Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0704.1951

Zeta function and cryptographic exponent of supersingular curves of genus 2

Gabriel Cardona·Enric Nart
View original source ↗Read source document ↗Export metadata ↓

Abstract

We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. We display these computations in an appendix where we select a family of representatives of all these curves up to geometric isomorphism and we exhibit equations and the zeta function of all their twists. As an application we obtain a direct computation of the cryptographic exponent of the Jacobians of these curves.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gabriel Cardona, Enric Nart. 2007-04-16. Zeta function and cryptographic exponent of supersingular curves of genus 2. https://arxiv.org/abs/0704.1951

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

KEEP EXPLORING

Related papers

arXiv2026

Spectral mod p Satake isomorphism for GL_n

Let $K/\mathbb{Q}_p$ be a finite extension with residue field $k$. By a work of Emerton--Gee, irreducible components inside the reduced special fiber of the moduli stack of rank $n$ étale $(φ,Γ)$-modules are labeled by Serre weights of $\mathrm{GL}_n(k)$. Let $σ$ be a non-Steinberg Serre weight and $\mathcal{C}_σ$ be the corresponding irreducible component. Motivated by the categorical $p$-adic local Langlands program, we construct a natural injective map $\mathcal{O}(\mathcal{C}_σ) \hookrightarrow \mathcal{H}(σ)$ from the ring of global functions on $\mathcal{C}_σ$ to the Hecke algebra of $σ$ compatible with the mod $p$ Satake isomorphism by Herzig and Henniart--Vignéras in a suitable sense. For sufficiently generic $σ$, we prove that it is an isomorphism. As an application, we obtain a natural stratification of the irreducible component whose strata are equipped with a parabolic structure. Our main input is a construction of a morphism from an integral Hecke algebra of a generic tame type to the ring of global functions on a tamely potentially crystalline Emerton--Gee stack.

math.NT↗
arXiv2026

Serre weight conjectures for $\mathrm{GSp}_4$

We prove the weight part of Serre's conjecture for Galois representations valued in $\mathrm{GSp}_4$ that are tamely ramified with explicit genericity at places above $p$ as conjectured by Herzig--Tilouine and Gee--Herzig--Savitt. This improves on arXiv:2304.13879 where an inexplicit genericity hypothesis was required. As an application, we prove a modularity lifting theorem for $\mathrm{GSp}_4$ under similar assumptions.

math.NT↗
arXiv2026

Multivariate and quantitative Erdős-Kac laws for Beatty sequences

The classical Erdős-Kac theorem states that for $n$ chosen uniformly at random from $1, \dots, N$, the random variable $(ω(n) - \log\log N)/\sqrt{\log\log N}$ converges in distribution to the standard Gaussian as $N$ tends to infinity. Banks and Shparlinski showed that this Gaussian convergence holds for any Beatty sequence $\lfloorαn + β\rfloor$ in place of $n$. Continuing in this spirit, Crnčević, Hernández, Rizk, Sereesuchart and Tao considered the joint distribution of $ω(n)$ and $ω(\lfloorαn\rfloor)$, which they showed to be asymptotically independent for irrational values of $α$. Generalising both results, we show that for any positive integer $k$, real numbers $α_1, \dots, α_k > 0$ and $β_1, \dots, β_k$, where $α_i/α_j$ is irrational for $i\neq j$, the joint distribution of $(ω(\lfloorα_in + β_i\rfloor) - \log\log N)/\sqrt{\log\log N}$ converges to the $k$-dimensional standard Gaussian. We next discuss quantitative bounds on the rate of convergence which do not depend on the values taken by the relevant parameters. Banks and Shparlinski remarked that such quantitative bounds may be given for a single Beatty sequence $\lfloorαn + β\rfloor$ under Diophantine type assumptions on $α$. We show that such assumptions are in fact unnecessary. Specifically, for any real numbers $α> 0, β$, we show that the Kolmogorov distance between the random variable $(ω(\lfloorαn + β\rfloor) - \log\log N)/\sqrt{\log\log N}$ and the standard Gaussian is bounded above by $O(\log\log\log N/\sqrt{\log\log N})$ as $N$ tends to infinity. On the other hand, we show that universal quantitative bounds of this kind do not exist for higher-degree generalised polynomials or for the joint convergence for multiple Beatty sequences.

math.NT↗