Cubic twin prime polynomials are counted by a modular form

TL;DR

This paper links the counting of cubic twin prime polynomials over finite fields to modular forms through geometric and cohomological methods, revealing connections to elliptic curves and Hardy-Littlewood heuristics.

Contribution

It introduces a geometric approach using cohomology and trace formulas to count twin prime polynomials, connecting the problem to modular forms and elliptic curves.

Findings

Counting formulas align with Hardy-Littlewood heuristic for certain q

Results agree with predictions for q ≡ 2 mod 3

Anomalies observed for q ≡ 1 mod 3

Abstract

We present the geometry lying behind counting twin prime polynomials in $\mathbb{F}_q[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^3 = Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder. The formula we get in degree $3$ is compatible with the Hardy-Littlewood heuristic on average, agrees with the prediction for $q \equiv 2 \pmod 3$ but shows anomalies for $q \equiv 1 \pmod 3$.