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 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 On some dimension formula for automorphic forms of weight one I

1982 ◽  
Vol 85 ◽  
pp. 213-221 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Fuchsian Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
The Other ◽  
Dimension Formula ◽  
Linearly Independent ◽  
Other Hand ◽  
Algebraic Properties ◽  
Lecture Notes ◽  
Weight One

Let Γ be a fuchsian group of the first kind not containing the element . We shall denote by d0 the number of linearly independent automorphic forms of weight 1 for Γ. It would be interesting to have a certain formula for d0. But, Hejhal said in his Lecture Notes 548, it is impossible to calculate d0 using only the basic algebraic properties of Γ. On the other hand, Serre has given such a formula of d0 recently in a paper delivered at the Durham symposium ([7]). His formula is closely connected with 2-dimensional Galois representations.

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 IWASAWA THEORY FOR THE SYMMETRIC SQUARE OF A CM MODULAR FORM AT INERT PRIMES

2011 ◽  
Vol 54 (2) ◽  
pp. 241-259
Author(s):  
ANTONIO LEI
Keyword(s):  
Modular Form ◽  
Complex Multiplication ◽  
Iwasawa Theory ◽  
The Other ◽  
Selmer Groups ◽  
Supersingular Primes ◽  
Supersingular Prime ◽  
Symmetric Square ◽  
The One ◽  
Invent Math

AbstractLet f be a modular form with complex multiplication (CM) and p an odd prime that is inert in the CM field. We construct two p-adic L-functions for the symmetric square of f, one of which has the same interpolating properties as the one constructed by Delbourgo and Dabrowski (A. Dabrowski and D. Delbourgo, S-adic L-functions attached to the symmetric square of a newform, Proc. Lond. Math. Soc. 74(3) (1997), 559–611), whereas the other one has a similar interpolating properties but corresponds to a different eigenvalue of the Frobenius. The symmetry between these two p-adic L-functions allows us to define the plus and minus p-adic L-functions à la Pollack (R. Pollack, on the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118(3) (2003), 523–558). We also define the plus and minus p-Selmer groups analogous to the ones defined by Kobayashi (S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152(1) (2003), 1–36). We explain how to relate these two sets of objects via a main conjecture.

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).

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