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29 March 2024
 
  » arxiv » 1907.9422

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Geometry of the eigencurve at CM points and trivial zeros of Katz $p$-adic $L$-functions
Adel Betina ; Mladen Dimitrov ;
Date 22 Jul 2019
AbstractThe primary goal of this paper is to study the geometry of the $p$-adic eigencurve at a point $f$ corresponding to a weight $1$ theta series $ heta_psi$ which is irregular at $p$. We show that $f$ belongs to exactly three or four irreducible components and study their mutual congruences. In particular, we show that the congruence ideal of one of the CM components has a simple zero at $f$ if, and only if, a certain $mathscr{L}$-invariant $mathscr{L}_-(psi_-)$ does not vanish. Further, using Roy’s Strong Six Exponential Theorem we show that at least one amongst $mathscr{L}_-(psi_-)$ and $mathscr{L}_-(psi_-^{-1})$ is non-zero. Combined with a divisibility proved by Hida and Tilouine, we deduce that the anti-cyclotomic Katz $p$-adic $L$-function of $psi_-$ has a simple (trivial) zero at $s=0$, if $mathscr{L}_-(psi_-)$ is non-zero, which can be seen as an anti-cyclotomic analogue of a result of Ferrero and Greenberg. Finally, we propose a formula for the linear term of the two-variable Katz $p$-adic $L$-function of $psi_-$ at $s=0$ which can be seen as an extension of a conjecture of Gross.
Source arXiv, 1907.9422
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