On the non-existence of simple congruences for quotients of Eisenstein series
We show that the slopes of the operator acting on 5-adic overconvergent modular forms of weight with primitive Dirichlet character of conductor 25 are given by eitherdepending on and .We also prove that the space of classical cusp forms of weight and character has a basis of eigenforms for the Hecke operators and which is defined over .
The purpose of this work is to carry out the first step in our four-step program towards the main conjecture for by the method of Eisenstein congruence on , where is an imaginary quadratic field. We construct a -adic family of ordinary Eisenstein series on the group of unitary similitudes with the optimal constant term which is basically the product of the Kubota-Leopodlt -adic -function and a -adic -function for . This construction also provides a different point of view of -adic...
We give a geometric definition of overconvergent modular forms of any -adic weight. As an application, we reprove Coleman’s theory of -adic families of modular forms and reconstruct the eigencurve of Coleman and Mazur without using the Eisenstein family.
We study the critical values of the complex standard--function attached to a holomorphic Siegel modular form and of the twists of the -function by Dirichlet characters. Our main object is for a fixed rational prime number to interpolate -adically the essentially algebraic critical -values as the Dirichlet character varies thus providing a systematic control of denominators of critical values by generalized Kummer congruences. In order to organize this information we prove the existence of...
We study the -adic nearly ordinary Hecke algebra for cohomological modular forms on over an arbitrary number field . We prove the control theorem and the independence of the Hecke algebra from the weight. Thus the Hecke algebra is finite over the Iwasawa algebra of the maximal split torus and behaves well under specialization with respect to weight and -power level. This shows the existence and the uniqueness of the (nearly ordinary) -adic analytic family of cohomological Hecke eigenforms...