Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.1743

Relative log convergent cohomology and relative rigid cohomology II

Atsushi Shiho
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we develop the theory of relative log convergent cohomology of radius $λ$ ($0 < λ\leq 1$), which is a generalization of the notion of relative log convergent cohomology in the previous paper. By comparing this cohomology with relative log crystalline cohomology, relative rigid cohomology and its variants and by using some technique of hypercovering, we prove a version of Berthelot's conjecture on the overconvergence of relative rigid cohomology for proper smooth families.

Explore related subjects

math.NTmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Atsushi Shiho. 2008-05-21. Relative log convergent cohomology and relative rigid cohomology II. https://arxiv.org/abs/0707.1743

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

KEEP EXPLORING

Related papers

arXiv2026

Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$, revisited

Let $r(N)$ be the number of representations of $N$ as a sum of two squares. For every integer $n\ge 2$ we prove an asymptotic formula for $\sum_{x\equiv e \bmod 2}Φ(x/M)\,r(n^2+x^2)$, that is, for the number of integral solutions of $x_1^2+x_2^2-x_3^2=n^2$ with $x_3\equiv e \bmod 2$ in a range $|x_3|\ll M$, with smooth weight $Φ$. The formula has an explicit secondary term and a power saving in the error term, and it is non-trivial as soon as $M\ge n^{5/6+\varepsilon}$, that is, $M\ge D^{1/2-1/12+\varepsilon}$ with $D=n^2$. The leading term is $\frac{4}π(\log n)\intΦ(x/M)\,dx$, independently of the arithmetic of $n$; this corrects the main term stated in the published version (Ramanujan J. 63 (2024), 293-337). The proof factorises $n+ix_3$ in $\mathbb{Z}[i]$, which leads to Kloosterman sums. These are estimated by Weil's bound, combined with a Plancherel argument that exploits the curvature of the underlying quadric. This paper was written in collaboration with Claude (Opus 5.5, High), an AI model developed by Anthropic.

math.NT↗
arXiv2026

Bessel Periods on $U(2,1) \times U(1,1)$, Relative Trace Formula and Non-Vanishing of Central $L$-values

In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations $(π, π')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over $π$). Using these, we obtain quantitative non-vanishing results for the Rankin-Selberg central $L$-values $L(1/2, π\times π')$, which are of degree twelve over ${\mathbb Q}$, with concomitant difficulty in applying standard methods, especially since we are in a `conductor dropping' situation. We use the relative trace formula, and the orbital integrals are evaluated rather than compared with others. Besides their intrinsic interest, non-vanishing of these critical values also leads, by known results, to deducing certain associated Selmer groups have rank zero.

math.NT↗
arXiv2026

Rational points on $X_0(N)^*$ when $N$ is non-squarefree

Let $N$ be a positive integer and $X_0(N)^*$ be the quotient of the modular curve $X_0(N)$ by its full group of Atkin-Lehner involutions. Elkies conjectures that the rational points on $X_0(N)^*$ are only cusps or CM points when $N$ is large enough. We establish an integrality result for the $j$-invariants of non-cuspidal rational points on $X_0(N)^*$ when $N$ is non-squarefree, representing a significant step toward resolving a key subcase of Elkies' conjecture. To this end, we prove the existence of rank-zero quotients of certain modular Jacobians $J_0(pq)$. Furthermore, in the non-squarefree case we provide a complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$, when they are finite. In the process we identify exceptional rational points on $X_0(75)^*$ and $X_0(147)^*$ which were not known before.

math.NT↗