Function Field Analogue of Shimura's Conjecture on Period Symbols

TL;DR

This paper introduces and studies Shimura's period symbols over function fields, proving an analogue of Shimura's conjecture on their algebraic independence, extending previous work on abelian t-modules with complex multiplication.

Contribution

It formulates and proves a function field analogue of Shimura's conjecture, establishing algebraic independence of period symbols for certain abelian t-modules.

Findings

Proved algebraic independence of period symbols in specified cases

Extended Yu's work on Hilbert-Blumenthal t-modules

Established fundamental properties of Shimura's period symbols over function fields

Abstract

In this paper we introduce the notion of Shimura's period symbols over function fields in positive characteristic and establish their fundamental properties. We further formulate and prove a function field analogue of Shimura's conjecture on the algebraic independence of period symbols. Our results enable us to verify the algebraic independence of the coordinates of any nonzero period vector of an abelian t-module with complex multiplication whose CM type is non-degenerate and defined over an algebraic function field. This is an extension of Yu's work on Hilbert-Blumenthal t-modules.