[Paper Review] An explicit formula for the Euler product of Hecke polynomials

Xian‐Jin Li|arXiv (Cornell University)|Mar 9, 2004

Analytic Number Theory Research3 citations

TL;DR

This paper derives an explicit formula for the Euler product of Hecke polynomials using the Eichler–Selberg trace formula, enabling a criterion to locate nontrivial zeros of Hecke L-functions attached to weight 2 cusp forms. The key contribution is a number-theoretic criterion linking trace formula computations to zero-location analysis in families of modular L-functions.

ABSTRACT

Abstract. An explicit formula is given for a sequence of numbers by using the Eichler-Selberg trace formula. A criterion is obtained by using these numbers for the location of nontrivial zeros of Hecke L-functions, which are associated with a family of cusp forms of weight 2. 1.

Motivation & Objective

• To derive an explicit formula for the Euler product of Hecke polynomials using trace formula techniques.
• To establish a number-theoretic criterion for locating nontrivial zeros of Hecke L-functions associated with weight 2 cusp forms.
• To connect the Eichler–Selberg trace formula to the analytic behavior of L-functions in families of modular forms.
• To provide a computational framework for studying zero distribution in Hecke L-functions through trace-derived invariants.

Proposed method

• The Eichler–Selberg trace formula is applied to compute trace data for Hecke operators on spaces of cusp forms of weight 2.
• The trace data are used to construct a sequence of numbers that encode arithmetic information about the Hecke polynomials.
• These numbers are then used to derive an explicit formula for the Euler product of the Hecke polynomials.
• A criterion for the location of nontrivial zeros of Hecke L-functions is formulated based on the sign and magnitude of these computed numbers.
• The method links spectral data from the trace formula to analytic properties of L-functions via algebraic number theory.

Experimental results

Research questions

• RQ1 How can the Eichler–Selberg trace formula be used to generate an explicit formula for the Euler product of Hecke polynomials?
• RQ2 What arithmetic invariants derived from trace computations can serve as criteria for locating nontrivial zeros of Hecke L-functions?
• RQ3 In what way do the computed numbers from the trace formula reflect the zero distribution of weight 2 cusp form L-functions?
• RQ4 Can the trace formula’s output be systematically transformed into a criterion for nontrivial zero location in families of modular L-functions?

Key findings

• An explicit formula is derived for the Euler product of Hecke polynomials using the Eichler–Selberg trace formula.
• A sequence of numbers is computed via the trace formula, which encodes essential arithmetic data of the Hecke operators.
• These numbers are used to formulate a criterion for the location of nontrivial zeros of Hecke L-functions associated with weight 2 cusp forms.
• The criterion provides a number-theoretic condition based on trace-derived invariants to predict the location of nontrivial zeros.