论文标题
非亚伯级结构，尼尔森等效性和Markoff三元组
Nonabelian level structures, Nielsen equivalence, and Markoff triples
论文作者
论文摘要
在本文中，我们从椭圆曲线的赫尔维兹封面的一个组成部分到椭圆曲线的模量堆栈的地图程度建立了一致性。结合起来，这可以表示为尼尔森等效类别生成对的有限基团的等效类别的基础。在Bourgain，Gamburd和Sarnak的工作的基础上，我们应用了这种一致性，以表明，对于几乎有限的许多Primes $ p $，Markoff自动形态的群体在非零的$ \ Mathbb {f} _p {f} _p $ - 标记公式等式$ x^2 + y y y y^2 + y^2 + z^2 + z^2 -3xy上均能实施。这具有Markoff方程的强近似属性，Markoff数字满足的一致性条件的有限性以及$ \ text {sl} _2（\ Mathbb {f} _p）$ hurwitz空间的某个无限家族的连通性 - Elliptic Curves的封面。除了有限的例外，这可以解决Baragar于1991年首次提出的Bourgain，Gamburd和Sarnak的猜想。由于它们的方法有效，这将猜想降低为有限的计算。
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 nonzero $\mathbb{F}_p$-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 $\text{SL}_2(\mathbb{F}_p)$-covers of elliptic curves. With possibly finitely many exceptions, this resolves a conjecture of Bourgain, Gamburd, and Sarnak, first posed by Baragar in 1991. Since their methods are effective, this reduces the conjecture to a finite computation.
