductive groups introduced by Heinloth, Ngô and Yun [Ann. of Math. (2) 177 (2013), 241–310]. As an application, we give proofs of conjectures of Evans [Proc. Amer. Math. Soc. 138 (2010), 517–531; Israel J. Math. 175 (2010), 349–362] expressing the seventh and eighth symmetric power moments of the classical Kloosterman sum in terms of Fourier coefficients of explicit modular forms. The proof for the eighth symmetric power moment conjecture relies on the computation done in Appendix B by C. Vincent.

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.

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