Computation of Harmonic Weak Maass Forms

Bruinier, Jan Hendrik; Strömberg, Fredrik

doi:10.1080/10586458.2012.645778

Cited by 9 publications

(11 citation statements)

References 32 publications

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“…These computations were done using Sage[55] by Bruinier and Strömberg in[24]. Stephan Ehlen obtained the same numbers using our results (also using Sage).…”

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confidence: 68%

Weierstrass mock modular forms and elliptic curves

et al. 2015

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Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves E/Q. We show that mock modular forms which arise from Weierstrass ζ -functions encode the central L-values and L-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by Hövel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values for the quadratic twists of E. We employ results of Bruinier and the third author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger, and Zagier. We also obtain p-adic formulas for the corresponding weight 2 newform using the action of the Hecke algebra on the Weierstrass mock modular form.

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“…These computations were done using Sage[55] by Bruinier and Strömberg in[24]. Stephan Ehlen obtained the same numbers using our results (also using Sage).…”

mentioning

confidence: 68%

Weierstrass mock modular forms and elliptic curves

et al. 2015

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“…2]). For every set of integers h > 0, e ∞ > 0, e 2 ≥ 0, e 3 ≥ 0, g ≥ 0 consistent with (12) there exists a subgroup of the modular group with signature (h; g, e ∞ , e 2 , e 3 ) .…”

Section: Subgroupsmentioning

confidence: 99%

“…For the last equality we use the Gauss-Bonnet theorem (see (12) for the formulation in our setting). Using the residue theorem again and combining ( 10) and ( 11) we find that…”

Section: Counting Eigenvalues and The Average Weyl's Lawmentioning

confidence: 99%

Noncongruence subgroups and Maass waveforms

Strömberg

2019

Journal of Number Theory

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The main topic of the paper is spectral theory for noncongruence subgroups of the modular group. We have studied a selection of the main conjectures in the area: the Roelcke-Selberg and Phillips-Sarnak conjectures and Selberg's conjecture on exceptional eigenvalues. The first two concern the existence and nonexistence of an infinite discrete spectrum for certain types of Fuchsian groups and last states that there are no exceptional eigenvalues for congruence subgroups, or in other words, there is a specific spectral gap in the cuspidal spectrum.Our main theoretical result states that if the corresponding surface has a reflectional symmetry which preserves the cusps then the Laplacian on this surface has an infinite number of "new" discrete eigenvalues. We define old and new spaces of Maass cusp forms for noncongruence subgroups in a way that provides a natural generalization of the usual definition from congruence subgroups and give a method for determining the structure of the old space algorithmically.In addition to the theoretical result we also present computational data, including a table of subgroups of the modular group and eigenvalues of Maass forms for noncongruence subgroups. We also present, for the first time, numerical examples of both exceptional and (non-trivial) residual eigenvalues. To be able to (even heuristically) cerify lists of computed eigenvalues we also proved an explicit average Weyl's law in this setting.

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“…The computational aspects were foremost in mind when we obtained these formulas; we needed efficient algorithms for the Weil representation in order to compute vector-valued Poincaré series [39] and harmonic weak Maass forms [6]. The formula stated in the Main Theorem is implemented as part of a package [1] written in Sage [48] for computing with finite quadratic modules.…”

Section: Introductionmentioning

confidence: 99%

Weil representations associated with finite quadratic modules

Strömberg

2013

Math. Z.

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ABSTRACT. To a finite quadratic module, that is, a finite abelian group D together with a non-singular quadratic form Q : D → Q/Z, it is possible to associate a representation of either the modular group, SL 2 (Z), or its metaplectic cover, Mp 2 (Z), on C [D], the group algebra of D. This representation is usually called the Weil representation associated to the finite quadratic module. The main result of this paper is a general explicit formula for the matrix coefficients of this representation. The formula, which involves the p-adic invariants of the quadratic module, is given in a way which is easy to implement on a computer. The result presented completes an earlier result by Scheithauer for the Weil representation associated to a discriminant form of even signature.

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Computation of Harmonic Weak Maass Forms

Cited by 9 publications

References 32 publications

Weierstrass mock modular forms and elliptic curves

Weierstrass mock modular forms and elliptic curves

Noncongruence subgroups and Maass waveforms

Weil representations associated with finite quadratic modules

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