Cubic Twin Prime Polynomials are Counted by a Modular Form

Bary-Soroker, Lior ; Stix, Jakob

Canadian journal of mathematics, Tome 71 (2019) no. 6, pp. 1323-1350

Résumé

We present the geometry 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 $q\equiv 2$ (mod 3), but shows anomalies for $q\equiv 1$ (mod 3).

Détail

DOI : 10.4153/CJM-2018-018-9

Mots-clés : twin primes, finite field, polynomial

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