Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions

Article, Preprint English OPEN
Bringmann, Kathrin ; Diamantis, Nikolaos ; Raum, Martin (2013)
  • Publisher: Elsevier
  • Journal: Advances in Mathematics, volume 233, issue 1, pages 115-134 (issn: 0001-8708)
  • Related identifiers: doi: 10.1016/j.aim.2012.09.025
  • Subject: Mathematics(all) | Mathematics - Number Theory | 11F67, 11F03
    arxiv: Mathematics::Number Theory

We introduce a new technique of completion for 1-cohomology which parallels the corresponding technique in the theory of mock modular forms. This technique is applied in the context of non-critical values of L-functions of GL(2) cusp forms. We prove that a generating series of non-critical values can be interpreted as a mock period function we\ud define in analogy with period polynomials. Further, we prove that non-critical values can be encoded into a sesquiharmonic Maass form. Finally, we formulate and prove an Eichler-Shimura-type isomorphism for the space of mock period functions.
  • References (36)
    36 references, page 1 of 4

    [1] G. Andrews, On the theorems of Watson and Dragonette for Ramanujan's mock theta functions, Amer. J. Math. 88 (1966), 454-490.

    [2] K. Bringmann, Asymptotics for rank partition functions, Trans. Amer. Math. Soc. 361 (2009), 3483-3500.

    [3] K. Bringmann, On certain congruences for Dyson's ranks, Int. J. Number Theory 5 (2009), 573-584.

    [4] K. Bringmann and B. Kane, Inequalities for differences of Dyson's rank for all odd moduli, Math. Res. Lett. 17 (2010), 927-942.

    [5] K. Bringmann, B. Kane, and R. Rhoades, Duality and differential operators for harmonic Maass forms, submitted for publication.

    [6] K. Bringmann, A. Folsom, and K. Ono, q-series and weight 3/2 Maass forms, Compositio Math. 145 (2009), 541-552.

    [7] K. Bringmann, P. Guerzhoy, Z. Kent, and K. Ono, Eichler-Shimura theory for mock modular forms, submitted for publication.

    [8] K. Bringmann and K. Ono, The f (q) mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), 243-266.

    [9] K. Bringmann and K. Ono, Lifting cusp forms to Maass forms with an application to partitions, Proc. Nat. Acad. Sci. (USA) 104 (2007), 3725-3731.

    [10] J. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), 45-90.

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