Geometric modularity for algebraic and non-algebraic weights

TL;DR

This paper explores the relationship between geometric modularity of algebraic and non-algebraic weights, establishing new implications that connect previous conjectures and generalizations in number theory.

Contribution

It generalizes a result linking geometric modularity of algebraic and non-algebraic weights, revealing new implications and connections in the theory of modular forms.

Findings

Geometric modularity of non-algebraic weights implies modularity of multiple algebraic weights.

Modularity of multiple algebraic weights sometimes implies modularity of a non-algebraic weight.

Connects work of Diamond--Sasaki to generalizations of Serre's conjecture.

Abstract

In this short paper we generalise a result of Diamond--Sasaki connecting geometric modularity of algebraic weights to geometric modularity of non-algebraic weights and vice versa. In particular, we show that geometric modularity of non-algebraic weights implies geometric modularity of multiple algebraic weights, which is easy to see. More significantly, we show that geometric modularity of multiple algebraic weights sometimes implies geometric modularity of a non-algebraic weight. This connects work of Diamond--Sasaki to earlier work on generalisations of the weight part of Serre's conjecture.