Reductions of Galois representations and the theta operator (original) (raw)
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2016
Let F be a totally real field and χ an abelian totally odd character of F. In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt's conjecture holds, assuming that either there are at least two primes above p in F , or that a certain condition relating the L-invariants of χ and χ −1 holds. This condition on L-invariants is always satisfied when χ is quadratic. Contents S. DASGUPTA, H. DARMON, and R. POLLACK 4. Galois representations 477 4.1. Representations attached to ordinary eigenforms 477 4.2. Construction of a cocycle 480 References 482
Modular forms of weight one: Galois representations and dimension
2009
The present notes are the expanded and polished version of three lectures given in Stanford, concerning the analytic and arithmetic properties of weight one modular forms. The author tried to write them in a style accessible to non-analytically oriented number theoritists: in particular, some effort is made to be precise on statements involving uniformity in the parameters. On the other hand, another purpose was to provide an introduction, together with a set of references, consciously kept small, to the realm of Galois representations, for non-algebraists -- like the author. The proofs are sketched, at best, but we tried to motivate the results, and to relate them to interesting conjectures.
20 06 Irreducibility and cuspidality
2006
Irreducible representations are the building blocks of general, semisimple Galois representations ρ, and cuspidal representations are the building blocks of automorphic forms π of the general linear group. It is expected that when an object of the former type is associated to one of the latter type, usually in terms of an identity of L-functions, the irreducibility of the former should imply the cuspidality of the latter, and vice-versa. It is not a simple matter at all to prove this expectation, and nothing much is known in dimensions > 2. We will start from the beginning and explain the problem below, and indicate a result (in one direction) at the end of the Introduction, which summarizes what one can do at this point. The remainder of the paper will be devoted to showing how to deduce this result by a synthesis of known theorems and some new ideas. We will be concerned here only with the so called easier direction of showing the cuspidality of π given the irreducibility of ρ,...
On congruences for the coefficients of modular forms and some applications
1997
We start with a brief overview of the necessary theory: Given any cusp form f=∑ n≥ 1 an (f) qn of weight k, we denote by L (f, s) the L-function of f. For Re (s)> k/2+ 1, the value of L (f, s) is given by L (f, s)=∑ n≥ 1 an (f) ns and, one can show that L (f, s) has analytic continuation to the entire complex plane. The value of L (f, s) at s= k/2 will be of particular interest to us, and we will refer to this value as the central critical value of L (f, s).