On local characterizations of Hida families of Siegel modular forms

Shaunak V. Deo; Bharathwaj Palvannan

Home/Archive/arXiv

arXiv · 2602.20737

On local characterizations of Hida families of Siegel modular forms

Shaunak V. Deo·Bharathwaj Palvannan

View original source ↗Read source document ↗Export metadata ↓

Abstract

We provide new local characterizations of Hida families of Siegel modular forms with genus two arising from stable Yoshida lifts, that is, automorphic inductions of nearly ordinary Hilbert modular eigenforms over real quadratic fields. Our characterizations involve (i) density of de Rham at $p$ specializations at the singular weights $(k,2)$ and (ii) local decomposability at $p$ of the associated $\Lambda$-adic Galois representation. These are analogous to the characterizations of Hida families of CM modular forms provided by Ghate--Vatsal. Our approach is similar to that of Castella--Wang-Erickson who provided an alternate strategy to reproving Ghate--Vatsal's main results by applying Ribet's method when an anti-cyclotomic class group is assumed to be pseudo-null and cyclic as a $\Lambda$-module. Along these lines, one key input to our methods involves an assumption of pseudo-nullity of Selmer groups that are defined by imposing stricter conditions at $p$ than those imposed for the usual Greenberg Selmer groups appearing in the Asai main conjectures over real quadratic fields.

Explore related subjects

math.NT

Keep this discovery

BibTeX RIS

Shaunak V. Deo, Bharathwaj Palvannan. 2026-02-24. On local characterizations of Hida families of Siegel modular forms. https://arxiv.org/abs/2602.20737

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

KEEP EXPLORING

Related papers

arXiv2026

Ordinary 3-Isogeny Graphs and Improvement of Supersingularity Testing for Twisted Hessian Curves over Prime Fields

For any primes $p \neq \ell$, $\ell$-isogeny graphs of ordinary elliptic curves defined over $\mathbb{F}_{p^2}$ have a typical structure called $\ell$-volcanoes, and the structure is the core of Sutherland's supersingularity testing algorithm for elliptic curves. In this paper, by exploiting the properties of $3$-isogenies between twisted Hessian curves, we show that when $p \equiv 2 \pmod{3}$ and $\ell = 3$, every ordinary twisted Hessian curve defined over $\mathbb{F}_p$ lies on the surface of the $3$-volcano. As an application, we give an improved version of Sutherland's supersingularity testing algorithm specialized to twisted Hessian curves defined over $\mathbb{F}_p$ with $p \equiv 2 \pmod{3}$. We also give a generalization of the known fact that any supersingular $j$-invariant is a cube in $\mathbb{F}_{p^2}$; we show that for any twisted Hessian curve $H(a,d)$ defined over $\mathbb{F}_{p^2}$, its $j$-invariant is not a cube in $\mathbb{F}_{p^2}$ if and only if $H(a,d)$ is ordinary and lies on the floor of a $3$-volcano.

math.NT↗

arXiv2026

Effective estimates for exponential sums with multiplicative coefficients

Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$ for every $x\ge1$. If $|\alpha-a/q|\le q^{-2}$, $(a,q)=1$, and $3\le R\le q\le N/R$, we prove \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha) \ll_A \frac{N}{\log N} +\frac{N}{\sqrt R}\sqrt{\log\log(3R)} \] with effective implied constants. Montgomery and Vaughan proved this with second term $NR^{-1/2}(\log R)^{3/2}$, and, for $1$-bounded functions, Bachman replaced it by $NR^{-1/2}\sqrt{\log R\log\log R}$. We remove the factor $\sqrt{\log R}$ from Bachman's second term while retaining the original coefficient hypotheses of Montgomery and Vaughan. A more precise estimate records the distance from a rational number. The proof combines the Brun-Titchmarsh inequality on short intervals with maximal Fourier estimates derived from the Carleson-Hunt theorem; the local bounds permit arbitrary prime-dependent prefixes. We also prove sharpness of the square-root displacement dependence.

math.NT↗

arXiv2026

Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers

In this paper, we will study a trivariate extension of the Cauchy numbers of both the first kind (also called Gregory coefficients) and the second kind (also called N\"orlund numbers) via the Laurent expansion of the reciprocal of any positive integer power (which is called the order) of multiple polylogarithms. In the case of logarithm, we will show by the WZ method that for each order $\ell>1$ some Gregory coefficient of order $\ell$ must vanish, in contrast to the fact that all classical Gregory coefficients are nonzero. We also prove in this higher order logarithm case that the sequence is eventually alternating for each fixed order, a property enjoyed by the classical Gregory coefficients. In the most general setting, we conjecture that these new sequences are all eventually positive, which is supported by strong numerical evidence. Finally, we confirm this conjecture in the special case of polylogarithms and double polylogarithms. As a by product, for each zeta value and double zeta value, we find an infinite family of identities expressing its reciprocal as a sum of a rational number and an improper integral.

math.NT↗