Functiones et Approximatio Commentarii Mathematici

Non-vanishing of derivatives of certain modular $L$-functions

Narasimha Kumar

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This paper is to show a non-vanishing property of the derivatives of certain class of $L$-functions. We study the non-vanishing and transcendence of special values of $L$-functions and their derivatives, attached to (cuspidal) Siegel-Hecke eigenforms of genus $2$, quadratic twists of classical Hecke eigenforms, and half-integral weight modular forms.

Article information

Funct. Approx. Comment. Math., Volume 51, Number 1 (2014), 121-132.

First available in Project Euclid: 24 September 2014

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Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11J81: Transcendence (general theory)
Secondary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F11: Holomorphic modular forms of integral weight 11F37: Forms of half-integer weight; nonholomorphic modular forms

Siegel modular forms twists of cusp forms half-integral weight modular forms $L$-functions non-vanishing special values transcendence


Kumar, Narasimha. Non-vanishing of derivatives of certain modular $L$-functions. Funct. Approx. Comment. Math. 51 (2014), no. 1, 121--132. doi:10.7169/facm/2014.51.1.6.

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  • A.N. Andrianov, Euler products that correspond to Siegel's modular forms of genus $2$, (Russian) Uspehi Mat. Nauk 29 (1974), no. 3 (177), 43–110.
  • A. Andrianov, Introduction to Siegel modular forms and Dirichlet series, Universitext. Springer, New York, 2009. xii+182 pp. ISBN: 978-0-387-78752-7
  • S. Böcherer, Bemerkungen über die Dirichletreihen von Koecher und Maass, Math. Gottingensis, Schriftenr. d. Sonderforschungsbereichs Geom. Anal. 68, (1986).
  • C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over $\ADOFQ$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939.
  • D. Bump, S. Friedberg, J. Hoffstein, Nonvanishing theorems for $L$-functions of modular forms and their derivatives, Invent. Math. 102 (1990), no. 3, 543–618.
  • F. Diamond, J. Shurman, A first course in modular forms, Graduate Texts in Mathematics, 228. Springer-Verlag, New York, 2005. xvi+436 pp.
  • G. van der Geer, Siegel modular forms and their applications, The 1-2-3 of modular forms, 181–245, Universitext, Springer, Berlin, 2008.
  • S. Gun, M.R. Murty, P. Rath, Transcendental nature of special values of $L$-functions, Canad. J. Math. 63 (2011), no. 1, 136–152.
  • W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32–72.
  • W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237–268.
  • W. Kohnen, On Hecke eigenforms of half-integral weight, Math. Ann. 293 (1992), no. 3, 427–431.
  • W. Kohnen, M. Kuss, Some numerical computations concerning spinor zeta functions in genus 2 at the central point, Math. Comp. 71 (2002), no. 240, 1597–1607.
  • K. Ono, C. Skinner, Non-vanishing of quadratic twists of modular $L$-functions, Invent. Math. 134 (1998), no. 3, 651–660.
  • M.R. Murty, V.K. Murty, Mean values of derivatives of modular $L$-series, Ann. of Math. (2) 133 (1991), no. 3, 447–475.
  • G. Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481.
  • N. Tanabe, Non-vanishing of derivatives of $L$-functions attached to Hilbert modular forms. Int. J. Number Theory 8 (2012), no. 4, 1099–1105.
  • A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551.