Endomorphism rings and algebras of Jacobians of certain superelliptic curves
Open Access
Author:
Eritsyan, Tigran
Graduate Program:
Mathematics (PHD)
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
August 17, 2022
Committee Members:
Mihran Papikian, Major Field Member Mark Strikman, Outside Unit & Field Member Yuriy Zarkhin, Chair & Dissertation Advisor John Lesieutre, Major Field Member Alexei Novikov, Professor in Charge/Director of Graduate Studies
Keywords:
Abelian variety Ree groups Superelliptic jacobians Trigonal curves Modular Representations Galois Groups
Abstract:
In this article we explore the endomorphism rings and algebras of Jacobians of trigonal curves of the form $y^3=f(x)$, where $f(x)$ is a separable polynomial with coefficients in $k$, irreducible over field $k$ of characteristic zero and with $Gal(f)$ isomorphic to one of the 2-transitive Ree groups in the series ${}^2G_2(q)$ with $q=3^{2m+1}$ for positive integers $m$. We show that the endomorphism algebras $End^0(J)$ of such Jacobians are simple and their centers are isomorphic to the number field $\QQ(\zeta_3)$, where $\zeta_3$ (the third root of unity) is the solution to to the polynomial equation over the ring of integers $x^2+x+1$. For $m\in ABS_m \cup COMP_m$ (see page 2), we determine that the endomorphism algebras of such Jacobians are absolutely simple and are isomorphic to the number field $\QQ(\zeta_3)$, while the rings of endomorphisms of such Jacobians are isomorphic to the ring $\ZZ[\zeta_3]$.
