Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms.
Gauss–Manin connections for -adic families of nearly overconvergent modular forms
We interpolate the Gauss–Manin connection in -adic families of nearly overconvergent modular forms. This gives a family of Maass–Shimura type differential operators from the space of nearly overconvergent modular forms of type to the space of nearly overconvergent modular forms of type with -adic weight shifted by . Our construction is purely geometric, using Andreatta–Iovita–Stevens and Pilloni’s geometric construction of eigencurves, and should thus generalize to higher rank groups.
Gauss's ₂F₁ hypergeometric function and the congruent number elliptic curve
Geometric and -adic Modular Forms of Half-Integral Weight
In this paper we introduce a geometric formalism for studying modular forms of half-integral weight. We then use this formalism to define -adic modular forms of half-integral weight and to construct -adic Hecke operators.
Hasse invariant and group cohomology.
Hida families, -adic heights, and derivatives
This paper concerns the arithmetic of certain -adic families of elliptic modular forms. We relate, using a formula of Rubin, some Iwasawa-theoretic aspects of the three items in the title of this paper. In particular, we examine several conjectures, three of which assert the non-triviality of an Euler system, a -adic regulator, and the derivative of a -adic -function. We investigate sufficient conditions for the first conjecture to hold and show that, under additional assumptions, the first...
Higher congruence companion forms
Induced representations of GL (n, A) for p-adic division algebras A.
Iwasawa modules attached to congruences of cusp forms
Kummer's Criterion for the Special Values of Hecke L-Functions of Imaginary Quadratic Fields and Congruences Among Cusp Forms.
La filtration canonique des points de torsion des groupes -divisibles
Étant donnés un entier et un groupe de Barsotti-Tate tronqué d’échelon et de dimension sur un anneau de valuation d’inégales caractéristiques, nous donnons une borne explicite sur son invariant de Hasse qui implique que sa filtration de Harder-Narasimhan possède un sous-groupe libre de rang . Lorsque nous redémontrons également le théorème d’Abbes-Mokrane ([120]) et de Tian ([164]) par des méthodes locales. On applique cela aux familles -adiques de tels objets et en particulier à certaines...
l-adic representations associated to modular forms over imaginary quadratic fields. II.
Mod p Hecke Operators and Congruences Between Modular Forms.
Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences.
Modular representations of PGL2 and automorphic forms for Shimura curves.
Modular symbols, Eisenstein series, and congruences
Let and be an Eisenstein series and a cusp form, respectively, of the same weight and of the same level , both eigenfunctions of the Hecke operators, and both normalized so that . The main result we prove is that when and are congruent mod a prime (which we take in this paper to be a prime of lying over a rational prime ), the algebraic parts of the special values and satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions,where the...
Modularity of fibres in rigid local systems.
Modularity of -adic Galois representations via -adic approximations
In this short note we give a new approach to proving modularity of -adic Galois representations using a method of -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 -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,...
Nearly ordinary deformations of irreducible residual representations
Non-optimal levels of mod l modular representations.