Rank--two Euler systems for symmetric squares

TL;DR

This paper constructs a rank-two Euler system for the symmetric square motive of a modular form with specific properties, linking its non-triviality to special values of L-functions and p-adic periods.

Contribution

It introduces a novel rank-two Euler system for symmetric square motives, connecting algebraic and analytic aspects of modular forms.

Findings

Constructed a rank-two Euler system for symmetric square motives.

Established non-triviality conditions based on L-value non-vanishing.

Linked p-adic period non-vanishing to the Euler system's non-triviality.

Abstract

Let $p\ge 7$ be a prime number and $f$ a normalized eigen-newform with good reduction at $p$ such that its $p$-th Fourier coefficient vanishes. We construct a rank-two Euler system attached to the $p$-adic realization of the symmetric square motive of $f$. Furthermore, we show that the non-triviality is guaranteed by the non-vanishing of the leading term of the relevant $L$-value and the non-vanishing of a certain $p$-adic period modulo $p$.