# $p$-adic Cohomology and Arithmetic Applications (17w5118)

Arriving in Banff, Alberta Sunday, October 1 and departing Friday October 6, 2017

## Organizers

Christopher Lazda (Universita di Padova)

Ambrus Pal (Imperial College London)

Tomoyuki Abe (Kavli Institute for the Physics and Mathematics of the Universe)

Kiran Kedlaya (University of California, San Diego)

## Objectives

These exciting new trends emerging in the field are of course deeply interwoven, as we already mentioned, and by hosting this workshop we hope to encourage new progress in these areas by promoting both predictable and unpredictable synergies between them. For example, extending the scope of $p$-adic cohomology will require a more sophisticated view of the foundations of the subject in order to cope with these more general situations, and will in turn feed into many of the other areas of interest, in particular representation theory and the local Langlands correspondence by providing a more powerful language in which to discuss these questions. It is important to note that p-adic cohomology is often characterised by a plethora of different approaches to the subject, each of which has its own particular perspective and scope of application. By drawing together people working on all aspects of the theory, and building on the successful conference hosted by 2 of the organisers at Imperial College London in March 2015, we will provide a platform for a cross-fertilisation of the raft of new ideas in all these different approaches, and stimulate new developments across the whole breadth of the subject. Here we list a few topics and the expected interactions which we hope to foster via the workshop.

There is a closely related work of Huyghe, Patel, Schmidt and Strauch on localisation theorems in the setting of arithmetic $\mathcal D$-modules of Berthelot (see [16]) which proves that there is an equivalence of categories between the category of locally analytic admissible representations of some split reductive group over a finite extension of $\mathbb Q_p$, and the category of coadmissible arithmetic $\mathcal D$-modules over the rigid analytic space attached to the flag variety of the group. Similarly, the 6 operations formalism has been used by Abe [1] to prove a $p$-adic Langlands correspondence in the function field setting, and thus prove Deligne's ``petits camarades cristallins" conjecture on the existence of $p$-adic companions to compatible systems of $\ell$-adic Galois representations (at least over curves). Finally let us mention the work of Christian Johannsson, who studied the classicality for small slope overconvergent automorphic forms on certain higher dimensional Shimura varieties (see [19]), a work whose primary innovation is to use a robust formalism of $p$-adic cohomology. These works all represent different aspects of the $p$-adic Langlands program, both over number fields and function fields, and all rely heavily on the methods of $p$-adic cohomology. Promoting co-operation between the experts of this subject and the leaders of the foundational theory of $p$-adic cohomology will therefore be essential in progressing this exciting new direction of research.

This is very closely related to the study of $p$-adic properties of $L$-functions in characteristic $p$ where most of the work recently has been done on $1$-dimensional families of abelian varieties, for example [20], [29] and [33] which look at the refined Birch--Swinnerton-Dyer conjecture, the integrality of $p$-adic $L$-functions and the equivariant Tamagawa number conjecture, respectively. What is common in these works is the crucial use of integral $p$-adic cohomology theories predating the construction in [11], typically log crystalline cohomology. Therefore they are forced either to reduce the general case to the semi-stable one, or worse, restrict to the situation when the abelian scheme is semi-stable and the considered Galois covers of the base are tame. This demonstrate the limitations of these methods, but with sufficient progress on the finiteness properties of the the overconvergent de Rham--Witt complex we expect that this area would start to develop very rapidly.

#### Foundations and theory over non-perfect fields

Traditionally, $p$-adic cohomology theories have been expressed for varieties over perfect ground fields of characteristic $p$. While much of the theory still works over non-perfect fields, arithmetic considerations (in particular the general phenomenon of semistable reduction, as well as analogies with the $\ell$-adic theory) lead one to expect certain refinements of existing $p$-adic cohomologies (such as rigid cohomology) when working over such non-perfect fields. As a first step in this direction, the basics of this picture have been recently worked out over the simplest of non-perfect fields, namely Laurent series field in one variable, which has paved the way for a whole host of applications, such as a $p$-adic version of the weight monodromy conjecture and good reduction criteria for curves. This approach appears to be a rich source of new arithmetic results on varieties in characteristic $p$, although there is still much more foundational work to be done, both in the case of Laurent series fields and in terms of moving towards other examples such as global fields or higher dimensional local fields. On there other hand the groundbreaking work of Caro in the last decade has culminated in the proof of the existence of a 6 operations formalism in $p$-adic cohomology, including a full theory of weights [2]. We expect to see interactions between these two strides of research, and the workshop will provide the perfect environment to achieve this.#### The Langlands program and links with representation theory

