Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.10626

Eisenstein-Kronecker classes on Hilbert moduli spaces and a new construction of Katz's $p$-adic measure

Guillermo Gamarra Segovia
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this work, we extend the construction of the Eisenstein-Kronecker classes of Kings-Sprang to the universal abelian schemes lying over Hilbert moduli spaces parametrizing objects with $(Γ(l), Γ_{00}(p^\infty))$-level. We then compare the associated ($p$-adic) Hilbert modular forms to the Eisenstein series constructed by Katz. Similarly to the procedure of Kings-Sprang, we produce a $p$-adic measure which interpolates the Eisenstein-Kronecker classes, and use the mentioned comparison results to show that this recovers essentially the $p$-adic Eisenstein measure constructed by Katz.

Explore related subjects

math.NTmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Guillermo Gamarra Segovia. 2026-06-09. Eisenstein-Kronecker classes on Hilbert moduli spaces and a new construction of Katz's $p$-adic measure. https://arxiv.org/abs/2606.10626

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

KEEP EXPLORING

Related papers

arXiv2026

On vector valued automorphic forms for the Weil representation

We develop a theory of vector valued automorphic forms associated to the Weil representation $ω_f$ and corresponding to vector valued modular forms transforming with the ``finite'' Weil representation $ρ_L$. For each prime $p$ we determine the structure of a vector valued spherical Hecke algebra depending on $ω_f$, which acts on the space of automorphic forms.

math.NT↗
arXiv2026

Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields

Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in $\mathbb{Z}_p$-towers of imaginary quadratic fields $K$. For a odd prime $p$, the lines $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ are identified with $\mathbb{Z}_p$-extensions $ K_{a,b}/K $. Under certain conditions on $ K $ that involve explicit elliptic curves, we identify a line $(a_0,b_0) \in \mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ such that for all $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ with $(a, b)\not\equiv (a_0, b_0)\pmod{p}$, Hilbert's tenth problem has a negative answer in all finite layers of $ K_{a,b} $. Using results of Bhargava et al., we prove unconditionally that a positive proportion of imaginary quadratic fields meet our criterion when $p=3$. For $p=11,13,31,37$, the analogous conclusions obtained from the rank-zero twist families of Kriz--Li are conditional on the vanishing of the $p$-primary Tate--Shafarevich groups for a positive relative proportion of those twists.

math.NT↗
arXiv2026

The standard $L$-function attached to a vector valued modular form

We define two $L$-functions associated to a common vector valued eigenform $f$ transforming with the ``finite'' Weil representation. The first one can be seen as a standard zeta function defined by the eigenvalues of $f$. The second one can be interpreted as standard $L$-function defined as an Euler product where each $p$-factor is a rational function in terms of two unramified characters of the $p$-adic field $\Q_p$. We show that both $L$-functions are related and prove further that they both can be continued meromorphically to the whole complex $s$-plane.

math.NT↗