## Smoothness of the Hecke curve of $$\text{GL}_2$$ at critical Eisenstein points. (Lissité de la courbe de Hecke de $$\text{GL}_2$$ aux points Eisenstein critiques.)(English)Zbl 1095.11025

Let $$p$$ be a prime number and $$C$$ be the $$p$$-adic tame level 1 eigencurve introduced by Coleman and Mazur [R. F. Coleman, Invent. Math. 127, No. 3, 417–479 (1997; Zbl 0918.11026), R. Coleman and B. Mazur, in Galois representations in arithmetic algebraic geometry. Lond. Math. Soc. Lect. Note Ser. 254, 1–113 (1998; Zbl 0932.11030)]. $$C$$ is separable and reduced.
The authors prove (Théorème 1) that $$C$$ is smooth at the critical Eisenstein points (points $$x$$ satisfying, in particular, $$U_p(x)= p^{k-1}$$). The key ingredients in the proof are: the determination at these points of the schematic reducibility locus of the pseudo-character carried by $$C$$ restricted to a decomposition group at $$p$$ (Proposition 5), the fact that the Dirichlet $$L$$-functions only have simple zeros at integers, and Lemma 6.
Let $$k:C\to W$$ denote the (natural) morphism into the rigid space parametrizing continuous even $$p$$-adic characters on $$\mathbb{Z}^\times_p$$. It is flat and locally finite. The authors give necessary and sufficient conditions for étaleness of $$k$$ at critical Eisenstein points in terms of $$p$$-adic zeta values (Théorème 3).

### MSC:

 11F11 Holomorphic modular forms of integral weight 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11F80 Galois representations 11F85 $$p$$-adic theory, local fields

### Citations:

Zbl 0918.11026; Zbl 0932.11030
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