Monomial Quadratic Identities of Hecke Eigenforms

Author

• Trevor Vilardi, Clemson UniversityFollow

Date of Award

5-2026

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Committee Chair/Advisor

Hui Xue

Committee Member

Michael Burr

Committee Member

James Coykendall

Committee Member

Robert Dicks

Abstract

Duke and Ghate independently studied the question of when it is possible for the product of two eigenforms to be an eigenform. In this dissertation, we take up a generalization of that question, namely is it possible for the product of two eigenforms to be equal to a different product of two eigenforms? Under this formulation, the question becomes closer to one about unique factorization, i.e., how closely do eigenforms work like irreducible elements? Our conjecture is that there are only finitely many cases where the product of two eigenforms is equal to a different product of two eigenforms, and these cases are all classically known and forced to occur due to either M_k or S_k being one-dimensional.

This problem naturally splits into three cases.

1. When all eigenforms are Eisenstein series. 
2. When there is an Eisenstein series and a cuspidal eigenform on each side of the equation. 
3. When all eigenforms are cuspidal.

The first case has been completely solved. In this dissertation, we solve the second case under the assumption of Maeda's Conjecture, and partially solve the third case with a hypothesis similar to Maeda's Conjecture.

Recommended Citation

Vilardi, Trevor, "Monomial Quadratic Identities of Hecke Eigenforms" (2026). All Dissertations. 4229.
https://open.clemson.edu/all_dissertations/4229