On Zeta Functions and Families of Siegel Modular Forms

Alexei Panchishkin

Home/Archive/arXiv

arXiv · 0709.1645

On Zeta Functions and Families of Siegel Modular Forms

Alexei Panchishkin

View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $p$ be a prime, and let $Γ=\Sp_g(\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic $L$-functions attached to $\Sp_g$, and their analytic properties are given. Critical values for the spinor $L$-functions and $p$-adic constructions are discussed. Rankin's lemma of higher genus is established. A general conjecture on a lifting from $ GSp_{2m} \times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modular forms are given using Ikeda-Miyawaki constructions.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Alexei Panchishkin. 2007-09-11. On Zeta Functions and Families of Siegel Modular Forms. https://arxiv.org/abs/0709.1645

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

KEEP EXPLORING

Related papers

arXiv2026

Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas

Let L be the transfer operator associated with the Gauss' continued fraction map, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banach space. In this work we prove a two-term asymptotic formula for the eigenvalues of L, show their algebraic simplicity, sign alternation pattern, and decrease in absolute value. This settles, in a stronger form, the conjectures of D. Mayer and G. Roepstorff (1988), A.J. MacLeod (1992), Ph. Flajolet and B. Vallee (1995), also supported by several other authors. Further, we find an exact series for the eigenvalues, which also gives the canonical decomposition of trace formulas due to D. Mayer (1976) and K.I. Babenko (1978). This crystallizes the contribution of each individual eigenvalue in the trace formulas.

math.NT↗

arXiv2026

Implications of Breuil-Herzig-Hu-Morra-Schraen's conjectures on Zábrádi's functor

Let $ρ$ be a smooth $n$-dimensional representation of $\mathcal{G}_{\mathbb{Q}_p}$ over $\overline{\mathbb{F}_p}$. When $ρ$ is generic and a good conjugate, the article "Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$", by Breuil-Herzig-Hu-Morra-Schraen, introduces the notion of an admissible representation $Π$ of $\mathrm{GL}_n(\mathbb{Q}_p)$ compatible with $ρ$. In loc. cit., the five authors also question whether there exists some $Π$ compatible with $ρ$ from which Zábrádi's functor $\mathbf{V}_Δ$ recovers a specific representation $\overline{L}^{\boxtimes}(ρ)$ of $\mathcal{G}_{\mathbb{Q}_p}^{n-1}$, constructed from $ρ$. We give a range of results about how badly $\mathbf{V}_Δ(Π)$ behaves for an arbitrary $Π$ satisfying some weaker compatibilities with $ρ$. In particular, when $ρ$ is reducible and $n\geq 3$, no representation $Π$ compatible with $\widetilde{P}_ρ$ can satisfy $\mathbf{V}_Δ(Π)\simeq \overline{L}^{\boxtimes}(ρ)$.

math.NT↗

arXiv2026

Curves of genus two with maps of every degree to a fixed elliptic curve

We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that for every genus-$2$ curve $C$, there is an integer $n$ with $1 < n \le 59$ such that there is no minimal degree-$n$ map from $C$ to an elliptic curve.

math.NT↗