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Cubic twin prime polynomials are counted by a modular form. (arXiv:1711.05564v1 [math.NT])
来源于:arXiv
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 $q \equiv 1 \pmod 3$. 查看全文>>