Joseph H. Silverman: Field of Moduli and Fields of Definition in Arithmetic [...] (NTWS 187)
Channel: Number Theory Web Seminar
Date: 2023-11-02

Title: Field of Moduli and Fields of Definition in Arithmetic Geometry and Arithmetic Dynamics

Abstract: Let X/Qbar be an algebraic variety defined over the field of algebraic numbers. We say that a number field K is a field of definition (FOD) for X if there is a variety Y/K such that Y is Qbar-isomorphic to X.
The field of moduli (FOM) of X is the fixed field of

          { s \in G_Q : s(X) is Qbar-isomorphic to X }.

It is easy to check that every FOD for X contains the FOM of X, but there are many situations where the FOM of X is not a FOD. I will briefly discuss the FOM versus FOD problem in the classical case of abelian varieties, and then turn to the the analogous question for morphisms f : P^N \rightarrow P^N defined over Qbar, where two maps are (dynamically) isomorphic if they are conjugate by a linear fractional transformation. I will describe what is known for N=1, including examples of maps for which the FOM is not an FOD. I will then discuss recent results for higher dimensional projective spaces in which we show that every map f has a FOD whose degree over its FOM is bounded by a function depending only on N and deg(f).  (Joint work with John Doyle.)

Link to slides: https://drive.google.com/file/d/1Ej4YPLZXs9x68XkhAtGWmOYbJCD-Ar7r

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, November 2, 2023 (8am PDT, 11am EDT, 3pm GMT, 4pm CET, 5pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)Friday, November 3, 2023 (2am AEDT, 4am NZDT)