On a completed generating function of locally harmonic Maass forms
نویسندگان
چکیده
منابع مشابه
Algebraicity of Harmonic Maass Forms
In 1947 D. H. Lehmer conjectured that Ramanujan’s tau-function never vanishes. In the 1980s, B. Gross and D. Zagier proved a deep formula expressing the central derivative of suitable Hasse-Weil L-functions in terms of the Neron-Tate height of a Heegner point. This expository article describes recent work (with J. H. Bruinier and R. Rhoades) which reformulates both topics in terms of the algebr...
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Harmonic Maass forms have recently been related to many different topics in number theory: Ramanujan’s mock theta functions, Dyson’s rank generating functions, Borcherds products, and central values and derivatives of quadratic twists of modular L-functions. Motivated by these connections, we obtain exact formulas for the coefficients of harmonic Maass forms of non-positive weight, and we obtai...
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Harmonic weak Maass forms of half-integral weight are the subject of many recent works. They are closely related to Ramanujan’s mock theta functions, their theta lifts give rise to Arakelov Green functions, and their coefficients are often related to central values and derivatives of Hecke L-functions. We present an algorithm to compute harmonic weak Maass forms numerically, based on the automo...
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The object of this paper is to initiate a study of the Fourier coefficients of a weight one harmonic Maass form and relate them to the complex Galois representation associated to a weight one newform, which is the form’s image under a certain differential operator. In this paper, our focus will be on weight one dihedral newforms of prime level p ≡ 3 (mod 4). In this case we give properties of t...
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This short course is an introduction to the theory of harmonic Maass forms, mock modular forms, and quantum modular forms. These objects have many applications: black holes, Donaldson invariants, partitions and q-series, modular forms, probability theory, singular moduli, Borcherds products, central values and derivatives of modular L-functions, generalized Gross-Zagier formulae, to name a few....
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ژورنال
عنوان ژورنال: Compositio Mathematica
سال: 2014
ISSN: 0010-437X,1570-5846
DOI: 10.1112/s0010437x13007719