Abstract

We show that a certain family of the coefficients of a Drinfeld-Goss modular form with certain power eigenvalues for the Hecke operators at degree 1 primes the can be expressed as polynomial multiples of the first possible non-zero coefficient of that form. Along the way, we obtain some interesting combinatorial properties regarding difference operators in finite characteristic.

Highlights

  • We present a result in the same vein of Theorem 2.1 which applies to eigenforms in Mk2,m for any type m and any weight k

  • If a1+l = 0, the theorem asserts that certain coefficients of the normalized form f /al+1 are given as polynomials depending on the eigenvalues, which can be viewed a function field analogue to how the p-th coefficient of a normalized eigenform is itself the eigenvalue of the p-th Hecke operator in the classical case

  • We prove the formula by induction on n

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Summary

Introduction

If a1+l = 0, the theorem asserts that certain coefficients of the normalized form f /al+1 are given as polynomials depending on the eigenvalues, which can be viewed a function field analogue to how the p-th coefficient of a normalized eigenform is itself the eigenvalue of the p-th Hecke operator in the classical case. Theorem 2.3 implies that a1+3i (F) = a1+3i (f ) for any type 0 double-cuspidal normalized form F satisfying Tθ (F) = θ 4F, regardless of its weight. This can be verified directly for F := h2g6 + (θ 3 − θ )h4g2 ∈ M220,0.

It follows that the minimum value of X is
Hence the maximum value of X is
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