On Mazur-Tate type conjectures for quadratic imaginary fields and elliptic curves

In this thesis, we formulate and partially prove conjectures à la MazurTate for two cases of L-functions. Suppose given a L-function L(M,K, s) attached to an arithmetic object M over a number field K (a “motive”) so that L can be twisted by characters of the Galois group G of an extension of K. To this pair (L,G), one may hope to associate a theta element Θ(L,G) ∈ A[G], where A is a well defined ring, that interpolates the special values L(M,K,χ, 1) of L twisted by characters χ of G. The notion of interpolation means that the evaluation of Θ(L,G) at χ gives the value L(M,K,χ, 1) or at least a simple and explicit transformation of the value L(M,K,χ, 1). Following the ideas of Mazur and Tate, the theta element should capture the arithmetic properties of L or more precisely the arithmetic properties of the geometric objects M and K from which L is constructed. The first chapter is devoted to the case of Artin L-functions associated to a quadratic imaginary field K and twisted by characters of the Galois groups of ray class field extensions K(m) of K. The theta element captures information about the class number formulas of the fields K(m). For the second chapter, the underlying geometric object is a pair (K,E) of a quadratic imaginary field K and an elliptic curve E defined over Q. The theta element should capture there information about the rank of the Mordell-Weil groups E(K). The key ingredient in the study of the theta elements in both cases is the existence of a set of cohomology classes in appropriate cohomology groups that satisfy local and global compatibilities. These systems, sometimes called Euler system or Kolyvagin system even if the literature is not unified on this appellation, arise from purely geometric considerations. In chapter 1, these classes, called in that case elliptic units are constructed by considering units in the ray class fields of K, whereas in chapter 2, they arise by considering the Heegner points over K in a Shimura curves associated to E and K. Each chapter has the form of an article and can be read independently from the other one.