Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.26500

Stable base change from unitary groups in three variables and integral relation of automorphic periods

Tristan Ricoul
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $E$ be a real quadratic field and let $U_E$ be the quasi-split unitary group in three variables associated with $E$. We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation $π_U$ of $U_E$ and the periods of its Rogawski stable base change to $\mathrm{GL}_3(E)$. This generalizes earlier works of Tilouine-Urban and Hida in the case of $\mathrm{GL}_2$. The divisibility we prove involves a new kind of automorphic periods for self-conjugate cuspidal automorphic representations of $\mathrm{GL}_3(E)$, defined within the middle degree of the cuspidal cohomology rather than the top or bottom degrees. Moreover, we prove an à la Hida adjoint $L$-value formula for $\mathrm{GL}_3(E)$, relating these newly defined middle-degree periods to the usual top and bottom automorphic periods. Finally, we also prove a similar formula for $U_E$, which is the first instance of such a result for quasi-split unitary groups.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tristan Ricoul. 2026-09-22. Stable base change from unitary groups in three variables and integral relation of automorphic periods. https://arxiv.org/abs/2609.26500

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

KEEP EXPLORING

Related papers

arXiv2026

Documentation for the ratpoints program

This note explains how to obtain, install, and use the ratpoints program. The program finds rational points up to a specified height on hyperelliptic curves using a highly optimized quadratic sieving algorithm.

math.NT↗
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

Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

Let $\mathbb{WP}^n_{\mathbf{q}}$ be a weighted projective space with weights $\mathbf{q} = (q_0, \dots, q_n)$, $q = \operatorname{lcm}(q_i)$, and let $ϕ\colon \mathbb{WP}^n_{\mathbf{q}} \to \mathbb{P}^n$, $[x_i] \mapsto [x_i^{q/q_i}]$, be the Veronese morphism. A point of $\mathbb{P}^n(\mathbb{Q})$ is the image of a rational point of $\mathbb{WP}^n_{\mathbf{q}}$ only if its valuation vector at every prime satisfies a Kummer congruence. We count the rational points of $\mathbb{WP}^n_{\mathbf{q}}$ of bounded weighted height $\mathfrak{h} = H(ϕ(\,\cdot\,))^{1/q}$ and prove that, on the locus where all coordinates are nonzero, $$ Z^{\circ}_{\mathfrak{h}}\big( \mathbb{WP}^n_{\mathbf{q}}(\mathbb{Q}), X \big) = X^{q\,a(\mathbf{q})} P_{\mathbf{q}}(\log X) + O\big( X^{q\,a(\mathbf{q}) - θ} \big), \qquad θ> 0, $$ with $P_{\mathbf{q}}$ of exact degree $β(\mathbf{q})$, where $a(\mathbf{q})$ and $β(\mathbf{q})$ are the value and the dimension of the optimal face of a linear program determined by the Kummer congruences. The exponent satisfies $Q \leq q\,a(\mathbf{q}) \leq q(n+1)$, $Q = \sum q_i$, with equality on the right if and only if the exponents $q/q_i$ are pairwise coprime; the difference $q(n+1) - q\,a(\mathbf{q})$ measures the sparsity of the rational points of $\mathbb{WP}^n_{\mathbf{q}}$ relative to those of its Veronese image. The leading constant is evaluated when the dual optimum is diagonal and for the weights $(2,2,3,3)$. The full counting function follows by stratification, and we formulate the conjecture over an arbitrary number field.

math.NT↗