Über die Darstellung der ganzen Spitzenformen zu den Idealstufen der Hilbertschen Modulgruppe und die Abschätzung ihrer Fourierkoeffizienten

Gundlach, Karl-Bernhard

doi:10.1007/bf02392707

Cited by 36 publications

(12 citation statements)

References 11 publications

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“…With the theorem and the estimates from [2], [5] quoted above r(t, va~ιL) -r(t, spnίm" 1 !/)) + r{t, m~ιL) -r(t, spnm" 1 !) thus gives an asymptotic formula for r(t, m~xL).…”

Section: Let L P M X Be As In Lemma 4 Ord P T = S + V (V E Tv) Thenmentioning

confidence: 87%

Ternary quadratic forms and Brandt matrices

Schulze-Pillot

1986

Nagoya Mathematical Journal

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In a recent paper [9] the author showed (among other results) estimates on the asymptotic behaviour of the representation numbers of positive definite integral ternary quadratic forms, in particular, that for n in a fixed square class tZ2 and lattices L, K in the same spinor genus one has . The main tool utilized for the proof was the theory of modular forms of weight 3/2, especially Shimura’s lifting from the space of cusp forms of weight 3/2 to the space of modular forms of weight 2.

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“…With the theorem and the estimates from [2], [5] quoted above r(t, va~ιL) -r(t, spnίm" 1 !/)) + r{t, m~ιL) -r(t, spnm" 1 !) thus gives an asymptotic formula for r(t, m~xL).…”

Section: Let L P M X Be As In Lemma 4 Ord P T = S + V (V E Tv) Thenmentioning

confidence: 87%

Ternary quadratic forms and Brandt matrices

Schulze-Pillot

1986

Nagoya Mathematical Journal

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“…These bounds and (2.12) enable one to deduce the bounds on the generalised Kloosterman sum S a ′ ,a ′ (ω 1 , ω 2 ; c ′ ) that are contained in Lemma 2.5. Lemma 2.5 is directly analogous to Lemma 2.6 of [9]: another precedent for this type of result may be found in work of Gundlach in Section 4 of [14], which includes what is essentially a 'Weil-Estermann bound' for the analogue of the sum S a,b (ω 1 , ω 2 ; c) in the theory of principal congruence subgroups of Hilbert's modular group for any totally real algebraic number field.…”

Section: Index Of Notationmentioning

confidence: 91%

Spectral large sieve inequalities for Hecke congruence subgroups ofSL(2,Z[i])

Watt¹

2014

Journal of Number Theory

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We prove, in respect of an arbitrary Hecke congruence subgroup Γ = Γ 0 (q 0 ) ≤ SL(2, Z[i]), some new upper bounds (or 'spectral large sieve inequalities') for sums involving Fourier coefficients of Γ-automorphic cusp forms on SL(2, C). The Fourier coefficients in question may arise from the Fourier expansion at any given cusp c of Γ (our results are not limited to the case c = ∞). For this reason, our proof is reliant upon an extension, to arbitrary cusps, of the spectral-Kloosterman sum formula for Γ\SL(2, C) obtained by Hristina Lokvenec-Guleska in her doctoral thesis (generalising the sum formulae of Roelof Bruggeman and Yoichi Motohashi for P SL(2, Z[i])\P SL(2, C) in several respects, though not as regards the choice of cusps). A proof of the required extension of the sum formula is given in an appendix.

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“…If n > l , (9) is a consequence of (7), (8) (see [1], p. 323). M r (T K (p), v) is a finitedimensional vector space over C. The cusps of F K (p) are the points (£ ( 1 ) ,..., £ (n) ) of the boundary of H for £eJ£U{<»}.…”

Section: Introductionmentioning

confidence: 97%

“…Hilbert's modular group for K is the group The n different injections of K into R map K onto the conjugate fields K m ,..., K M . To each K 0) one assigns a complex variable r 0) , the /th conjugate of T = (T (1) , . .…”

Section: Introductionmentioning

confidence: 99%

On the representation of a number as a sum of squares

Gundlach

1978

Glasgow Math. J.

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It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the formwhere Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact thatis a modular form for a certain congruence subgroup of the modular group, andwithwhere Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).

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Über die Darstellung der ganzen Spitzenformen zu den Idealstufen der Hilbertschen Modulgruppe und die Abschätzung ihrer Fourierkoeffizienten

Cited by 36 publications

References 11 publications

Ternary quadratic forms and Brandt matrices

Ternary quadratic forms and Brandt matrices

Spectral large sieve inequalities for Hecke congruence subgroups ofSL(2,Z[i])

On the representation of a number as a sum of squares

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