Cubic twin prime polynomials are counted by a modular form

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 $q\equiv 2(\mathrm{mod}3)$ but shows anomalies for $q\equiv 1(\mathrm{mod}3)$.

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References in corpus (3)

• The variance of the number of prime polynomials in short intervals and in residue classes
• Higher moments of arithmetic functions in short intervals: a geometric perspective
• Bounded gaps between primes in number fields and function fields

Cited by in corpus (1)

• Correlation of arithmetic functions over ${\mathbb{F}}_q[T]$