任意算术维数的半算术曲面的长度谱中平均重数的指数增长

中文摘要 AI 辅助

我们研究有限共体积的半算术富克斯群Γ的测地线长度谱中平均重数的指数增长（EGMM），该群具有算术维数r≥1，并允许广义模嵌入到(±H)^(r−1)中。我们引入两个新要素：一是**多维施瓦茨-皮克收缩引理**：广义模嵌入F:H→H^(r−1)是全纯的，且关于乘积小林度量严格收缩，对所有z∈H，满足‖DF_z‖_op ≤ √(r−1)·(1−δ)，其中δ=δ(Γ)是与Γ相关的常数；二是**数论几何的范数形式估计**：将闵可夫斯基定理应用于不变迹域K中的代数整数格，对所有N，有#(L(Γ)∩[N−1,N]) ≤ CN^((r−1)^(3/2)(1−δ))，且该指数与r有关，这与r≤2的情况不同。结合这两个要素可证明：当r≤2时，Γ具有EGMM；当r≥3且δ满足**强收缩条件**δ>1−(√2·(r−1))⁻¹时，Γ也具有EGMM，该阈值是本方法得到的最尖锐阈值，由改进的数论几何论证（命题ref{propsharp}）得到，优于直接从范数形式得到的较粗指数(r−1)^(3/2)，二者在r=3时完全一致。

英文摘要

For a semi-arithmetic Fuchsian group $Γ$ of arithmetic dimension $r$ with a generalized modular embedding, EGMM holds for $r\leq2$, and for $r\geq3$ under an explicit \emph{strong contraction condition}, via a multi-dimensional Schwarz-Pick bound and a geometry-of-numbers estimate. A covering construction turns every known example into an infinite explicit family with the same exponent via principal congruence towers. We add new $r\leq2$ examples (genus-$3$/$4$ Prym eigenform loci, a different commensurability class from the known triangle groups and McMullen genus-two family) and attempt a direct flat-geometric bound on the contraction constant $δ$ for the latter via its Lyapunov exponents ($δ=2/3$ on ergodic average, by Bainbridge exact computation), explaining why an average bound does not yet certify the pointwise one \eqref{eqstrong-contraction} needs.

发表机构

• Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）

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