You are cordially invited to the Number Theory Seminar organized by the Department of Mathematics.<\/p>\n
Speaker: Sung Min Lee (DAKOTA STATE UNIVERSITY)<\/p>\n
\u201cOn Average Congruence Class Biases for Cyclicity and Koblitz problem\u201d<\/p>\n
Abstract: For an elliptic curve $E\/\\mathbb{Q}$ and a good prime $p$, $\\widetilde{E}_p$ denotes the reduction of $E$ modulo $p$. We call $p$ a prime of cyclic reduction if $\\widetilde{E}_p(\\mathbb{F}_p)$ is cyclic, and of Koblitz reduction if the group has a prime order. Determining the asymptotics of the counting functions for these primes are known as cyclicity problem and Koblitz problem, respectively. The former was proven by J.-P. Serre under the Generalized Riemann Hypothesis, while the latter remains open. Recent work has proven both problems unconditionally \u201con average,\u201d and Jones, assuming an affirmative answer to Serre\u2019s uniformity question, showed that the average of conjectured asymptotics align with the average results. In this talk, we first propose Koblitz problem for primes in arithmetic progressions, building upon Zywina\u2019s argument. Then, adapting Jones\u2019s methods, we refine his results by removing the assumption and restricting primes to lie in arithmetic progressions. Additionally, we present a rather counterintuitive observation that primes of cyclic reduction and Koblitz reduction are statistically oppositely biased over congruence classes, despite the fact that primes of Koblitz reduction must also be primes of cyclic reduction. This work is a joint research with Jacob Mayle and Tian Wang.<\/p>\n
Date: Thursday, October 24, 2024
\nTime: 18:00
\nPlace: ZOOM<\/p>\n
To request the event link, please send a message to\u00a0guloglua@fen.bilkent.edu.tr<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" You are cordially invited to the Number Theory Seminar organized by the Department of Mathematics. Speaker: Sung Min Lee (DAKOTA STATE UNIVERSITY) \u201cOn Average Congruence Class Biases for Cyclicity and Koblitz problem\u201d Abstract: For an elliptic curve $E\/\\mathbb{Q}$ and a good prime $p$, $\\widetilde{E}_p$ denotes the reduction of $E$ modulo [\u2026] <\/p>\n