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Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture

Krzysztof Klosin (2009)

Annales de l’institut Fourier

Let k be a positive integer divisible by 4, p > k a prime, f an elliptic cuspidal eigenform (ordinary at p ) of weight k - 1 , level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives ad 0 M ( - 1 ) and ad 0 M ( 2 ) , where M is the motif attached to f . More precisely, we prove that under certain conditions the p -adic valuation of the algebraic part of the symmetric square L -function of f evaluated at k provides a lower bound for the p -adic valuation of the order of the Pontryagin...

Congruences between modular forms and lowering the level mod ℓ n

Luis Dieulefait, Xavier Taixés i Ventosa (2009)

Journal de Théorie des Nombres de Bordeaux

In this article we study the behavior of inertia groups for modular Galois mod ℓ n representations and in some cases we give a generalization of Ribet’s lowering the level result (cf. [9]).

Congruences between Siegel modular forms on the level of group cohomology

Karsten Buecker (1996)

Annales de l'institut Fourier

Vector-valued Siegel modular forms may be found in certain cohomology groups with coefficients lying in an irreducible representation of the symplectic group. Using functoriality in the coefficients, we show that the ordinary components of the cohomology are independent of the weight parameter. The meaning of ordinary depends on a choice of parabolic subgroup of G S p ( 4 ) , giving a particular direction in the change of weight. Our results complement those of Taylor and Tilouine-Urban for the two other possible...

Congruences for certain binomial sums

Jung-Jo Lee (2013)

Czechoslovak Mathematical Journal

We exploit the properties of Legendre polynomials defined by the contour integral 𝐏 n ( z ) = ( 2 π i ) - 1 ∮ ( 1 - 2 t z + t 2 ) - 1 / 2 t - n - 1 d t , where the contour encloses the origin and is traversed in the counterclockwise direction, to obtain congruences of certain sums of central binomial coefficients. More explicitly, by comparing various expressions of the values of Legendre polynomials, it can be proved that for any positive integer r , a prime p ⩾ 5 and n = r p 2 - 1 , we have ∑ k = 0 ⌊ n / 2 ⌋ 2 k k ≡ 0 , 1 or - 1 ( mod p 2 ) , depending on the value of r ( mod 6 ) .

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