@article{35acf11873994fdeb65c93f0f5daa793,

title = "The number of monodromy representations of Abelian varieties of low p-rank",

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 {\'e}t(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.",

keywords = "Abelian varieties, Local systems, Profinite groups",

author = "Brett Frankel",

note = "Funding Information: This paper is derived from my Ph.D. thesis, and as such there are far too many people who have impacted this work than can be listed here. It is a pleasure to thank Ted Chinburg, my thesis advisor, for suggesting this problem and for countless discussions and guidance. Thanks also to David Harbater, Zach Scherr, and Ching-Li Chai for helpful discussions and answering my questions. I am grateful to Professor Fernando Rodriguez-Villegas for hosting me for a very productive week at the International Centre for Theoretical Physics. It was he who observed Corollary 3, and introduced me to both his theorem with Cameron Gordon [5] and the theorem of Frobenius, [3] from which one easily deduces Corollary 3. He also made me aware of Proposition 2. Bob Guralnick very helpfully referred me to both his paper with Sethuraman [6] and earlier work of Gerstenhaber [4]. Additional thanks are due to Nir Avni for comments on the draft and Sebastian Sewerin for help reading [3]. The exposition here is much improved thanks to the referee's careful reading. Many others have had a great influence on my mathematical development, which no doubt manifests itself in some of the work that follows. There are too many to list here, but I have attempted to acknowledge many of these people in the thesis document [2]. This work was partially supported by NSF FRG grant 1265290 and NSA grant H98230-14-1-0145. Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

year = "2018",

month = sep,

day = "15",

doi = "10.1016/j.jalgebra.2018.05.024",

language = "English (US)",

volume = "510",

pages = "393--412",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press Inc.",

}