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Modular parametrizations of certain elliptic curves

Matija Kazalicki, Koji Tasaka (2014)

Acta Arithmetica

Kaneko and Sakai (2013) recently observed that certain elliptic curves whose associated newforms (by the modularity theorem) are given by the eta-quotients can be characterized by a particular differential equation involving modular forms and Ramanujan-Serre differential operator. In this paper, we study certain properties of the modular parametrization associated to the elliptic curves over ℚ, and as a consequence we generalize and explain some of their findings.

Modular symbols, Eisenstein series, and congruences

Jay Heumann, Vinayak Vatsal (2014)

Journal de Théorie des Nombres de Bordeaux

Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k 2 and of the same level N , both eigenfunctions of the Hecke operators, and both normalized so that a 1 ( f ) = a 1 ( E ) = 1 . The main result we prove is that when E and f are congruent mod a prime 𝔭 (which we take in this paper to be a prime of ¯ lying over a rational prime p > 2 ), the algebraic parts of the special values L ( E , χ , j ) and L ( f , χ , j ) satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, τ ( χ ¯ ) L ( f , χ , j ) ( 2 π i ) j - 1 Ω f sgn ( E ) τ ( χ ¯ ) L ( E , χ , j ) ( 2 π i ) j Ω E ( mod 𝔭 ) where the...

Modularity of a nonrigid Calabi-Yau manifold with bad reduction at 13

Grzegorz Kapustka, Michał Kapustka (2007)

Annales Polonici Mathematici

We identify the weight four newform of a modular Calabi-Yau manifold studied by Hulek and Verrill. The main obstacle is that this Calabi-Yau manifold is not rigid and has bad reduction at prime 13. Replacing the original fiber product of elliptic fibrations with a fiberwise Kummer construction we reduce the problem to studying the modularity of a rigid Calabi-Yau manifold with good reduction at primes p ≥ 5.

Modularity of an odd icosahedral representation

Arnaud Jehanne, Michael Müller (2000)

Journal de théorie des nombres de Bordeaux

In this paper, we prove that the representation ρ from G in GL 2 ( ) with image A 5 in PGL 2 ( A 5 ) corresponding to the example 16 in [B-K] is modular. This representation has conductor 5203 and determinant χ - 43 ; its modularity was not yet proved, since this representation does not satisfy the hypothesis of the theorems of [B-D-SB-T] and [Tay2].

Modularity of Galois representations

Chris Skinner (2003)

Journal de théorie des nombres de Bordeaux

This paper is essentially the text of the author’s lecture at the 2001 Journées Arithmétiques. It addresses the problem of identifying in Galois-theoretic terms those two-dimensional, p -adic Galois representations associated to holomorphic Hilbert modular newforms.

Modularity of p -adic Galois representations via p -adic approximations

Chandrashekhar Khare (2004)

Journal de Théorie des Nombres de Bordeaux

In this short note we give a new approach to proving modularity of p -adic Galois representations using a method of p -adic approximations. This recovers some of the well-known results of Wiles and Taylor in many, but not all, cases. A feature of the new approach is that it works directly with the p -adic Galois representation whose modularity is sought to be established. The three main ingredients are a Galois cohomology technique of Ramakrishna, a level raising result due to Ribet, Diamond, Taylor,...

Motives over totally real fields and p -adic L -functions

Alexei A. Panchishkin (1994)

Annales de l'institut Fourier

Special values of certain L functions of the type L ( M , s ) are studied where M is a motive over a totally real field F with coefficients in another field T , and L ( M , s ) = 𝔭 L 𝔭 ( M , 𝒩 𝔭 - s ) is an Euler product 𝔭 running through maximal ideals of the maximal order 𝒪 F of F and L 𝔭 ( M , X ) - 1 = ( 1 - α ( 1 ) ( 𝔭 ) X ) · ( 1 - α ( 2 ) ( 𝔭 ) X ) · ... · ( 1 - α ( d ) ( 𝔭 ) X ) = 1 + A 1 ( 𝔭 ) X + ... + A d ( 𝔭 ) X d being a polynomial with coefficients in T . Using the Newton and the Hodge polygons of M one formulate a conjectural criterium for the existence of a p -adic analytic continuation of the special values. This conjecture is verified in a number of cases related to...

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