Speaker : Jun-Wen Pen (NCTS)
Title : Toward dynamical Sato-Tate conjecture
Time : 2021-10-15 (Fri) 13:30 - 14:45
Place : Seminar Room 617, Institute of Mathematics (NTU Campus)
Abstract:
The Sato-Tate conjecture is a statistic statement about the family of elliptic curves $E_p$ over the finite field with $p$ elements, with $p$ a prime number, obtained from an elliptic curve $E$ over the rational field, by the process of reduction modulo a prime for all most all $p$. Many problems in arithmetic dynamics generalize questions and results from abelian varieties. With this motivation, we desired to formulate a statistic statement about the family of the dynamical system $f_p:\mathbb{F_p}\to\mathbb{F_p}$ obtained from a rational map $f$ over the rational field, by the process of reduction modulo a prime for all most all $p$. In this talk, we will present data of the number of periodic points in $\mathbb{F}_p$ under $f_p$ and the cardinality of the image set $f_p(\mathbb{F}_p)$. When $f$ is a monomial or a Chebyshev polynomial, we can apply the prime number theorem to state the number of periodic points. However, it isn't easy to give a complete statistic statement for generic polynomials. I will provide a possible direction of proving a statistic statement.

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