## Abstract

Let A_{g} 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 π_{1} ^{ét}(A_{g},a) to GL_{n}(F_{q}), 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 language | English (US) |
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Pages (from-to) | 393-412 |

Number of pages | 20 |

Journal | Journal of Algebra |

Volume | 510 |

DOIs | |

State | Published - Sep 15 2018 |

## Keywords

- Abelian varieties
- Local systems
- Profinite groups

## ASJC Scopus subject areas

- Algebra and Number Theory