On the non-triviality of arithmetic invariants and its applications

One fruitful approach in algebraic number theory and arithmetic geometry is to study arithmetic invariants varying over families. In Iwasawa theory, one obtains p-adic families, for example, by varying number fields (as in classical Iwasawa theory) or by considering p-adic families of modular forms (as in Hida theory). The fact that studying arithmetic invariants over families is an important perspective in number theory is highlighted by Galois deformation theory, which played a crucial role in Fermat’s last theorem. One of the central themes of this program, is whether interesting arithmetic invariants (such as Iwasawa mu-invariant) associated to L-functions are generically non-trivial over families, which are often encoded by whether analytic invariants (such as special values of L-functions) are generically non-trivial modulo p. There have been numerous developments on the non-triviality of these arithmetic/analytic invariants, which have had various arithmetic applications (for instance, towards proving Iwasawa main conjectures). It is time to bring experts together to explore new ideas for the non-vanishing of L-values. Therefore, we plan to devote the first week of the program to a rigorous exposition of the ideas in the pioneering works, and the second week of the program will be devoted to the recent developments by some of the leading experts in the area.