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##### This chapter has been cited by the following publications. This list is generated based on data provided by CrossRef.

Ludwig, Judith 2018. -INDISTINGUISHABILITY ON EIGENVARIETIES. Journal of the Institute of Mathematics of Jussieu, Vol. 17, Issue. 02, p. 425.

Hansen, David and Thorne, Jack A. 2017. On the $$\mathrm {GL}_n$$ GL n -eigenvariety and a conjecture of Venkatesh. Selecta Mathematica, Vol. 23, Issue. 2, p. 1205.

Ludwig, Judith 2017. A p-adic Labesse–Langlands transfer. manuscripta mathematica, Vol. 154, Issue. 1-2, p. 23.

Loeffler, David and Zerbes, Sarah Livia 2016. Rankin–Eisenstein classes in Coleman families. Research in the Mathematical Sciences, Vol. 3, Issue. 1,

Andreatta, Fabrizio Iovita, Adrian and Pilloni, Vincent 2015. p-adic families of Siegel modular cuspforms. Annals of Mathematics, p. 623.

Bellaïche, Joël and Dasgupta, Samit 2015. The -adic -functions of evil Eisenstein series. Compositio Mathematica, Vol. 151, Issue. 06, p. 999.

Bellovin, Rebecca 2015. p-adic Hodge theory in rigid analytic families. Algebra & Number Theory, Vol. 9, Issue. 2, p. 371.

Newton, James 2015. Towards local-global compatibility for Hilbert modular forms of low weight. Algebra & Number Theory, Vol. 9, Issue. 4, p. 957.

BRASCA, RICCARDO 2014. QUATERNIONIC MODULAR FORMS OF ANY WEIGHT. International Journal of Number Theory, Vol. 10, Issue. 01, p. 31.

Pottharst, Jonathan 2013. Analytic families of finite-slope Selmer groups. Algebra & Number Theory, Vol. 7, Issue. 7, p. 1571.

Urban, Eric 2011. Eigenvarieties for reductive groups. Annals of Mathematics, Vol. 174, Issue. 3, p. 1685.

Ramsey, Nick 2008. The half-integral weight eigencurve. Algebra & Number Theory, Vol. 2, Issue. 7, p. 755.

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• Print publication year: 2007
• Online publication date: April 2010

### Eigenvarieties

Summary

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.

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