Cup products and L-values of cusp forms

TL;DR

This paper explores a conjecture linking cup product pairings of cyclotomic p-units to L-values of newforms, highlighting deep connections between algebraic and analytic number theory in the context of cyclotomic fields.

Contribution

It proposes a new conjecture relating cup product pairings in cyclotomic fields to special L-values of modular forms, bridging algebraic and analytic aspects.

Findings

Formulates a conjecture connecting cup products and L-values

Highlights the role of modulo p congruences with Eisenstein series

Suggests potential pathways for future proofs or computational verification

Abstract

In this note, we describe a conjecture, that, for an odd prime p, relates special values of a cup product pairing on cyclotomic p-units in the pth cyclotomic field to the L-values of newforms satisfying modulo p congruences with Eisenstein series.