Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2311.08361

Eisenstein points on the Hilbert cuspidal eigenvariety

Adel Betina·Mladen Dimitrov·Sheng-Chi Shih
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a comprehensive study of the geometry of Hilbert $p$-adic eigenvarieties at classical parallel weight one intersection points of their cuspidal and Eisenstein loci. For instance, we determine all such points at which the weight map is étale. The Galois theoretic approach presents genuine difficulties due to the lack of good deformation theory for pseudo-characters irregular at $p$ and reflects the richness of the local geometry. We believe that our geometric results lead to deeper insight into the arithmetic of Hilbert automorphic forms and we produce in support several applications in Iwasawa theory.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Adel Betina, Mladen Dimitrov, Sheng-Chi Shih. 2026-01-12. Eisenstein points on the Hilbert cuspidal eigenvariety. https://arxiv.org/abs/2311.08361

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

KEEP EXPLORING

Related papers

arXiv2026

Asymptotic density of k-almost primes

Landau's well known asymptotic formula $$N_k(x):=\ \mid\{n\leq x : Ω(n)=k\}\mid \ \sim \left( \frac{x}{\log x} \right) \frac{(\log\log x)^{k-1}}{(k - 1)!}\ \ (x \rightarrow \infty),$$ which also holds for $$π_k(x):=\ \mid\{n\leq x : ω(n)=k\}\mid,$$ is known to be fairly poor for $k > 1$, and when $k$ is allowed to tend to infinity with $x$, the study of $N_k(x)$ and $π_k(x)$ becomes very technical [1, Chapter II.6, $§$ 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as $k$ tends to infinity.

math.NT↗
arXiv2026

Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel's base change to some real quadratic field $E$. This generalizes earlier works of Tilouine-Urban and of Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods, defined using the middle degree of the cuspidal cohomology of $\mathrm{GL}_3(E)$, instead of the top or bottom degrees. We also investigate the Rogawski's stable base change from the quasi-split unitary group $U_E$ associated with $E$ to $\mathrm{GL}_3(E)$. In this situation, we also obtain some results toward a $p$-adic divisibility of automorphic periods.

math.NT↗