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Chen, William Y.

Nonabelian level structures, Nielsen equivalence, and Markoff triples. (English) Zbl 07782633

Ann. Math. (2) 199, No. 1, 301-443 (2024).
Summary: In this paper we establish a congruence on the degree of the map from a component of a Hurwitz space of covers of elliptic curves to the moduli stack of elliptic curves. Combinatorially, this can be expressed as a congruence on the cardinalities of Nielsen equivalence classes of generating pairs of finite groups. Building on the work of Bourgain, Gamburd, and Sarnak, we apply this congruence to show that for all but finitely many primes \(p\), the group of Markoff automorphisms acts transitively on the non-zero \(\mathbb{F}_p\)-points of the Markoff equation \(x^2+y^2+z^2-3xyz=0\). This yields a strong approximation property for the Markoff equation, the finiteness of congruence conditions satisfied by Markoff numbers, and the connectivity of a certain infinite family of Hurwitz spaces of \(\mathrm{SL}_2(\mathbb{F}_p)\)-covers of elliptic curves. With possibly finitely many exceptions, this resolves a conjecture of Bourgain, Gamburd, and Sarnak, first posed by Baragar in 1991, and a question of Frobenius, posed in 1913. Since their methods are effective, this reduces the conjecture to a finite computation.

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MSC:

11D25 Cubic and quartic Diophantine equations
11J06 Markov and Lagrange spectra and generalizations
14H10 Families, moduli of curves (algebraic)
14H30 Coverings of curves, fundamental group
14D20 Algebraic moduli problems, moduli of vector bundles
14G12 Hasse principle, weak and strong approximation, Brauer-Manin obstruction
14M35 Character varieties
20D60 Arithmetic and combinatorial problems involving abstract finite groups

Keywords:

Nielsen equivalence; Hurwitz stacks; elliptic curves; non-congruence modular curves; admissible covers; character varieties; Markoff triples
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References:

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