Number Theory Web Seminar: William Chen (Institute for Advanced Study)

Markoff triples and connectivity of Hurwitz spaces

In this talk we will show that the integral points of the Markoff equation x^2 + y^2 + z^2 - xyz = 0 surject onto its F_p-points for all but finitely many primes p. This essentially resolves a conjecture of Bourgain, Gamburd, and Sarnak, and a question of Frobenius from 1913. The proof relates the question to the classical problem of classifying the connected components of the Hurwitz moduli spaces H(g,n) classifying finite covers of genus g curves with n branch points. Over a century ago, Clebsch and Hurwitz established connectivity for the subspace classifying simply branched covers of the projective line, which led to the first proof of the irreducibility of the moduli space of curves of a given genus. More recently, the work of Dunfield-Thurston and Conway-Parker establish connectivity in certain situations where the monodromy group is fixed and either g or n are allowed to be large, which has been applied to study Cohen-Lenstra heuristics over function fields. In the case where (g,n) are fixed and the monodromy group is allowed to vary, far less is known. In our case we study SL(2,p)-covers of elliptic curves, only branched over the origin, and establish connectivity, for all sufficiently large p, of the subspace classifying those covers with ramification indices 2p. The proof builds upon asymptotic results of Bourgain, Gamburd, and Sarnak, the key new ingredient being a divisibility result on the degree of a certain forgetful map between moduli spaces, which provides enough rigidity to bootstrap their asymptotics to a result for all sufficiently large p.

For further information about the seminar, please visit this webpage.

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