The number of monodromy representations of Abelian varieties of low p-rank

Brett Frankel

Research output: Contribution to journalArticlepeer-review

Abstract

Let Ag be an abelian variety of dimension g and p-rank λ≤1 over an algebraically closed field of characteristic p>0. We compute the number of homomorphisms from π1 ét(Ag,a) to GLn(Fq), where q is any power of p. We show that for fixed g, λ, and n, the number of such representations is polynomial in q, and give an explicit formula for this polynomial. We show that the set of such homomorphisms forms a constructible set, and use the geometry of this space to deduce information about the coefficients and degree of the polynomial. In the last section we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups. As a corollary, we deduce that when λ=0, [Formula presented] is a Laurent polynomial in q.

Original languageEnglish (US)
Pages (from-to)393-412
Number of pages20
JournalJournal of Algebra
Volume510
DOIs
StatePublished - Sep 15 2018

Keywords

  • Abelian varieties
  • Local systems
  • Profinite groups

ASJC Scopus subject areas

  • Algebra and Number Theory

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