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Twin Prime Questions For Elliptic Curves

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posted on 01.08.2021, 00:00 authored by McKinley T Meyer
For an elliptic curve $E$ defined over $\Q$ and for a rational prime $p$ of good reduction, one can define an integer $a_p$ related to the number of $\F_p$-points lying on the reduction of $E$ modulo $p$ as $a_p = p + 1 - \#E(\F_p)$. The integer $a_p$, called the Frobenius trace of $E$ modulo $p$, lies in the interval $(- 2 \sqrt{p}, 2 \sqrt{p})$ and has several other remarkable properties. In this thesis, we study the arithmetic properties of $a_p$, specifically how often $a_p$ is prime. Using heuristical reasoning similar to that used in formulating the Hardy-Littlewood Conjecture regarding the number of twin primes up to a bound $x$, it is natural to formulate a conjecture for the asymptotic growth of the number of primes $p \leq x$ for which $a_p$ is also prime. As evidence in support of this conjecture, we prove two main results, each in the case when $E$ is without complex multiplication and under the $\theta$-quasi Generalized Riemann Hypothesis. First, we establish an upper bound for the number of primes $p \leq x$ for which $a_p$ is prime; this bound has the correct order of magnitude, as predicted by the aforementioned conjecture. Then we prove a lower bound, also with the correct order of magnitude, for the number of primes $p \leq x$ such that $a_p$ is ``almost" prime, in the sense of having at most a certain fixed number of prime factors, distinct or indistinct.

History

Advisor

Cojocaru, Alina Carmen

Chair

Cojocaru, Alina Carmen

Department

Mathematics, Statistics, and Computer Science

Degree Grantor

University of Illinois at Chicago

Degree Level

Doctoral

Degree name

PhD, Doctor of Philosophy

Committee Member

Jones, Nathan Mubayi, Dhruv Pollack, Paul Takloo-Bighash, Ramin

Submitted date

August 2021

Thesis type

application/pdf

Language

en

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