El problema del nombre de classes 1

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Artur Travesa i Grau[en] The ring of integers is a unique factorization domain, but, in general, this isn’t the case for the ring of integers of a number field. The class number 1 problem consists in giving a complete list of all imaginary quadratic fields whose ring of integers is a unique factorization domain. In this thesis we provide an adaptation of Kurt Heegner’s original solution including an overview of the required theoretical tools, namely class field theory and the theory of elliptic curves with complex multiplication

p-arithmetic cohomology and p-adic automorphic forms

The cohomology of an arithmetic group with coefficients in finite-dimensional representations can be described in terms of automorphic representations of the group. In this thesis, we prove similar results for the cohomology of an *S*-arithmetic groups (where *S* is a finite set of primes) with coefficients in different types of representations. For example, we show that the cohomology of (duals of) locally algebraic representations of the local groups at places in *S* can be described in terms of automorphic representations satisfying certain conditions determined by the locally algebraic representation. We show that the cohomology with coefficients in (duals of) locally analytic representations can be used to define *p*-adic automorphic forms and families of them (eigenvarieties). In particular, we are able to give constructions of these objects in many new cases, such as when the reductive group is not quasi-split at *p*. We also prove that these constructions are equivalent, in the cases where they are defined, to those obtained using overconvergent cohomology and to the Bernstein eigenvarieties constructed by Breuil-Ding.Cambridge Trust DPMMS, University of Cambridg