Displaying similar documents to “Equidistribution in S -arithmetic and adelic spaces”

Commutator subgroups of the extended Hecke groups H ¯ ( λ q )

Recep Şahin, Osman Bizim, I. N. Cangul (2004)

Czechoslovak Mathematical Journal

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Hecke groups H ( λ q ) are the discrete subgroups of P S L ( 2 , ℝ ) generated by S ( z ) = - ( z + λ q ) - 1 and T ( z ) = - 1 z . The commutator subgroup of H ( λ q ) , denoted by H ' ( λ q ) , is studied in [2]. It was shown that H ' ( λ q ) is a free group of rank q - 1 . Here the extended Hecke groups H ¯ ( λ q ) , obtained by adjoining R 1 ( z ) = 1 / z ¯ to the generators of H ( λ q ) , are considered. The commutator subgroup of H ¯ ( λ q ) is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the H ( λ q ) case, the index of H ' ( λ q ) is changed by q , in the case of H ¯ ( λ q ) , this number is...

Hecke operators in half-integral weight

Soma Purkait (2014)

Journal de Théorie des Nombres de Bordeaux

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In [], Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm’s bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators T p ℓ with p prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators...

Koecher-Maass series of a certain half-integral weight modular form related to the Duke-Imamoḡlu-Ikeda lift

Hidenori Katsurada, Hisa-aki Kawamura (2014)

Acta Arithmetica

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Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k - n/2 + 1/2 for Γ₀(4), let f be the corresponding primitive form of weight 2k-n for SL₂(ℤ) under the Shimura correspondence, and Iₙ(h) the Duke-Imamoḡlu-Ikeda lift of h to the space of cusp forms of weight k for Spₙ(ℤ). Moreover, let ϕ I ₙ ( h ) , 1 be the first Fourier-Jacobi coefficient of Iₙ(h), and σ n - 1 ( ϕ I ₙ ( h ) , 1 ) be the cusp form in the generalized Kohnen plus space of weight k - 1/2 corresponding to...

Filling Radius and Short Closed Geodesics of the 2 -Sphere

Stéphane Sabourau (2004)

Bulletin de la Société Mathématique de France

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We show that the length of the shortest nontrivial curve among the simple closed geodesics of index zero or one and the figure-eight geodesics of null index provides a lower bound on the area and the diameter of the Riemannian 2 -spheres.

Shimura varieties with Γ 1 ( p ) -level via Hecke algebra isomorphisms: the Drinfeld case

Thomas J. Haines, Michael Rapoport (2012)

Annales scientifiques de l'École Normale Supérieure

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We study the local factor at  p of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at  p by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining their images under the Hecke algebra isomorphisms of Goldstein, Morris, and Roche. ...

A quadratic form with prime variables associated with Hecke eigenvalues of a cusp form

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let f be a normalized primitive holomorphic cusp form of even integral weight k for the full modular group SL ( 2 , ℤ ) , and denote its n th Fourier coefficient by λ f ( n ) . We consider the hybrid problem of quadratic forms with prime variables and Hecke eigenvalues of normalized primitive holomorphic cusp forms, which generalizes the result of D. Y. Zhang, Y. N. Wang (2017).

Systole growth for finite area hyperbolic surfaces

Florent Balacheff, Eran Makover, Hugo Parlier (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this note, we observe that the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature ( g , n ) is greater than a function that grows logarithmically in terms of the ratio g / n .

Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras

Peng Shan (2011)

Annales scientifiques de l'École Normale Supérieure

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We define the i -restriction and i -induction functors on the category 𝒪 of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

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 ∥ ∞ ≪ ϵ λ 5 24 + ϵ N 1 3 + ϵ for all ϵ > 0 . This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.

The distribution of Fourier coefficients of cusp forms over sparse sequences

Huixue Lao, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

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Let λ f ( n ) be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform f ( z ) ∈ S k ( Γ ) . We establish that ∑ n ≤ x λ f 2 ( n j ) = c j x + O ( x 1 - 2 / ( ( j + 1 ) 2 + 1 ) ) for j = 2,3,4, which improves the previous results. For j = 2, we even establish a better result.

On q-orders in primitive modular groups

Jacek Pomykała (2014)

Acta Arithmetica

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We prove an upper bound for the number of primes p ≤ x in an arithmetic progression 1 (mod Q) that are exceptional in the sense that ℤ * p has no generator in the interval [1,B]. As a consequence we prove that if Q > e x p [ c ( l o g p ) / ( l o g B ) ( l o g l o g p ) ] with a sufficiently large absolute constant c, then there exists a prime q dividing Q such that ν q ( o r d p b ) = ν q ( p - 1 ) for some positive integer b ≤ B. Moreover we estimate the number of such q’s under suitable conditions.

Higher simple structure sets of lens spaces with the fundamental group of arbitrary order

L’udovít Balko, Tibor Macko, Martin Niepel, Tomáš Rusin (2019)

Archivum Mathematicum

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Extending work of many authors we calculate the higher simple structure sets of lens spaces in the sense of surgery theory with the fundamental group of arbitrary order. As a corollary we also obtain a calculation of the simple structure sets of the products of lens spaces and spheres of dimension grater or equal to 3 .