Unearthing the visions of a master: harmonic Maass forms and number theory
summary
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Underline (red), Dec 21, 2013, 2:38 AM: Section 14 (Borcherds Products). Recently, Borcherds [42, 43, 44] provided a striking description for the exponents in the naive infinite product expansion of many modular forms, those forms with a Heegner divisor. He proved that the exponents in these infinite product expansions are certain coefficients of modular forms of weight 1/2
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Underline (red), Dec 21, 2013, 2:38 AM: In the 1980s, Waldspurger [205], and Kohnen and Zagier [135, 136, 137] established that certain half-integral weight modular forms serve as generating functions of a new type. Using the Shimura correspondence [192], they proved that certain coefficients of half-integral weight cusp forms essentially are square-roots of central values of quadratic twists of modular L-functions.
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Underline (red), Dec 21, 2013, 2:38 AM: Here we present an example which numerically illustrates the most general form of Theorem 5.1 for the weight 2 newform G which corresponds to the conductor 37 elliptic curve
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Underline (red), Dec 21, 2013, 2:38 AM: The table below includes some of the coefficients of a suitable f which were numerically computed by F. Str¨ omberg (also see [69])
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Underline (red), Dec 21, 2013, 2:38 AM: Theorem 6.1. (Zwegers) The function H(z) is a vector-valued real analytic modular form of weight 1/2 satisfying
Underline (red), Dec 21, 2013, 2:38 AM: 6.2. Zwegers's weight 1/2 non-holomorphic Jacobi form. In his thesis, Zwegers constructed weight 1/2 harmonic Maass forms by making use of the transformation prop-erties of Lerch sums. Here we briefly recall some of these important results which address the difficult problem of constructing weight 1/2 harmonic Maass forms.
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Underline (red), Dec 21, 2013, 2:38 AM: This observation was critical in Zwegers's work on Ramanujan's mock theta functions.
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Underline (red), Dec 21, 2013, 2:38 AM: Example 14.8 (A generalized Borcherds product for Ramanujan's ω(q)). Here we give an example of a generalized Borcherds product arising from Ramanujan's mock theta function
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Underline (red), Dec 21, 2013, 2:38 AM: In view of the Birch and Swinnerton-Dyer Conjecture, and Theorem 15.2, we are com-pelled to study central values and derivatives of weight 2 modular L-functions. In this direction we have the celebrated works of Kohnen, Zagier and Waldspurger, and also the work of Gross and Zagier [108]. It turns out that the Fourier coefficients of half-integral weight cusp forms often inter-polate the “square-roots” of the central critical values of the L-functions of the quadratic twists of even weight newforms.
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Underline (red), Dec 30, 2013, 4:48 PM: There is a unique newform, say f(z) ∈ Snew 2k (N), associated to g(z) under Shimura's correspondence. The coefficients of g(z) determine the central critical values of many of the quadratic twists L(f, χD, s).
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Underline (red), Dec 30, 2013, 5:16 PM: The author and Bruinier [67] have generalized this theorem of Waldspurger and Kohnen to prove that the Fourier coefficients of weight 1/2 harmonic Maass forms encode the vanishing and nonvanishing of both the central values and derivatives of quadratic twists of weight 2 modular L-functions. Here we describe a special case of the main result of [67].
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Underline (red), Dec 30, 2013, 5:16 PM: [135] W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982) 32-72. [136] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237-268. [137] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, pages 175–198.
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Underline (red), Dec 30, 2013, 5:16 PM: [205] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, pages 375–484.