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

Bellaïche, J.; Chenevier, G.

J. Inst. Math. Jussieu 5, No. 2, 333-349 (2006).

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).

Reviewer: Andrzej Dąbrowski (Szczecin)

MSC:

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

Keywords:

eigencurve; \(p\)-adic modular form; \(p\)-adic zeta function; Eisenstein series; Galois representation; Hodge-Tate theory

Citations:

Zbl 0918.11026; Zbl 0932.11030