Searcharxiv⌕ Search

arXiv · 0704.0662

Bounds for Multiplicities of Unitary Representations of Cohomological Type in Spaces of Cusp Forms

Abstract

Let $\Goo$ be a semisimple real Lie group with unitary dual $\Ghat$. The goal of this note is to produce new upper bounds for the multiplicities with which representations $π\in \Ghat$ of cohomological type appear in certain spaces of cusp forms on $\Goo$.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Frank Calegari, Matthew Emerton. 2007-04-04. Bounds for Multiplicities of Unitary Representations of Cohomological Type in Spaces of Cusp Forms. https://arxiv.org/abs/0704.0662

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

KEEP EXPLORING

Related papers

Sub-Weyl bound for $GL(2)$ via trivial delta

For a $SL(2,\mathbb{Z})$ form $f$, we obtain the sub-Weyl bound \begin{equation*} L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-δ+\varepsilon}, \end{equation*} where $δ=1/174$, thereby crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the `trivial' delta method.

math.NT↗

A Shiu Theorem for Larger and Smoother Functions

In this paper, we broaden Shiu's Brun-Titchmarsh theorem to allow for functions that are larger and/or smooth-supported. In particular, let $x$ be a large parameter, and let $f_x(n)$ be a nonnegative multiplicative function parametrized by $x$. We prove that if there exists a small constant $β$ and a constant $A_1$ such that $f_x(p^l)\leq (A_1\log\log x)^{lβ}$ for every prime $p\leq x$ and every $l\geq 1$, and if $f_x(n)\ll \max\{n^ε,(\log x)^ε\}$ for every $ε>0$ and every $n\leq x$, then $$\sum_{\substack{x-y\leq n\leq x \\ n\equiv a\pmod k}}f_x(n)\ll_{ε_0} \frac{y}{ϕ(k)(\log x)^{1-ε_0}}\exp\left(\sum_{\substack{p\leq x \\ p\nmid k}}\frac{f(p)}{p}\right)$$ for every $ε_0>0$, where $x^κ\leq y\leq x$, $k<y^{1-α}$, and $(a,k)=1$ for relatively small $α$ and $κ$, and the implied constant does not depend upon the parameter $x$. Moreover, we prove that if $f$ is a $Q$-smooth-supported function then there exists a constant $C$ for which $$\sum_{\substack{x-y\leq n\leq x \\ n\equiv a\pmod k}}f_x(n)\ll_{ε_0} \frac{y}{ϕ(k)(\log x)^{1-ε_0}}\exp\left(\sum_{\substack{p\leq x \\ p\nmid k}}\frac{f(p)}{p}\right)ρ(u)^C,$$ where $u=\frac{\log x}{\log Q}$, $ρ$ is the Dickman-de Bruijn function, and $C$ depends on whether $f_x(p)$ is bounded by a constant or by a slowly-growing function of $x$. We also give applications to both the divisor function to large powers and to smooth numbers in short intervals.

math.NT↗

Improved, sublinear projective Schwartz-Zippel and (sub)quadratic dimension growth bounds in arbitrary codimension

We introduce a sublinear sharpened version of a classical projective variant of the Schwartz-Zippel bounds. Here, the sublinearity is in the degree, whereas the classical Schwartz-Zippel bounds are linear in the degree. We prove such sublinear bounds in several cases, including for unions of linear varieties. These bounds lead to a subquadratic dependence on the degree for projective dimension growth bounds in low dimensions. This improves on the quadratic dependence obtained by Binyamini, Cluckers and Kato in [BCK25] and thus brings the dimension growth upper bounds and lower bounds for the worst case closer together, a question asked by Cluckers and Glazer in [CG25]. We also address another question from [CG25], to treat the case of general codimension. To this end, we introduce a natural projection argument with pull-backs. We also give our main results for any global field instead of just $\mathbb{Q}$.

math.NT↗