Local indecomposability of modular Galois representation

2021 ◽  
pp. 341-380
Keyword(s):  
Galois Representation

scholarly journals Tautological rings of spaces of pointed genus two curves of compact type

2016 ◽  
Vol 152 (7) ◽  
pp. 1398-1420 ◽  
Author(s):  
Dan Petersen
Keyword(s):  
Hodge Structure ◽  
Galois Representation ◽  
Mixed Hodge Structure ◽  
Poincaré Duality ◽  
General Study ◽  
Compact Type ◽  
Poincare Duality ◽  
Tautological Ring ◽  
Genus Two ◽  
Tautological Rings

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$, the moduli space of $n$-pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$. This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$. This improves and simplifies results of the author and Orsola Tommasi.

scholarly journals On the Notion of Conductor in the Local Geometric Langlands Correspondence

2017 ◽  
Vol 69 (1) ◽  
pp. 107-129
Author(s):  
Masoud Kamgarpour
Keyword(s):  
Irreducible Representation ◽  
Galois Representation ◽  
Loop Group ◽  
Langlands Correspondence ◽  
Categorical Representation ◽  
Swan Conductor ◽  
Geometric Langlands ◽  
Meromorphic Connection ◽  
Local Langlands Correspondence ◽  
Geometric Analogue

AbstractUnder the local Langlands correspondence, the conductor of an irreducible representation of Gln(F) is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

scholarly journals Nearly ordinary Galois deformations over arbitrary number fields

2008 ◽  
Vol 8 (1) ◽  
pp. 99-177 ◽  
Author(s):  
Frank Calegari ◽  
Barry Mazur
Keyword(s):  
Arbitrary Number ◽  
Quadratic Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Base Change ◽  
Deformation Spaces ◽  
Totally Real ◽  
Set Of Points ◽  
Dense Set

AbstractLet K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

ARITHMETIC ON A QUINTIC THREEFOLD

2001 ◽  
Vol 12 (08) ◽  
pp. 943-972 ◽  
Author(s):  
CATERINA CONSANI ◽  
JASPER SCHOLTEN
Keyword(s):  
Modular Form ◽  
Theta Series ◽  
Continuous System ◽  
Galois Representation ◽  
Hilbert Modular Form ◽  
Galois Action ◽  
Hilbert Modular ◽  
Common Eigenvector ◽  
Quintic Threefold ◽  
Real Quadratic

This paper investigates some aspects of the arithmetic of a quintic threefold in Pr 4 with double points singularities. Particular emphasis is given to the study of the L-function of the Galois action ρ on the middle ℓ-adic cohomology. The main result of the paper is the proof of the existence of a Hilbert modular form of weight (2, 4) and conductor 30, on the real quadratic field [Formula: see text], whose associated (continuous system of) Galois representation(s) appears to be the most likely candidate to induce the scalar extension [Formula: see text]. The Hilbert modular form is interpreted as a common eigenvector of the Brandt matrices which describe the action of the Hecke operators on a space of theta series associated to the norm form of a quaternion algebra over [Formula: see text] and a related Eichler order.

On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

2005 ◽  
Vol 48 (1) ◽  
pp. 16-31 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Ernst Kani
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Upper Bounds

AbstractLet E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l, let ϕl be the Galois representation associated to the l-division points of E. From a celebrated 1972 result of Serre we know that ϕl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕl is not surjective.

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 An algorithm for computing the reduction of 2-dimensional crystalline representations of Gal(ℚ¯p/ℚp)

2018 ◽  
Vol 14 (07) ◽  
pp. 1857-1894 ◽  
Author(s):  
Sandra Rozensztajn
Keyword(s):  
Galois Representation ◽  
Langlands Correspondence ◽  
Dimension Formula ◽  
Crystalline Representations ◽  
Reduction Modulo

We describe an algorithm to compute the reduction modulo [Formula: see text] of a crystalline Galois representation of dimension [Formula: see text] of [Formula: see text] with distinct Hodge–Tate weights via the semi-simple modulo [Formula: see text] Langlands correspondence. We give some examples computed with an implementation of this algorithm in SAGE.

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