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Cubic Twin Prime Polynomials are Counted by a Modular Form

Part of: Finite fields and commutative rings (number-theoretic aspects) Arithmetic algebraic geometry

Published online by Cambridge University Press:  09 January 2019

Lior Bary-Soroker  and
Jakob Stix
Lior Bary-Soroker
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 6997801 Tel Aviv, Israel Email: [email protected]
Jakob Stix
Affiliation:
Institut für Mathematik, Goethe–Universität Frankfurt, Robert-Mayer-Straße 6–8, 60325 Frankfurt am Main, Germany Email: [email protected]
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Abstract

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).

Keywords

twin primesfinite fieldpolynomial

MSC classification

Primary: 11T55: Arithmetic theory of polynomial rings over finite fields
Secondary: 11G25: Varieties over finite and local fields
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 6 , December 2019 , pp. 1323 - 1350
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

The authors acknowledge support provided by DAAD-Programm 57271540 Strategische Partnerschaften (supported by BMBF). The first author was partially supported by a grant from the Israel Science Foundation.

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