An algebraic property of Hecke operators and two indefinite theta series

TL;DR

This paper proves an algebraic property of Hecke operators on period polynomials, showing Hecke equivariance of the Petersson scalar product, and derives two indefinite theta series identities related to classical theta formulas.

Contribution

It introduces a new algebraic property of Hecke operators on period polynomials and connects it to indefinite theta series identities, extending classical theta function results.

Findings

Hecke operators are shown to have an algebraic property on period polynomials.

The Petersson scalar product pairing is Hecke equivariant.

Two indefinite theta series identities are derived as analogues of Jacobi's formula.

Abstract

We prove an algebraic property of the elements defining Hecke operators on period polynomials associated with modular forms, which implies that the pairing on period polynomials corresponding to the Petersson scalar product of modular forms is Hecke equivariant. As a consequence of this proof, we derive two indefinite theta series identities which can be seen as analogues of Jacobi's formula for the theta series associated with the sum of four squares.