The Behavior of General Theta Functions Under the Modular Group and the Characters of Binary Modular Congruence Groups. II

Kloosterman, H. D.

doi:10.2307/1969083

Cited by 8 publications

(9 citation statements)

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“…These facts allow one to do the calculations more explicitly. Proof: This fact is easily shown using (9) and is essentially proven (for the theta functions) in [4] and [8]. In particular for a matrix ( a b 4m d ) (using the useful fact above) in (4m) the…”

Section: Transformations Under Congruence Subgroupsmentioning

confidence: 84%

“…We will use these properties to investigate how the different vector components are connected. In order to efficiently compute the matrix representations for general elements of Sl 2 (Z) we use the results of [4,8] (Chap. IX, Section 3, (24)), among others.…”

Section: Basicsmentioning

confidence: 99%

“…If it was found using 0 (4m/t) then the calculations above show it as the sum of some of the other components. These components have different Fourier expansions (see (4)) and therefore cannot cancel. The components generated by 0 (4m) similarly cannot be zero unless the component used to generate them was identically zero.…”

Section: Jacobi Forms Of Square-free Indexmentioning

confidence: 99%

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Generating Jacobi Forms

Skogman

2005

Ramanujan J

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In this paper we explore the relationship between vector-valued modular forms and Jacobi forms and give explicit relations over various congruence subgroups. The main result is that a Jacobi form of square-free index on the full Jacobi group is uniquely determined by any of the associated vector components. In addition, an explicit construction is given to determine the other vector components from this single component. In other words, we give an explicit construction of a Jacobi form from a subset of its Fourier coefficients. This leads to results about how the transformation properties are affected by congruence restrictions on the Fourier expansion.

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Section: Transformations Under Congruence Subgroupsmentioning

confidence: 84%

Section: Basicsmentioning

confidence: 99%

Section: Jacobi Forms Of Square-free Indexmentioning

confidence: 99%

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Generating Jacobi Forms

Skogman

2005

Ramanujan J

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“…This gives, in the notation of section one, the matrix An for each element g in the group. Matrix coefficients for this group have been calculated before, in [4] for example. We need coefficients with respect to a specific basis, so it is easiest to start from scratch.…”

Section: Congruence Representationsmentioning

confidence: 99%

Some explicit cases of the Selberg trace formula for vector valued functions

Stopple¹

1989

Trans. Amer. Math. Soc.

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Abstract.The trace formula for SL{2,Z) can be developed for vector-valued functions which satisfy an automorphic condition involving a group representation n . This paper makes this version explicit for the class of representations which can be realized as representations of the finite group PSL(2,Z/q) for some prime q . The body of the paper is devoted to computing, for the singular representations n , the determinant of the scattering matrix <&{s,n) on which the applications depend. The first application is a version of the Roelcke-Selberg conjecture. This follows from known results once the scattering matrix is given.The study of representations of SL{2, Z) in finite-dimensional vector spaces of (scalar-valued) holomorphic forms dates back to Hecke. Similar problems can be studied for vector spaces of Maass wave forms, with fixed level q and eigenvalue X . One would like to decompose the natural representation of SL(2,Z) in this space, and count the multiplicities of its irreducible components. The eigenvalue estimate obtained for vector-valued forms is equivalent to an asymptotic count, as A -► oo , of these multiplicities.The trace formula for SL(2,Z) can be developed for vector-valued functions which satisfy an automorphic condition involving a group representation n . In fact, Selberg's original paper stated the trace formula in this generality. This paper makes this version explicit for the class of representations which can be realized as representations of the finite group PSL(2,Z/q) for some prime q . We then present some applications. The body of the paper is devoted to computing, for the singular representations n , the determinant of the scattering matrix (s , n) on which the applications depend. The Dirichlet series coefficients of

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“…Attempts have been made to compute the values of the irreducible characters of SL(2, Z/p n Z) and complete results were obtained by Shur [11] in the case n -1 and Praetorius [9] and Rohrbach [10] in the case n = 2. Since that time analytic techniques involving theta-functions [7] and methods used for similar problems over locally compact groups [12] have been applied with partial results in the first case and a classification theorem in the second.…”

mentioning

confidence: 99%

The characters of the binary modular congruence group

Kutzko¹

1973

Bull. Amer. Math. Soc.

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The Behavior of General Theta Functions Under the Modular Group and the Characters of Binary Modular Congruence Groups. II

Cited by 8 publications

References 0 publications

Generating Jacobi Forms

Generating Jacobi Forms

Some explicit cases of the Selberg trace formula for vector valued functions

The characters of the binary modular congruence group

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