Galois representations for holomorphic Siegel modular forms

2012 ◽  
Vol 355 (1) ◽  
pp. 381-400 ◽  
Author(s):  
Andrei Jorza
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Siegel Modular Forms

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.

scholarly journals Images de représentations galoisiennes associées à certaines formes modulaires de Siegel de genre 2

2017 ◽  
Vol 13 (05) ◽  
pp. 1129-1144
Author(s):  
Salim Tayou
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Siegel Modular Forms ◽  
Vector Valued

We study the image of the [Formula: see text]-adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes [3]. These representations are symplectic of dimension 4. Following methods used by Dieulefait in [4], we determine the primes [Formula: see text] for which these representations are absolutely irreducible. In addition, we show that their image is “full” for all primes [Formula: see text] such that the associated residual representation is absolutely irreducible, except in two cases.

SYMMETRIC SQUARE L-FUNCTIONS AND SHAFAREVICH–TATE GROUPS, II

2009 ◽  
Vol 05 (07) ◽  
pp. 1321-1345 ◽  
Author(s):  
NEIL DUMMIGAN
Keyword(s):  
Modular Forms ◽  
Prime Order ◽  
Galois Representations ◽  
Critical Value ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Fourier Expansions ◽  
Symmetric Square ◽  
Vector Valued ◽  
Siegel Cusp Form

We re-examine some critical values of symmetric square L-functions for cusp forms of level one. We construct some more of the elements of large prime order in Shafarevich–Tate groups, demanded by the Bloch–Kato conjecture. For this, we use the Galois interpretation of Kurokawa-style congruences between vector-valued Siegel modular forms of genus two (cusp forms and Klingen–Eisenstein series), making further use of a construction due to Urban. We must assume that certain 4-dimensional Galois representations are symplectic. Our calculations with Fourier expansions use the Eholzer–Ibukiyama generalization of the Rankin–Cohen brackets. We also construct some elements of global torsion which should, according to the Bloch–Kato conjecture, contribute a factor to the denominator of the rightmost critical value of the standard L-function of the Siegel cusp form. Then we prove, under certain conditions, that the factor does occur.

scholarly journals On Galois Representations via Siegel Modular Forms of Genus Two

2001 ◽  
Vol 8 (4) ◽  
pp. 577-588 ◽  
Author(s):  
Michael Dettweiler ◽  
Ulf Kühn ◽  
Stefan Reiter
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Siegel Modular Forms ◽  
Genus Two

scholarly journals Higher pullbacks of modular forms on orthogonal groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Brandon Williams
Keyword(s):  
Modular Forms ◽  
Differential Operators ◽  
Jacobi Form ◽  
Siegel Modular Forms ◽  
Orthogonal Groups

Abstract We apply differential operators to modular forms on orthogonal groups O ⁢ ( 2 , ℓ ) {\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature ( 2 , 2 ) {(2,2)} and ( 2 , 3 ) {(2,3)} , for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

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