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Toward an explicit eigencurve for $G L (3)$

Avner Ash

Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA

E-mail Address: Avner.Ash@bc.edu

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and

David Pollack

Department of Mathematics and Computer Science, Wesleyan University Middletown, CT 06459, USA

E-mail Address: dpollack@wesleyan.edu

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https://doi.org/10.1142/S1793042117500658Cited by:0 (Source: Crossref)

Abstract

Starting with a numerically noncritical (at $p$ ) Hecke eigenclass $f$ in the homology of a congruence subgroup $Γ$ of ${SL}_{3} (ℤ)$ (where $p$ divides the level of $Γ$ ) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of $f$ to a Hecke eigenclass $F$ with coefficients in a module of $p$ -adic distributions. Then we show how to find to any desired degree of accuracy the germ of the projection $Z$ to weight space of the eigencurve around the point $z$ corresponding to the system of Hecke eigenvalues of $F$ . We do this under the conjecturally mild hypothesis that $Z$ is smooth at $z$ .

Keywords:

AMSC: 11F55, 11F85

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