Kyoto Journal of Mathematics

On mod p nonvanishing of special values of L-functions associated with cusp forms on GL2 over imaginary quadratic fields

Kenichi Namikawa

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Let f be a cusp form on GL2 over an imaginary quadratic field F of class number 1, and let p be an odd prime which satisfies some mild conditions. Then we show the existence of a finite-order Hecke character φ of FA× such that the algebraic part of the special value of L-functions of fφ at s=1 is a p-adic unit. This is an analogous result to the result of A. Ash and G. Stevens for GL2 over the field of rationals obtained in [AS].

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Kyoto J. Math., Volume 52, Number 1 (2012), 117-140.

First available in Project Euclid: 19 February 2012

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Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture)


Namikawa, Kenichi. On mod $p$ nonvanishing of special values of $L$ -functions associated with cusp forms on $\operatorname{GL}_{2}$ over imaginary quadratic fields. Kyoto J. Math. 52 (2012), no. 1, 117--140. doi:10.1215/21562261-1503782.

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  • [AS] A. Ash and G. Stevens, Modular forms in characteristic l and special values of their L-functions, Duke Math. J. 53 (1986), 849–868.
  • [Hi1] H. Hida, p-ordinary cohomology groups for SL(2) over number fields, Duke Math. J. 69 (1993), 259–314.
  • [Hi2] H. Hida, On the critical values of L-functions of GL(2) and GL(2)×GL(2), Duke Math. J. 74 (1994), 431–529.
  • [Hi3] H. Hida, “Non-critical values of adjoint L-functions for SL(2),” in Automorphic Forms, Automorphic Representations, and Arithmetic (Forth Worth, 1996), Proc. Symp. Pure Math. 66, Part I, Amer. Math. Soc., Providence, 1999, 123–175.
  • [OP] T. Ochiai and K. Prasanna, Two-variable Iwasawa theory for Hida families with complex multiplication, preprint.
  • [St] G. Stevens, The cuspidal group and special values of L-functions, Trans. Amer. Math. Soc. 291 (1985), 519–550.
  • [Su] H. S. Sun, Homological interpretation of a theorem of L. Washington, J. Number Theory 127 (2007), 47–63.
  • [Ur] E. Urban, Formes automorphes cuspidales pour GL(2) sur un corps quadratique imaginare. Valeurs spéciales de fonctions L et congruences, Compos. Math. 99 (1995), 283–324.
  • [Va] V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), 397–419.