##
**Eigenvarieties.**
*(English)*
Zbl 1230.11054

Burns, David (ed.) et al., \(L\)-functions and Galois representations. Based on the symposium, Durham, UK, July 19–30, 2004. Cambridge: Cambridge University Press (ISBN 978-0-521-69415-5/pbk). London Mathematical Society Lecture Note Series 320, 59-120 (2007).

From the text: 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.

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 non-ordinary 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, R. F. Coleman in [Invent. Math. 127, No. 3, 417–479 (1997; Zbl 0918.11026)] 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.

Firstly, we set up Coleman’s Fredholm theory and Riesz theory (in sections 2 and 3 respectively) in a slightly more general situation, so that they can be applied to spaces such as direct summands of orthonormalizable Banach modules; the motivation for this is that at times in the theory we meet Banach modules which are invariants of orthonormalizable Banach modules under the action of a finite group; such modules are not necessarily orthonormalizable, but we want to use Fredholm theory anyway.

And secondly we show in sections 4–5 that given a projective Banach module and a collection of commuting operators, one of which is compact, one can glue the local pieces constructed by Coleman to form an eigenvariety, in the case where the base ring is an arbitrary reduced affinoid. At one stage we are forced to use Raynaud’s theory of formal models; in particular, this generalisation is not an elementary extension of Coleman’s ideas.

The resulting machine can be viewed as a construction of a geometric object from a family of Banach spaces equipped with certain commuting linear maps. Once one has this machine, one can attempt to feed in Banach spaces of “overconvergent automorphic forms” into the machine, and get “eigenvarieties” out. We extend the results of R. F. Coleman and B. Mazur [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; Zbl 0932.11030)] in Part II of this paper (sections 6 and 7), constructing an eigencurve using families of overconvergent modular forms, and hence removing some of the assumptions on \(p\) and \(N\) in the main theorems of [Coleman–Mazur, loc. cit.]. Note that here we do not need the results of section 4, as weight space is 1-dimensional and Coleman’s constructions are enough.

There are still technical geometric problems to be resolved before one can give a definition of an overconvergent automorphic form on a general reductive group, but one could certainly hope for an elementary definition if the group in question is compact mod centre at infinity, as the geometry then becomes essentially non-existent. As a concrete example of this, we propose in Part III (sections 8–13) a definition of an overconvergent automorphic form in the case when the reductive group is a compact form of \(\text{GL}_2\) over a totally real field, and apply our theory to this situation to construct higher-dimensional eigenvarieties.

Chenevier has constructed Banach spaces of overconvergent automorphic forms for compact forms of \(\text{GL}_n\) over \(\mathbb Q\) and one can feed his spaces into the machine also to get eigenvarieties for these unitary groups.

This work began in 2001 during a visit to Paris-Nord. In fact the author believes that he was the first to coin the phrase “eigenvarieties”, in 2001. Part I of this paper was written at that time, as well as some of Part III. The paper then remained in this state for three years, and the author most sincerely thanks Chenevier for encouraging him to finish it off. In fact Theorem 4.6 of this paper is assumed both by G. Chenevier in [J. Reine Angew. Math. 570, 143–217 (2004; Zbl 1093.11036)], and A. Yamagami in [Proceedings of the Japan-Korea joint seminar on number theory, October 9–12, 2004, Kuju, Japan. Fukuoka: Kyushu University. 193–209 (2004; Zbl 1082.11031), J. Number Theory 123, No. 2, 363–387 (2007; Zbl 1160.11028)], who independently announced results very similar to those in Part III of this paper, the main difference being that Yamagami works with the \(U\) operator at only one prime above \(p\) and fixes weights at the other places, hence his eigenvarieties can have smaller dimension than ours, but they see more forms (they are only assumed to have finite slope at one place above \(p\)).

A lot has happened in this subject since 2001. Matthew Emerton has recently developed a general theory of eigenvarieties which in many cases produces cohomological eigenvarieties associated to a large class of reductive algebraic groups. As well as Coleman and Mazur, many other people (including Emerton, Ash and Stevens, Skinner and Urban, Mazur and Calegari, Kassaei, Kisin and Lai, Chenevier, and Yamagami), have made contributions to the area, all developing constructions of eigenvarieties in other situations. We finish this introduction with an explanation of the relationship between Emerton’s work and ours. Emerton’s approach to eigenvarieties is more automorphic and more conceptual than ours. His machine currently needs a certain spectral sequence to degenerate, but this degeneration occurs in the case of the Coleman-Mazur eigencurve and hence Emerton has independently given a construction of this eigencurve for arbitrary \(N\) and \(p\) as in Part II of this paper. However, Emerton’s construction is less “concrete” and in particular the results in [the author and F. Calegari, Doc. Math., J. DMV Extra Vol., 211–232 (2006; Zbl 1138.11015)] and [Compos. Math. 141, No. 3, 605–619 (2005; Zbl 1187.11020)] rely on the construction of the 2-adic eigencurve presented in this paper. On the other hand Emerton’s ideas give essentially the same construction of the eigenvariety associated to a totally definite quaternion algebra over a totally real field, in the sense that one can check that his more conceptual approach, when translated down, actually becomes equivalent to ours.

