Geometry of the eigencurve at critical Eisenstein series of weight 2

Dipramit Majumdar

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arXiv · 1307.4846

Geometry of the eigencurve at critical Eisenstein series of weight 2

Dipramit Majumdar

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Abstract

In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} is smooth in the full eigencurve C^{full}(l) and E_{2}^{crit_{p},ord_{l_{1}},ord_{l_{2}}} is non-smooth in the full eigencurve C^{full}(l_{1}l_{2}). Further, we show that, E_{2}^{crit_{p}}, is \'etale over the weight space in the eigencurve C(l). As a consequence, we show that level lowering conjecture of Paulin fails to hold at E_{2}^{crit_{p},ord_{l}}.

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Dipramit Majumdar. 2013-07-18. Geometry of the eigencurve at critical Eisenstein series of weight 2. https://arxiv.org/abs/1307.4846

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