One of the importance of the original theory of algebraic $D$-modules, which is over a field of characteristic zero, is that it has various application to representation theory. Beilinson--Bernstein correspondence is one of the most famous such examples. About 20 years ago Berthelot proposed a framework to establish a 6 functor formalism for schemes over fields of positive characteristics by pursuing an analogy with algebraic $D$-modules, and named it arithmetic $D$-module theory. With the above mentioned work of Caro, the foundations of the theory are essentially in place, and attention is turning to a new stage. As in the classical situation, it is hoped that the theory will prove a powerful tool for representation theory, including the $p$-adic Langlands program. A similar such application of $D$-module theory over rigid analytic spaces over $p$-adic fields has been already found by Ardakov--Wadsley [3], who used their theory to answer some representation theoretical problems which arose in the new $p$-adic local Langlands program.There is a closely related work of Huyghe, Patel, Schmidt and Strauch on localisation theorems in the setting of arithmetic $\mathcal D$-modules of Berthelot (see [16]) which proves that there is an equivalence of categories between the category of locally analytic admissible representations of some split reductive group over a finite extension of $\mathbb Q_p$, and the category of coadmissible arithmetic $\mathcal D$-modules over the rigid analytic space attached to the flag variety of the group. Similarly, the 6 operations formalism has been used by Abe [1] to prove a $p$-adic Langlands correspondence in the function field setting, and thus prove Deligne's ``petits camarades cristallins" conjecture on the existence of $p$-adic companions to compatible systems of $\ell$-adic Galois representations (at least over curves). Finally let us mention the work of Christian Johannsson, who studied the classicality for small slope overconvergent automorphic forms on certain higher dimensional Shimura varieties (see [19]), a work whose primary innovation is to use a robust formalism of $p$-adic cohomology. These works all represent different aspects of the $p$-adic Langlands program, both over number fields and function fields, and all rely heavily on the methods of $p$-adic cohomology. Promoting co-operation between the experts of this subject and the leaders of the foundational theory of $p$-adic cohomology will therefore be essential in progressing this exciting new direction of research.

#### The de Rham--Witt complex and Iwasawa theory

One of the original motivations of Grothendieck and Berthelot for inventing crystalline cohomology as a $p$-adic companion to the family of $\ell$-adic cohomologies produced by the étale theory was to explain $p$-torsion phenomenon. While integral crystalline cohomology achieves this for smooth and proper varieties, the extension to a `good' cohomology theory for arbitrary varieties, which reached its zenith in the proof of the 6 operations formalism by Caro, has been achieved only for rational coefficients, i.e. after tensoring with $Q$. This therefore still leaves open the question of what an integral $p$-adic theory should look like for open or singular varieties, which has been the subject of much recent work in the field, in particular the study of the overconvergent de Rham--Witt complex by Davis, Langer and Zink [10]. This now seems to provide a good candidate for smooth (but possibly open) varieties, although there are still many important open questions still to answer, including comparisons with other candidates such as integral Monsky--Washnitzer cohomology.This is very closely related to the study of $p$-adic properties of $L$-functions in characteristic $p$ where most of the work recently has been done on $1$-dimensional families of abelian varieties, for example [20], [29] and [33] which look at the refined Birch--Swinnerton-Dyer conjecture, the integrality of $p$-adic $L$-functions and the equivariant Tamagawa number conjecture, respectively. What is common in these works is the crucial use of integral $p$-adic cohomology theories predating the construction in [11], typically log crystalline cohomology. Therefore they are forced either to reduce the general case to the semi-stable one, or worse, restrict to the situation when the abelian scheme is semi-stable and the considered Galois covers of the base are tame. This demonstrate the limitations of these methods, but with sufficient progress on the finiteness properties of the the overconvergent de Rham--Witt complex we expect that this area would start to develop very rapidly.

#### Relations with function field arithmetic

We already mentioned the deep analogy and the cross-fertilisation which occurred between $p$-adic Hodge theory and its function field analogue at a crucial point of their development. However there are other areas of $p$-adic cohomology and function field arithmetic which are closely analogous and more intimate interaction would benefit both. For example a central object of study in function field arithmetic is Goss $L$-functions of function field motives (see for example [31]). These motives have a cohomological theory with a trace formula (see [8]), but the theory does not admit 6 operations. It would greatly benefit the topic if the methods of $p$-adic cohomology were successfully transported into it. On there hand the transcendence theory of special values of Goss $L$-functions is highly developed, but uses cohomological, Tannakian and analytic methods which would be very familiar to experts of $p$-adic cohomology (such as Dwork's trick) if they knew them. We hope that workshop could bring the birth of a brand new transcendence theory of $p$-adic periods in characteristic $p$.#### Other topics

Let us mention a few more topics which were intensively studied recently and which all have deep connections to the main topic of the proposed workshop, but which we could not describe in much detail for the lack of space: $p$-adic differential equations, crystalline fundamental groups and $p$-adic Simpson correspondence, $p$-adic Hodge theory and $p$-adic representations. We just remark in passing that $p$-adic differential equations play a fundamental role in the foundations of the theory, the study of crystalline fundamental groups is necessary for removing some of the thorny problems encountered in the Langlands program over function fields, and some form of a $p$-adic Simpson correspondence might be the way to overcome these, while $p$-adic Hodge theory remains perhaps the single most important application of $p$-adic cohomology via the theory of $p$-adic representations. So we expect that they will remain in the focus of research, and by inviting experts in these fields we will not only spread knowledge of some of the powerful new methods available in $p$-adic cohomology, but also to inspire those working in the field with potential new applications of their research.#### Bibliography

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