Shimura varieties with $\Gamma _1(p)$-level via Hecke algebra isomorphisms: the Drinfeld case

Thomas J. Haines; Michael Rapoport

Displaying similar documents to “Shimura varieties with $Γ_{1} (p)$ -level via Hecke algebra isomorphisms: the Drinfeld case”

The size of the Lerch zeta-function at places symmetric with respect to the line $ℜ (s) = 1 / 2$

Ramūnas Garunkštis, Andrius Grigutis (2019)

Czechoslovak Mathematical Journal

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Let $ζ (s)$ be the Riemann zeta-function. If $t \geq 6.8$ and $σ > 1 / 2$ , then it is known that the inequality $| ζ (1 - s) | > | ζ (s) |$ is valid except at the zeros of $ζ (s)$ . Here we investigate the Lerch zeta-function $L (λ, α, s)$ which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters $λ = α$ it is still possible to obtain a certain version of the inequality $| L (λ, λ, 1 - \bar{s}) | > | L (λ, λ, s) |$ .

On divisibility of the class number $h^{+}$ of the real cyclotomic fields $ℚ (ζ_{p} + ζ_{p}^{- 1})$ by primes $q < 10000$

Pavel Trojovský (2000)

Mathematica Slovaca

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Mean values related to the Dedekind zeta-function

Hengcai Tang, Youjun Wang (2024)

Czechoslovak Mathematical Journal

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Let $K / ℚ$ be a nonnormal cubic extension which is given by an irreducible polynomial $g (x) = x^{3} + a x^{2} + b x + c$ . Denote by $ζ_{K} (s)$ the Dedekind zeta-function of the field $K$ and $a_{K} (n)$ the number of integral ideals in $K$ with norm $n$ . In this note, by the higher integral mean values and subconvexity bound of automorphic $L$ -functions, the second and third moment of $a_{K} (n)$ is considered, i.e., $\sum_{n \leq x} a_{K}^{2} (n) = x P_{1} (log x) + O (x^{5 / 7 + ϵ}), \sum_{n \leq x} a_{K}^{3} (n) = x P_{4} (log x) + O (X^{321 / 356 + ϵ}),$ where $P_{1} (t)$ , $P_{4} (t)$ are polynomials of degree 1, 4, respectively, $ϵ > 0$ is an arbitrarily small number.

Representation growth of linear groups

Michael Larsen, Alexander Lubotzky (2008)

Journal of the European Mathematical Society

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Let $Γ$ be a group and $r_{n} (Γ)$ the number of its $n$ -dimensional irreducible complex representations. We define and study the associated representation zeta function $𝒵_{Γ} (s) = \sum_{n = 1}^{\infty} r_{n} (Γ) n^{- s}$ . When $Γ$ is an arithmetic group satisfying the congruence subgroup property then $𝒵_{Γ} (s)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite...

Hybrid sup-norm bounds for Hecke–Maass cusp forms

Nicolas Templier (2015)

Journal of the European Mathematical Society

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Let $f$ be a Hecke–Maass cusp form of eigenvalue $λ$ and square-free level $N$ . Normalize the hyperbolic measure such that $vol (Y_{0} (N)) = 1$ and the form $f$ such that ${∥ f ∥}_{2} = 1$ . It is shown that ${∥ f ∥}_{\infty} ≪_{ϵ} λ^{\frac{5}{24} + ϵ} N^{\frac{1}{3} + ϵ}$ for all $ϵ > 0$ . This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.

Quiver varieties and the character ring of general linear groups over finite fields

Emmanuel Letellier (2013)

Journal of the European Mathematical Society

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Given a tuple $(𝒳_{1}, ..., 𝒳_{k})$ of irreducible characters of $G L_{n} (F_{q})$ we define a star-shaped quiver $Γ$ together with a dimension vector $v$ . Assume that $(𝒳_{1}, ..., 𝒳_{k})$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product $𝒳_{1} \otimes \dots \otimes 𝒳_{k}$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(Γ, v)$ . The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied,...

On certain $G L (6)$ form and its Rankin-Selberg convolution

Amrinder Kaur, Ayyadurai Sankaranarayanan (2024)

Czechoslovak Mathematical Journal

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We consider $L_{G} (s)$ to be the $L$ -function attached to a particular automorphic form $G$ on $G L (6)$ . We establish an upper bound for the mean square estimate on the critical line of Rankin-Selberg $L$ -function $L_{G \times G} (s)$ . As an application of this result, we give an asymptotic formula for the discrete sum of coefficients of $L_{G \times G} (s)$ .

The Brauer group and the Brauer–Manin set of products of varieties

Alexei N. Skorobogatov, Yuri G. Zahrin (2014)

Journal of the European Mathematical Society

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Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$ , and let $\bar{X}$ and $\bar{Y}$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$ , respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br $(\bar{X}) \oplus$ Br( $\bar{Y})$ has finite index in the Galois invariant subgroup of Br $(\bar{X} \times \bar{Y})$ . This implies that the cokernel of the natural map Br $(X) \oplus$ Br $(Y) \to$ Br $(X \times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer–Manin set of the...

On varieties of Hilbert type

Lior Bary-Soroker, Arno Fehm, Sebastian Petersen (2014)

Annales de l’institut Fourier

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A variety $X$ over a field $K$ is of Hilbert type if $X (K)$ is not thin. We prove that if $f : X \to S$ is a dominant morphism of $K$ -varieties and both $S$ and all fibers $f^{- 1} (s)$ , $s \in S (K)$ , are of Hilbert type, then so is $X$ . We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.

On the spinor zeta functions problem: higher power moments of the Riesz mean

Haiyan Wang (2013)

Acta Arithmetica

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Let F be a Siegel cusp form of integral weight k on the Siegel modular group Sp₂(ℤ) of genus 2. The coefficients of the spinor zeta function $Z_{F} (s)$ are denoted by cₙ. Let $D_{ρ} (x; Z_{F})$ be the Riesz mean of cₙ. Kohnen and Wang obtained the truncated Voronoï-type formula for $D_{ρ} (x; Z_{F})$ under the Ramanujan-Petersson conjecture. In this paper, we study the higher power moments of $D_{ρ} (x; Z_{F})$ , and then derive an asymptotic formula for the hth (h=3,4,5) power moments of $D ₁ (x; Z_{F})$ by using Ivić’s large value arguments and other techniques. ...

Bases for certain varieties of completely regular semigroups

Mario Petrich (2021)

Commentationes Mathematicae Universitatis Carolinae

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Completely regular semigroups equipped with the unary operation of inversion within their maximal subgroups form a variety, denoted by $𝒞ℛ$ . The lattice of subvarieties of $𝒞ℛ$ is denoted by $ℒ (𝒞ℛ)$ . For each variety in an $⋂$ -subsemilattice $Γ$ of $ℒ (𝒞ℛ)$ , we construct at least one basis of identities, and for some important varieties, several. We single out certain remarkable types of bases of general interest. As an application for the local relation $L$ , we construct $𝐋$ -classes of all varieties in $Γ$ . Two...

Categorical equivalences of varieties generated by algebras $𝔄$ and $𝔄^{r}$

K. Denecke (1988)

Banach Center Publications

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Displaying similar documents to “Shimura varieties with Γ 1 ( p ) -level via Hecke algebra isomorphisms: the Drinfeld case”

Displaying similar documents to “Shimura varieties with $Γ_{1} (p)$ -level via Hecke algebra isomorphisms: the Drinfeld case”