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The eigencurve. (English) Zbl 0932.11030
Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9–18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 1-113 (1998).
Let \(p\) be an odd prime number and \(N\geq 1\) be an integer relatively prime to \(p\). Let \(\Lambda_N:= \mathbb{Z}_p [[(\mathbb{Z}/ N\mathbb{Z})^\times \times \mathbb{Z}_p^\times]]\). H. Hida has constructed a finite flat \(\Lambda_N\)-algebra which is universal for slope \(O\) (overconvergent) eigenforms of tame level \(N\), and such that the associated rigid analytic space \(C_{p,N}^0\) parametrizes \(p\)-adic analytic families of slope \(O\) eigenforms [Ann. Sci. Ec. Norm. Supér. (4) 19, 231-273 (1986; Zbl 0607.10022); Invent. Math. 85, 545-613 (1986; Zbl 0612.10021)]. R. Coleman [Invent. 127, 417-479 (1997; Zbl 0918.11026)] established a satisfactory analogue of Hida’s result for finite slope classical eigenforms.
It is the aim of this article to construct a rigid analytic curve (“the eigencurve”) \(C_p\) which parametrizes all finite slope overconvergent \(p\)-adic eigenforms of tame level 1 (Theorem E). It turns out that Hida’s rigid space \(C_{p,1}\) occurs as a component part of \(C_p\). The eigencurve \(C_p\) has a natural embedding into the rigid analytic space \(X_p\times \mathbb{A}^1\), where \(X_p\) is the rigid analytic space attached to the universal deformation ring of certain Galois (pseudo-) representations. If \(c\in C_p\) corresponds to the overconvergent eigenform \(f_c\), then the first coordinate of \(c\) is the Galois (pseudo-) representation attached to \(f_c\) and the second one is the converse of the \(U_p\)-eigenvalue of \(f_c\). The authors give two different constructions (by means of Fredholm determinants, and the Banach module theory respectively) of rigid analytic curves which parametrize the collection of all overconvergent eigenforms of tame level 1 and of finite slope. Consequences of the relationship between them are summarized in section 1.5 (Theorems A, B, C, G).
For the entire collection see [Zbl 0905.00052].

11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
11G20 Curves over finite and local fields
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)