On \(p\)-adic \(L\)-functions for Hilbert modular forms (Q6605396)

The goal of this book is to define canonical \(p\)-adic \(L\)-functions associated with \(p\)-refined cohomological cuspidal automorphic representations of \(\mathrm{GL}_2\) over totally real number fields under a mild hypothesis. The construction is canonical, varies naturally in \(p\)-adic families, and does not require any small slope or non-criticality assumptions on the \(p\)-refinement.\N\NThe book consists of the Introduction, seven main chapters, and two appendices.\N\NThe chapters 2, 3, and 4 are comprised of a verbose discussion of adelic (co)chains on locally symmetric spaces (cohomology of local systems on symmetric spaces which arise in the context of Hilbert modular forms), Hilbert modular forms (recollection of definitions and basic constructions), and Shimura's algebraicity theorem.\N\NIn chapter 5, the authors discuss generalities on certain \(p\)-adic Lie groups and define various modules of locally analytic functions and distributions.\N\NIn (the most technical) chapter 6, the authors define a certain eigenvariety of time level \(\mathfrak n\) (assuming that \(\mathfrak n\) is an integral ideal that is co-prime to \(p\)), and show that reasonable classical points are smooth. Such eigenvarieties are families, parametrized by weights, of systems of Hecke eigenvalues that generalize \(p\)-adic systems of eigenvalues appearing in classical spaces of automorphic forms.\N\NIn chapter 7, the authors introduce and analyze the period maps. The main result here (Theorem 7.23) is an ''abstract'' equality of functionals on a certain overconvergent cohomology group, which relates the Hecke action at \(p\) to the \(p\)-adic evaluation classes via the period maps. This is the key ingredient in proving the correct interpolation formula for \(p\)-adic \(L\)-functions.\N\NIn chapter 8, the authors prove the main results (Theorems 1.2 and 1.7 from the Introduction): they construct canonical \(p\)-adic \(L\)-functions associated with \(p\)-refined cohomological cuspidal automorphic representations of \(\mathrm{GL}_2\) (over totally real number fields, and under a mild hypothesis).\N\NThis book, addressed to graduate students and experts working in number theory and arithmetic geometry, is a welcome addition to this beautiful and difficult area of research. The mathematics discussed here is wonderful, but prerequisites for the reader are rather high.