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The workshop will bring together researchers currently investigating the p-adic variation of motives. The goal of the workshop is two-fold: to report on recent progress, and to focus attention on the many open problems that remain in the field.
Recent progress. There has been significant progress in the study of p-adic variation of motives in the last year or so, and one goal of the workshop is to report on this progress.
Relations with p-adic Hodge theory. The p-adic Galois representations attached to classical motives are potentially semi-stable in the sense of Fontaine, and so it is quite reasonable to hope for relations between Fontaine's theory and the theory of p-adic variation of motives.
Recently concrete progress has been made in establishing this connection.Iovita has associated a Fontaine-Dieudonn\'ee module to a Hida family and shown that it has rank one over the parameter space. More generally, Kisin has made an extensive analysis of the p-adic Hodge theory of the family of Galois representations attached to Coleman's finite-slope families. As a consequence, he has verified the Fontaine-Mazur conjecture for such Galois representations: if the Galois representation occurring in such a family is potentially semi-stable at p then it arises from a classical modular form.
Slopes of modular forms. Buzzard has extensive computational evidence suggesting some remarkable patterns in the variations of slopes of classical modular forms. Together with Frank Calegari he has proven them in a special case. More precisely, they have shown that if m\ge 0 and if n is sufficiently large then for any k>0 such that 2^n|k, the first few 2-adic slopes of weight k forms on \Gamma_0(2) are equal to 1+2\,\ord_2((3n)!/n!), each slope occurring with multiplicity one.
Gouvea has also performed extensive computations of slopes, and likewise has discovered suprising and so-far unexplained phenomena.
p-adic deformations of automorphic forms. One approach to constructing a family of p-motives containing a given classical motive is to first construct a p-adic analytic deformation of a corresponding automorphic form, and then to construct an accompanying deformation of the p-adic Galois representation attached to the given motive. Indeed, this was the approach used by Hida in order to deform the motives attached to ordinary modular forms, and by Coleman and Mazur in order to deform the motives attached to finite-slope modular forms.
Recently, Kassaei has developed a theory of overconvergent p-adic automorphic forms attached to certain quaternion algebras, which allows him to construct p-adic analytic families of finite slope eigenforms for these quaternion algebras. As with Coleman's construction of families of modular eigenforms of finite slope, his approach depends on exploiting the p-adic geometry of the appropriate Shimura curve, using techniques of rigid analysis.
Ash and Stevens have developed another approach to interpolating finite slope automorphic forms on GL_n, using cohomological methods. More precisely, they are able to find p-adic analytic families of group cohomology classes that are finite-slope eigenclasses for an appropriate Hecke operator at p.
Emerton has initiated yet another approach to the problem of p-adically interpolating automorphic forms, which uses methods from the emerging theory of locally p-adic analytic representations of p-adic reductive groups. Applying his results to the representations constructed out of the cohomology of congruence subgroups, he is again able to construct p-adic analytic families of finite slope eigenclasses. When the reductive group in question gives rise to a Shimura variety, the functorial nature of his approach allows him to also construct corresponding families of Galois representations.
p-adic variation over finite fields. Wan has recently established Dwork's conjecture on the meromorphicity of the fixed slope parts of the L-function attached to families of varieties defined over a finite field. Overconvergence and p-adic limiting techniques play a key role in the proof.
Open problems. There are many fundamental open questions related to the p-adic variation of motives, and a second goal of the workshop is to focus the attention of experts on these problems, and develop some lines of attack.
Some of the most pressing open questions are: Can one establish the existence of the universal family (or any non-trivial family) of p-motives containing a given classical motive? What is the geometric nature of this universal family? How are the classical motives situated in this universal family? Can one intrinsically characterize those p-adic Galois representations which are p-motives belonging to such a deformation space (for example, in terms of their p-adic Hodge theory)? How should one formulate the ``main conjecture of Iwasawa theory'' for a general family of p-motives?
In the case of the eigencurve, the computations of Buzzard provide a hint as to the answer to the second of these questions, and the results of Kisin go a long way to suggesting an answer to the fourth. The representation-theoretic methods of Emerton provides a new point of view on the first and third. It seems likely that before we can fully understand the form that a ``main conjecture'' should take, it will be crucial to understand more closely the relationship between the variation of p-motives and p-adic Hodge theory.
By bringing together those people who are investigating these and related questions, and sharing our knowledge of the most current developments and techniques available in the field, the workshop should lead to a significant improvement in our understanding of all three of these questions, and the many other important open questions in the field.
Mathematical implications of the workshop. Each improvement in our understanding of the p-adic variation of motives has led to concrete developments in number theory and arithmetic geometry. For example, Hida's theory provided a crucial tool in Greenberg and Stevens' proof of the elliptic curve case of the Mazur-Tate-Teitelbaum conjecture on the L-invariant. As mentioned above, it also led to Mazur's theory of deformations of Galois representations. This theory in turn was fundamental to Taylor and Wiles' proof of Fermat's last theorem, as well as to recent progress of Buzzard, Dickinson, Shepherd-Barron and Taylor on the Artin conjecture. More recently, Coleman's construction of one parameter families of non-ordinary finite-slope eigenforms has been used by Stevens to prove the general case of Mazur-Tate-Teitelbaum conjecture, and (as mentioned as above) by Kisin to establish cases of the Fontaine-Mazur conjecture.
In light of this history, we expect that further developments in the theory of p-adic variation will have similarly far-reaching consequences for number theory as a whole. The proposed workshop would play a pivotal role in stimulating such developments.
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