Topics in Classical Automorphic Forms

Abstract

Preface xi Chapter 0. Introduction 1 Chapter 1. The Classical Modular Forms 3 1.1. Periodic functions 3 1.2. Elliptic functions 6 1.3. Modular functions 10 1.4. The Fourier expansion of Eisenstein series 13 1.5. The modular group 15 1.6. The linear space of modular forms 18 Chapter 2. Automorphic Forms in General 23 2.1. The hyperbolic plane 23 2.2. The classification of motions 27 2.3. Discrete groups-Fuchsian groups 29 2.4. Congruence groups 34 2.5. Double coset decomposition 37 2.6. Multiplier systems 40 2.7. Automorphic forms 42 2.8. The eta-function and the theta-function 44 vu viii Contents Chapter 3. The Eisenstein and the Poincare Series 47 3.1. General Poincare series 47 3.2. Fourier expansion of Poincare series 49 3.3. The Hilbert space of cusp forms 52 Chapter 4. Kloosterman Sums 57 4.1. General Kloosterman sums 57 4.2. Kloosterman sums for congruence groups 58 4.3. The classical Kloosterman sums 59 4.4. Power-moments of Kloosterman sums 61 4.5. Sums of Kloosterman sums 65 4.6. The Salie sums 66 Chapter 5. Bounds for the Fourier Coefficients of Cusp Forms 69 5.1. General estimates 69 5.2. Estimates by Kloosterman sums 72 5.3. Coefficients of cusp forms with theta multiplier

On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups

“…Let S k (Γ 0 (N)) be the space of elliptic cusp forms of weight k and level N, f ∈ S k (Γ 0 (N)), and denote its Fourier coefficients by a f (n). where , denotes the usual Petersson inner product, δ mn is the Kronecker delta, S(m, n; c) is the Kloosterman sum, J k−1 (·) is the (k − 1)-th J-Bessel function, and the sum on the left-hand side runs over an orthogonal basis of S k (N) (see Theorem 3.6 in [3]). From this formula, evaluating the sum on the right-hand side, we can show…”

Codes associated with special linear groups and power moments of multi-dimensional Kloosterman sums

“…Salié obtained these results in [15] by determining M 1 , M 2 , M 3 , and Iwaniec got these ones in [5] by computing A 2 , A 3 ,A 4 .…”

“…Then we see from the definition of the code C = C(SL(n, q))(cf. (4), (5)) that u is a codeword with weight i if and only if β∈Fq ν β = i and β∈Fq ν β β = 0 (an identity in F q ). As there are β∈Fq n β ν β many such codewords with weight i, we obtain the following theorem.…”

Ternary Quadratic Forms, Modular Equations, and Certain Positivity Conjectures

“…His observation was crucial to our investigation. We understand that Kane used Siegel's weighted average theorem [7] together with some local calculations of Jones [8]. We note that the Corollary 1.3 has a twin There is nothing very special about the exponent 7 in (1.13).…”