On the symmetric square: unstable local transfer.
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 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...
We show that certain quadratic base change -functions for are non-negative at their center of symmetry.