Effective computations for weakly optimal subvarieties

TL;DR

This paper refines the understanding of weakly optimal subvarieties in Shimura varieties by constructing finite algebraic families, providing effective degree bounds, and developing an explicit procedure to identify the weakly optimal locus.

Contribution

It introduces a finite collection of algebraic families for weakly optimal subvarieties, offers effective degree bounds, and presents a method to compute the weakly optimal locus.

Findings

Finite algebraic families of weakly optimal subvarieties constructed.

Effective degree bounds established for the locus and its members.

An explicit procedure to determine the weakly optimal locus developed.

Abstract

Ren and the second author established that the weakly optimal subvarieties (e.g. maximal weakly special subvarieties) of a subvariety $V$ of a Shimura variety arise in finitely many families. In this article, we refine this theorem by (1) constructing a finite collection of algebraic families whose fibers are precisely the weakly optimal subvarieties of $V$; (2) obtaining effective degree bounds on the weakly optimal locus and its individual members; (3) describing an effective procedure to determine the weakly optimal locus.