For the entire collection see [Zbl 1130.11004].

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 non-ordinary 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, R. F. Coleman in [Invent. Math. 127, No. 3, 417–479 (1997; Zbl 0918.11026)] 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.

Firstly, we set up Coleman’s Fredholm theory and Riesz theory (in sections 2 and 3 respectively) in a slightly more general situation, so that they can be applied to spaces such as direct summands of orthonormalizable Banach modules; the motivation for this is that at times in the theory we meet Banach modules which are invariants of orthonormalizable Banach modules under the action of a finite group; such modules are not necessarily orthonormalizable, but we want to use Fredholm theory anyway.

And secondly we show in sections 4–5 that given a projective Banach module and a collection of commuting operators, one of which is compact, one can glue the local pieces constructed by Coleman to form an eigenvariety, in the case where the base ring is an arbitrary reduced affinoid. At one stage we are forced to use Raynaud’s theory of formal models; in particular, this generalisation is not an elementary extension of Coleman’s ideas.

The resulting machine can be viewed as a construction of a geometric object from a family of Banach spaces equipped with certain commuting linear maps. Once one has this machine, one can attempt to feed in Banach spaces of “overconvergent automorphic forms” into the machine, and get “eigenvarieties” out. We extend the results of R. F. Coleman and B. Mazur [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; Zbl 0932.11030)] in Part II of this paper (sections 6 and 7), constructing an eigencurve using families of overconvergent modular forms, and hence removing some of the assumptions on \(p\) and \(N\) in the main theorems of [Coleman–Mazur, loc. cit.]. Note that here we do not need the results of section 4, as weight space is 1-dimensional and Coleman’s constructions are enough.

There are still technical geometric problems to be resolved before one can give a definition of an overconvergent automorphic form on a general reductive group, but one could certainly hope for an elementary definition if the group in question is compact mod centre at infinity, as the geometry then becomes essentially non-existent. As a concrete example of this, we propose in Part III (sections 8–13) a definition of an overconvergent automorphic form in the case when the reductive group is a compact form of \(\text{GL}_2\) over a totally real field, and apply our theory to this situation to construct higher-dimensional eigenvarieties.

Chenevier has constructed Banach spaces of overconvergent automorphic forms for compact forms of \(\text{GL}_n\) over \(\mathbb Q\) and one can feed his spaces into the machine also to get eigenvarieties for these unitary groups.

This work began in 2001 during a visit to Paris-Nord. In fact the author believes that he was the first to coin the phrase “eigenvarieties”, in 2001. Part I of this paper was written at that time, as well as some of Part III. The paper then remained in this state for three years, and the author most sincerely thanks Chenevier for encouraging him to finish it off. In fact Theorem 4.6 of this paper is assumed both by G. Chenevier in [J. Reine Angew. Math. 570, 143–217 (2004; Zbl 1093.11036)], and A. Yamagami in [Proceedings of the Japan-Korea joint seminar on number theory, October 9–12, 2004, Kuju, Japan. Fukuoka: Kyushu University. 193–209 (2004; Zbl 1082.11031), J. Number Theory 123, No. 2, 363–387 (2007; Zbl 1160.11028)], who independently announced results very similar to those in Part III of this paper, the main difference being that Yamagami works with the \(U\) operator at only one prime above \(p\) and fixes weights at the other places, hence his eigenvarieties can have smaller dimension than ours, but they see more forms (they are only assumed to have finite slope at one place above \(p\)).

A lot has happened in this subject since 2001. Matthew Emerton has recently developed a general theory of eigenvarieties which in many cases produces cohomological eigenvarieties associated to a large class of reductive algebraic groups. As well as Coleman and Mazur, many other people (including Emerton, Ash and Stevens, Skinner and Urban, Mazur and Calegari, Kassaei, Kisin and Lai, Chenevier, and Yamagami), have made contributions to the area, all developing constructions of eigenvarieties in other situations. We finish this introduction with an explanation of the relationship between Emerton’s work and ours. Emerton’s approach to eigenvarieties is more automorphic and more conceptual than ours. His machine currently needs a certain spectral sequence to degenerate, but this degeneration occurs in the case of the Coleman-Mazur eigencurve and hence Emerton has independently given a construction of this eigencurve for arbitrary \(N\) and \(p\) as in Part II of this paper. However, Emerton’s construction is less “concrete” and in particular the results in [the author and F. Calegari, Doc. Math., J. DMV Extra Vol., 211–232 (2006; Zbl 1138.11015)] and [Compos. Math. 141, No. 3, 605–619 (2005; Zbl 1187.11020)] rely on the construction of the 2-adic eigencurve presented in this paper. On the other hand Emerton’s ideas give essentially the same construction of the eigenvariety associated to a totally definite quaternion algebra over a totally real field, in the sense that one can check that his more conceptual approach, when translated down, actually becomes equivalent to ours.

For the entire collection see [Zbl 1130.11004].

Reviewer: Olaf Ninnemann (Berlin)

### MSC:

11F33 | Congruences for modular and \(p\)-adic modular forms |

11F85 | \(p\)-adic theory, local fields |

11F32 | Modular correspondences, etc. |

11M41 | Other Dirichlet series and zeta functions |