Families of modular forms and their attached Galois representations are of fundamental importance in many arithmetic questions. The authors introduce a new kind of families: for a given prime \(p\) they introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over \(\mathbb Q\). Such a family can be taken to consist of classical holomorphic Hecke eigenforms or can be chosen to be made of Katz modular Hecke eigenforms defined over \(\overline{\mathbb F}_p\).
“Galois families are fundamentally different from other kinds of families of modular forms, such as Hida families. On the one hand, if we took the classical members of a Hida family, the field cut out by the mod \(p\) Galois representation would be the same in all cases. On the other hand, Galois families do not see \(p\)-adic deformations, so they miss the interesting information in Hida families. Galois families are rooted in field arithmetic and we see our paper as a step towards strengthening connections between field arithmetic and the automorphic theory.”
“We exhibit some examples and provide an infinite Galois family of non-liftable weight one Katz modular eigenforms over \(\overline{\mathbb F}_p\) for \(p \in \{3,5,7,11\}\).”