David Loeffler

Overconvergent algebraic automorphic forms

A general theory of overconvergent p-adic modular forms and eigenvarieties is presented for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions due to Buzzard, Chenevier and Yamagami for forms of GLn. This leads to some new phenomena, including the appearance of intermediate spaces of ‘semi-classical’ automorphic forms; this gives a hierarchy of interpolation spaces (eigenvarieties) interpolating classical automorphic forms satisfying different finite slope conditions (corresponding to a choice of parabolic subgroup of G at p). The construction of these spaces relies on methods of locally analytic representation theory, combined with the theory of compact operators on Banach modules.

Euler systems for Rankin-Selberg convolutions of modular forms

We construct a Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use these elements to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin–Selberg L-function is nonvanishing at s=1.

On the computation of local components of a newform

The problem. Let f be a cuspidal newform for Γ1(N) with weight k ≥ 2 and character e. There are well-established methods for computing such forms using modular symbols; see [Ste07]. Let πf be the corresponding automorphic representation of the adele group GL2(AQ).

Rankin–Eisenstein classes and explicit reciprocity laws

We construct three-variable

P-Adic Integration on Ray Class Groups and Non-ordinary p-Adic L -Functions

p

Images of adelic Galois representations for modular forms

-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated

Rankin–Eisenstein classes in Coleman families

L

Local epsilon isomorphisms

-value does not vanish.

Spectral Expansions of Overconvergent Modular Functions

We study the theory of p-adic finite-order functions and distributions on ray class groups of number fields, and apply this to the construction of (possibly unbounded) p-adic L-functions for automorphic forms on \(\mathop{\text{GL}}\nolimits _{2}\) which may be non-ordinary at the primes above p. As a consequence, we obtain a “plus-minus” decomposition of the p-adic L-functions of automorphic forms for \(\mathop{\text{GL}}\nolimits _{2}\) over an imaginary quadratic field with p split and Hecke eigenvalues 0 at the primes above p, confirming a conjecture of B.D. Kim.

Wach modules and critical slope p-adic L-functions

We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic subgroup of GL2 (defined by F. Momose). We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.