1 February 2003 Special values of anticyclotomic $L$-functions
V. Vatsal
Duke Math. J. 116(2): 219-261 (1 February 2003). DOI: 10.1215/S0012-7094-03-11622-1


The purpose of the paper is to extend and refine earlier results of the author on nonvanishing of the $L$-functions associated to modular forms in the anticyclotomic tower of conductor $p\sp \infty$ over an imaginary quadratic field. While the author's previous work proved that such $L$-functions are generically nonzero at the center of the critical strip, provided that the sign in the functional equation is $+1$, the present work includes the case where the sign is $-1$. In that case, it is shown that the derivatives of the $L$-functions are generically nonzero at the center. It is also shown that when the sign is $+1$, the algebraic part of the central critical value is nonzero modulo $\ell$ for certain $\ell$. Applications are given to the mu-invariant of the $p$-adic $L$-functions of M. Bertolini and H. Darmon. The main ingredients in the proof are a theorem of M. Ratner, as in the author's previous work, and a new "Jochnowitz congruence," in the spirit of Bertolini and Darmon.


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V. Vatsal. "Special values of anticyclotomic $L$-functions." Duke Math. J. 116 (2) 219 - 261, 1 February 2003. https://doi.org/10.1215/S0012-7094-03-11622-1


Published: 1 February 2003
First available in Project Euclid: 26 May 2004

zbMATH: 1065.11048
MathSciNet: MR1953292
Digital Object Identifier: 10.1215/S0012-7094-03-11622-1

Primary: 11F67
Secondary: 11F33

Rights: Copyright © 2003 Duke University Press


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Vol.116 • No. 2 • 1 February 2003
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