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Article overview
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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 | Abstract: | The 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 | Services: | Forum | Review | PDF | Favorites |
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