The 1-2-3 of Modular Forms

Bruinier, Jan Hendrik; Geer, Gerard van der; Harder, Günter; Zagier, Don

doi:10.1007/978-3-540-74119-0

Cited by 227 publications

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“…As a result of our investigation, we produce further evidence for the Eichler-Shimura conjecture, as well as for a conjecture of Brumer and Kramer [5] associating abelian varieties to paramodular Siegel modular forms on Sp (4).…”

Section: Introductionsupporting

confidence: 56%

Examples of abelian surfaces with everywhere good reduction

Dembélé

Kumar

2015

Math. Ann.

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We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler-Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms with paramodular level structures. Mathematics Subject Classification

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Section: Introductionsupporting

confidence: 56%

Examples of abelian surfaces with everywhere good reduction

Dembélé

Kumar

2015

Math. Ann.

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“…For that purpose, we recall the definition of Rankin-Cohen brackets as given in p. 53 of [22], which differs slightly (see below) from the original one in Theorem 7.1 of [5].…”

Section: Resultsmentioning

confidence: 99%

“…[5], Theorem 7.1) that if f , g transform like modular forms of weight k and , respectively, then f , g n transforms like a modular form of weight k + + 2n and that f , g 0 = f · g. The interaction of the first Rankin-Cohen bracket, which itself fulfills the Jacobi identity of Lie brackets, and the regular product of modular forms give the graded algebra of modular forms the additional structure of a Poisson algebra (cf. [22], p. 53).…”

Section: Resultsmentioning

confidence: 99%

Mock modular forms and class number relations

Mertens

2014

Res Math Sci

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Purpose: Almost 40 years ago, H. Cohen formulated a conjecture about the modularity of a certain infinite family of functions involving the generating function of the Hurwitz class numbers of binary quadratic forms. Methods: We use techniques from the theory of modular, mock modular, and Jacobi forms. Result: In this paper, we prove a slight improvement of Cohen's original conjecture. Conclusions: From our main result, we derive so far unknown recurrence relations for Hurwitz class numbers.

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“…In [1], the local divisor class group of a generic point of a one-dimensional boundary component has been treated, even more general in the context of the group O(2, n). Another case that has been treated by Bruinier are the cusps of Hilbert modular surfaces in [4]. Here we have to consider the zero-dimensional cusps of Siegel threefolds.…”

Section: Then the Equation σ(St ) = 1 Has A Solution T ∈ T Hence Imentioning

confidence: 99%

“…Due to the sign (H) the product is not invariant under the group P. The failure of the invariance will enable us to compute the cohomology class of the divisor H(S, d) in the local divisor class group. We refer to [1] and to the article of Bruinier in [4] for similar constructions.…”

Section: We Want To Construct a Holomorphic Function With Divisor H(smentioning

confidence: 99%

On Siegel three-folds with a projective Calabi–Yau model

Freitag

Manni

2011

Communications in Number Theory and Physics

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In two recent papers we described some Siegel modular threefolds which admit a weak Calabi-Yau model. Not all of them admit a projective model. In fact, Bert van Geemen, in a private communication, pointed out a significative example which cannot admit a projective model. A weak Calabi-Yau threefold is projective if, and only if, it admits a Kaehler metric. The purpose of this paper is to exhibit criteria for the projectivity, to treat several examples, and to compute their Hodge numbers. Some of these Calabi-Yau manifolds seem to be new. We obtain a rigid Calabi-Yau manifold with Euler number 4. IntroductionIn the papers [9, 10] we described some Siegel modular three-folds which admit a weak Calabi-Yau model. 1 Not all of them admit a projective model. In fact, Bert van Geemen, in a private communication, pointed out a significative example which cannot admit a projective model. His comment was a starting motivation for this paper. We mention that a weak CalabiYau three-fold is projective if, and only if, it admits a Kaehler metric. The purpose of this paper is to exhibit criteria for the projectivity, to treat several examples and to compute their Hodge numbers. Some of these CalabiYau manifolds seem to be new. For example, we obtain a rigid Calabi-Yau manifold with Euler number 4.

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The 1-2-3 of Modular Forms

Abstract: The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cited by 227 publications

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Examples of abelian surfaces with everywhere good reduction

Examples of abelian surfaces with everywhere good reduction

Mock modular forms and class number relations

On Siegel three-folds with a projective Calabi–Yau model

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