Edited by David Burns
Edited by Kevin Buzzard
Edited by Jan Nekovář
Publisher: Cambridge University Press
Print Publication Year: 2007
Online Publication Date:April 2010
Online ISBN:9780511721267
Paperback ISBN:9780521694155
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511721267.004
Subjects: Number theory
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Abstract
We axiomatise and generalise the “Hecke algebra” construction of the Coleman-Mazur Eigencurve. In particular we extend the construction to general primes and levels. Furthermore we show how to use these ideas to construct “eigenvarieties” parametrising automorphic forms on totally definite quaternion algebras over totally real fields.
Introduction
In a series of papers in the 1980s, Hida showed that classical ordinary eigenforms form p-adic families as the weight of the form varies. In the non-ordinary finite slope case, the same turns out to be true, as was established by Coleman in 1995. Extending this work, Coleman and Mazur construct a geometric object, the eigencurve, parametrising such modular forms (at least for forms of level 1 and in the case p > 2). On the other hand, Hida has gone on to extend his work in the ordinary case to automorphic forms on a wide class of reductive groups. One might optimistically expect the existence of nonordinary families, and even an “eigenvariety”, in some of these more general cases.
Anticipating this, we present in Part I of this paper (sections 2–5) an axiomatisation and generalisation of the Coleman-Mazur construction. In his original work on families of modular forms, Coleman in [10] developed Riesz theory for orthonormalizable Banach modules over a large class of base rings, and, in the case where the base ring was 1-dimensional, constructed the local pieces of a parameter space for normalised eigenforms. There are two places where we have extended Coleman's work.
pp. i-vi
pp. vii-viii
pp. ix-x
List of participants: Read PDF
pp. xi-xii
Stark–Heegner points and special values of L-series: Read PDF
pp. 1-23
Presentations of universal deformation rings: Read PDF
pp. 24-58
pp. 59-120
Nontriviality of Rankin-Selberg L-functions and CM points: Read PDF
pp. 121-186
A correspondence between representations of local Galois groups and Lie-type groups: Read PDF
pp. 187-206
Non-vanishing modulo p of Hecke L–values and application: Read PDF
pp. 207-269
Serre's modularity conjecture: a survey of the level one case: Read PDF
pp. 270-299
Two p-adic L-functions and rational points on elliptic curves with supersingular reduction: Read PDF
pp. 300-332
From the Birch and Swinnerton-Dyer Conjecture to non-commutative Iwasawa theory via the Equivariant Tamagawa Number Conjecture - a survey: Read PDF
pp. 333-380
The André-Oort conjecture - a survey: Read PDF
pp. 381-406
Locally analytic representation theory of p-adic reductive groups: a summary of some recent developments: Read PDF
pp. 407-437
Modularity for some geometric Galois representations - with an appendix by Ofer Gabber: Read PDF
pp. 438-470
The Euler system method for CM points on Shimura curves: Read PDF
pp. 471-547
Représentations irréductibles de GL(2, F) modulo p: Read PDF
pp. 548-563