Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.00560

Distribution of Kaneko's val function

Toshiki Matsusaka
View original source ↗Read source document ↗Export metadata ↓

Abstract

Kaneko's val function is defined as the normalized cycle integral of the elliptic modular $j$-function along closed geodesics on the modular surface. We prove that, when primitive hyperbolic conjugacy classes are ordered by geodesic length, its values concentrate at the single point 720. More generally, an analogous concentration result holds for every weakly holomorphic modular function $f$ of weight 0, with the concentration point given by Atkin's inner product $(f, 1)_{\mathrm{At}}$. The proof combines an equidistribution theorem following Pollicott with the ergodicity of a continued-fraction suspension flow and uses the decomposition formula of Bengoechea-Imamoglu to construct a bounded continuous observable.

Explore related subjects

math.NTmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Toshiki Matsusaka. 2026-09-01. Distribution of Kaneko's val function. https://arxiv.org/abs/2609.00560

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the Pair Correlation of Zeros of $L$-Functions for Non-CM Newforms in Shifted Ranges

We study the pair correlation between zeros of a shifted auxiliary $ L $-function attached to a non-CM newform, the scale of which is a fixed constant. We prove an unconditional asymptotic result for the pair correlation and introduce a simplicity hypothesis for the zeros of this function, which if true means that multiple zeros of the original $ L $-function cannot be separated by the same fixed distance. Our results provide macroscopic information in contrast to the pair correlation of the original $ L $-function which is of microscopic nature.

math.NT↗
arXiv2026

Prime Solutions to a Binary Additive Equation and Mixed Moments of Character Sums

We obtain an asymptotic formula with a power-saving error term for counting the integer points $(a,b,c,d)$ in an expanding box that satisfy the determinant equation $x_1x_2-x_3x_4 =r$ for $r \neq 0 $ with two of entries to be prime. Finally, these estimates are applied to evaluate mixed fourth moments of Dirichlet character sums over integers and primes, yielding non-trivial bounds. The method involves the Poisson summation formula and the estimation for the average of the sums of the Kloosterman fractions over primes.

math.NT↗
arXiv2026

On properness of moduli stacks of $D^{\times}$-shtukas over ramified legs

Given a maximal order $\mathcal{D}$ of a central division algebra $D$ over a global function field $F$, we prove an explicit sufficient condition for moduli stacks of $\mathcal{D}^\times$-shtukas to be proper over a finite field (modulo a suitable central action) in terms of the \emph{local invariants} of $D$ and \emph{bounds}. Our proof is a refinement of E.~Lau's result (Duke Math. J. \textbf{140} (2007)), which showed the properness of the \emph{leg morphism} (or \emph{characteristic morphism}) away from the ramification locus of $D$. %, by carefully measuring the contribution of ``ramified legs''. We also establish non-emptiness of Newton and Kottwitz--Rapoport strata for moduli stacks of $\mathcal{B}^\times$-shtukas, where $\mathcal{B}$ is a maximal order of a central simple algebra over $F$.

math.NT↗