Generalized Kloosterman sums and the Fourier coefficients of cusp forms

Parson, L. Alayne

doi:10.1090/s0002-9947-1976-0412112-8

Cited by 15 publications

(3 citation statements)

References 25 publications

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“…By way of contradiction, we assume that Xr is a congruence character. In [4] it is shown that any congruence character on a congruence subgroup of level n is identically one on F(48n2) when m = 2 and F(36n2) when tn 3 …”

Section: Xr (M) = Exp [Git(i -N)(bd + Ac/n)/m] Exp[ 7{(n -1)(m + 1)(amentioning

confidence: 99%

Dedekind Sums and Noncongruence Subgroups of the Hecke Groups G(√2) and G(√3)

Parson¹

1976

Proceedings of the American Mathematical Society

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Section: Xr (M) = Exp [Git(i -N)(bd + Ac/n)/m] Exp[ 7{(n -1)(m + 1)(amentioning

confidence: 99%

Dedekind Sums and Noncongruence Subgroups of the Hecke Groups G(√2) and G(√3)

Parson¹

1976

Proceedings of the American Mathematical Society

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“…These two groups are the only Hecke groups, aside from the modular group, whose elements are completely known (see Remark 3.2). Also, H q is commensurable with SL(2, Z) if and only if q = 3, 4 or 6 (see [14,19,20,27]). We will consider the group H 4 at the end of Section 3.…”

Section: Introductionmentioning

confidence: 99%

Convex Standard Fundamental Domain For subgroups of Hecke Groups

Ibrahimou¹,

Yayenie²

2010

Bull. Aust. Math. Soc.

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It is well known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of SL(2, R), then its translates by the group form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such a hyperbolically convex fundamental domain for any discrete subgroup can be obtained by using Dirichlet's and Ford's polygon constructions. However, these two results are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction. A third, and most important and practical, method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. If 2 is a subgroup of 1 such that 1 = 2 · {L 1 , L 2 , . . . , L m } and F is the closure of a fundamental domain of the bigger group 1 , then the setis a fundamental domain of 2 . One can ask at this juncture, is it possible to choose the right coset suitably so that the set R is a convex hyperbolic polygon? We will answer this question affirmatively for Hecke modular groups.2000 Mathematics subject classification: primary 11F06; secondary 11F03, 20H05, 20H10.

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“…This estimate is a considerable improvement upon Hecke's [29,Satz 8], [30, p. 484] well-known estimate a(n) = O(n k/2 ) for the Fourier coefficients of f . Note that Weil's bound for Kloosterman sums [114] implies that a(n) = O (n k/2−1/4+ ) for any > 0 (see L. A. Parson's paper [47] for conditions on f ensuring this bound), while the celebrated theorem of Deligne [19], proving the Ramanujan-Petersson conjecture, asserts that a(n) = O (n k/2−1/2+ ) ( > 0) on subgroups Γ 0 (N ), where k is any positive integer, and the multiplier system is any Dirichlet character modulo N . The latter estimate, by the way, is best possible as was demonstrated in another paper by Rankin [84].…”

Section: Rankin-cohen Brackets and Differential Operators On Modular mentioning

confidence: 99%

10.1093/gao/9781884446054.013.7002294443

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Robert Alexander Rankin, an eminent Scottish number theorist and, for several decades, one of the world's foremost experts in modular forms, died on January 27, 2001 in Glasgow at the age of 85. He was one of the founding editors of The Ramanujan Journal. For this and the next issue of the The Ramanujan Journal, many well-known mathematicians have prepared articles in Rankin's memory. In this opening paper, we provide a short biography of Rankin and discuss some of his major contributions to mathematics. At the conclusion of this article, we provide a complete list of Rankin's doctoral students and a complete bibliography of all of Rankin's writings divided into five categories: 1) Research and Expository Papers; 2) Books; 3) Books Edited; 4) Obituaries; 5) Other Writings.

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Generalized Kloosterman sums and the Fourier coefficients of cusp forms

Cited by 15 publications

References 25 publications

Dedekind Sums and Noncongruence Subgroups of the Hecke Groups G(√2) and G(√3)

Dedekind Sums and Noncongruence Subgroups of the Hecke Groups G(√2) and G(√3)

Convex Standard Fundamental Domain For subgroups of Hecke Groups

10.1093/gao/9781884446054.013.7002294443

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