First moments of Rankin-Selberg convolution of Automorphic Forms on $GL(2)$

TL;DR

This paper derives a first moment formula for Rankin-Selberg convolutions of automorphic forms on $GL(2)$, providing new uniform bounds and a novel approach to error term estimation, with implications for form equality criteria.

Contribution

It introduces a new method for handling error terms in first moment formulas, enabling optimal estimates and paving the way for second moment analysis.

Findings

Established a first moment formula for Rankin-Selberg convolutions on $GL(2)$

Achieved the best known uniform bounds for form equality conditions

Developed a novel error term treatment method

Abstract

We obtain a first moment formula for Rankin-Selberg convolution $L$-series of holomorphic modular forms or Maass forms of arbitrary level on $GL(2)$, with an orthonormal basis of Maass forms. One consequence is the best result to date, uniform in level, spectral value and weight, for the equality of two Maass or holomorphic cusp forms if their Rankin-Selberg convolutions with the orthonormal basis of Mass forms $u_j$ is equal at the center of the critical strip for sufficiently many $u_j$. The main novelty of our approach is the new way the error terms are treated. They are brought into an exact form that provides optimal estimates for the first moment case, and also provide a basis for an extension to second moments, which will appear in another work.