$\mathrm{SL}_2(\mathbf{Z})$中Schottky半群的一致同余计数
Uniform congruence counting for Schottky semigroups in $\mathrm{SL}_2(\mathbf{Z})$
摘要：
设$Γ$是$\mathrm{SL}_2(\mathbf{Z})$中的一个Schottky半群，对于$q\in \mathbf N$，设$Γ(q):=\{γ\in Γ: γ= e \text{ (mod $ q $)}\}$是它的同馀子半群.我们证明了以下关于半径为$R$的$M_2(\mathbf{R})$中的欧几里德范球族$B_R$的一致同余计数定理：对于所有不具有小素因子的$q$，对于某些独立于$q$的$c_Γ>0, C>0, ε>0$，$ (Γ(q) \cap B_R )= c_Γ\frac{R^{2δ}}{ (\mathrm{SL}_2(\mathbf{Z}/q\mathbf{Z}))} +O(q^C R^{2δ-ε})$作为$R\to \infty$。我们的技巧也适用于给出$\mathrm{SL}_2(\mathbf{Z})$的连分数半群的一个类似的计数结果，这是在研究关于连分数的Zaremba猜想中出现的。
英文摘要：
Let $Γ$ be a Schottky semigroup in $\mathrm{SL}_2(\mathbf{Z})$, and for $q\in \mathbf N$, let $Γ(q):=\{γ\in Γ: γ= e \text{ (mod $q$)}\}$ be its congruence subsemigroup of level $q$. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls $B_R$ in $M_2(\mathbf{R})$ of radius $R$: for all $q$ with no small prime factors, $ (Γ(q) \cap B_R )= c_Γ\frac{R^{2δ}}{ (\mathrm{SL}_2(\mathbf{Z}/q\mathbf{Z}))} +O(q^C R^{2δ-ε})$ as $R\to \infty$ for some $c_Γ>0, C>0, ε>0$ which are independent of $q$. Our technique also applies to give a similar counting result for the continued fractions semigroup of $\mathrm{SL}_2(\mathbf{Z})$, which arises in the study of Zaremba's conjecture on continued fractions.
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