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| Matt Baker | Brian Conrad | Kiran Kedlaya | Jeremy Teitelbaum |
Berkovich spaces are very well-suited to studying the problem in general, and it is natural to consider the purely analytic problem of making quotients for etale equivalence relations with quasi-compact diagonal. The local structure theory for Berkovich spaces can be used to reduce the general existence problem to the case of finite etale equivalence relations, as well as to investigate existence criteria in the finite etale case. (In contrast with the scheme case, affinoids are not open in a Berkovich space, and a Berkovich space may not admit a separated open neighborhood around each of its points.) Overall, the study of many facets of this quotient construction problem will provide a lot of hands-on experience with Berkovich spaces and their many features. At the Winter School the problem will be broken into several parts, and perhaps different groups will work on different parts so that by the end some good progress will be made on the total problem.
To work on this problem, some familiarity with the etale topology and algebraic spaces will have to be acquired in advance. An excellent reference for learning the basics of the etale topology (and related notions in descent theory) is Chapters 1 and 6 of the book "Neron Models", and Knutson's book "Algebraic spaces" is a nice reference on the basics of algebraic spaces. (Read enough to become comfortable with what an algebraic space is, and to understand Knutson's discussion of analytification over C early in the book. It is not at all necessary to read most of the book.) Some further background reading in rigid geometry and Berkovich spaces will be provided later.
We will start with the Monsky-Washnitzer cohomology of a smooth affine variety over a finite field. This includes constructing the cohomology using a lift to characteristic zero, checking independence of the construction from choices, proving an excision formula, and checking the Lefschetz trace formula. (This all follows the original papers of Monsky-Washnitzer.) By sheafifying, we will obtain a cohomology theory for general smooth varieties, which is Berthelot's rigid cohomology (also known as crystalline cohomology with field coefficients).
We will then formulate the comparison theorem between rigid cohomology and de Rham cohomology for a smooth proper scheme over $\mathfrak{o}_K$, the ring of integers in a $p$-adic field $K$. This means that the de Rham cohomology of a variety over $K$ with good reduction carries a Frobenius action, even though the Frobenius map on the special fibre does not typically lift to characteristic zero. It also means that we can use what we know about the de Rham cohomology of the generic fibre to compute the zeta function of the special fibre; this paradigm, applied to hyperelliptic curves as in \cite{kedlaya}, forms the basis of applications of $p$-adic cohomology in cryptography (the ``practice'' of the title).
We will then consider Gauss-Manin connections from the algebraic point of view (following \cite{katz-oda}), with an eye towards using these to compute Frobenius actions in de Rham cohomology (following Lauder). This theme will be carried further in the project.
We will first compute the Hyodo-Kato Frobenius and monodromy actions on elliptic curves using the Tate uniformation (following \cite{lestum, coleman-iovita}). We will then recompute it (at least conjecturally; if time permits, we may try to prove correctness of these results using \cite{coleman-iovita}) using Gauss-Manin connections, particularly in the Legendre family (the universal family of elliptic curves with rational 2-torsion). One key question we will be investigating, to which I don't know the answer: if one encounters an elliptic curve with multiplicative reduction in a family such as the Legendre family, one apparently gets some Hyodo-Kato Frobenius on its de Rham cohomology from the global Frobenius action on the connection, but which one?
The $p$-adic upper half plane $\X$ over a $p$-adic field $K$ is a one-dimensional rigid analytic space whose points in any complete extension $L/K$ are given by
$$ \X(L)={\bf P}^{1}(L)\backslash{\bf P}^{1}(K). $$
This space was first introduced by Mumford, where it plays a key role in the generalization to higher genus of Tate's theory of $p$-adic uniformization of elliptic curves with semistable reduction. Slightly later, Drinfeld and Cerednik showed that appropriate quotients of this space by discrete arithmetic subgroups of $\PGL_{2}(K)$ coming from quaternion algebras yield Shimura curves. Since that time, through work of Morita, Schneider, Bertolini, Darmon, Iovita, the authors, and many others, this space and its relationships to arithmetic have been the subject of intensive study. In this course, we will study some aspects of this recent work.
The first part of the course will be a construction of the $p$-adic upper half plane $\X$ as a rigid space and a study of its relationship to the Bruhat-Tits tree $T$ of $\PGL_{2}$. This tree $T$ classifies norms on a fixed two dimensional vector space $V$ up to scaling, and there is a map $$ r:\X\to T$$ that commutes with the action of $\PGL_{2}$ on both spaces. We will also study the compactification of $T$ obtained by adding its set of ends, a set homeomorphic to ${\bf P}^{1}(K)$.
In the second part of the course, we will explore the relationship, first described by Schneider, between rigid analytic one-forms on $\X$, ``harmonic" functions on the edges of the tree $T$, and distributions on the (common) boundary ${\bf P}^{1}(K)$ of $\X$ and $T$. The key ideas in this part of the course are the residue map, which carries rigid one-forms to harmonic functions on $T$, and the integral transform, or ''Poisson Kernel," which carries measures on ${\bf P}^{1}(K)$ back to rigid one-forms.
In the third part, we will discuss a second, entirely different, construction of measures on ${\bf P}^{1}(K)$ due to Darmon that also uses the tree, but that blends classical harmonic forms and rigid geometry in a remarkable way.
In the last part of the course, we will examine some of the arithmetic applications of the theory developed so far. In particular, we will describe the construction of two different $\mathcal{L}$ invariants. Such invariants, first observed experimentally in \cite{MTT} , relate the special values of $p$-adic $L$-functions and classical $L$-functions of a modular form $f$ in cases where the two functions have functional equations of different signs; in that case, the $p$-adic $L$-function has an extra order of vanishing, and its critical value differs from that of the classical one by a $p$-adic number $\mathcal{L}(f)$. We will describe the quaternionic construction of an $\mathcal{L}(f)$-invariant due to the second author, and the ``double-integral" construction due to Darmon and Orton. As time permits, we will discuss Orton's proof that her $\mathcal{L}$-invariant depends only on the Galois representation of $f$ locally at $p$, as originally conjectured in \cite{MTT}. Also time permitting, we will describe Breuil's definition of the $\mathcal{L}$-invariant \cite{b}. \newpage
\bibitem{b} C. Breuil, S\'erie Sp\'eciale $p$-adique et Cohomolologie \'Etale Compl\'et\'ee. IHES preprint, 2003.
\bibitem{darmonHpH} H. Darmon, Integration on $\mathcal{H}\sb p\times\mathcal{H}$ and arithmetic applications. Ann. of Math. (2) 154 (2001), no. 3, 589--639.
\bibitem{GvDP} L. Gerritzen\ and\ M. van der Put, {\it Schottky groups and Mumford curves}, Lecture Notes in Math., 817, Springer , Berlin, 1980; (see esp. Chapter IX).
\bibitem{MTT}B. Mazur, J. Tate\ and\ J. Teitelbaum, On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer , Invent. Math. {\bf 84} (1986), no.~1, 1--48.
\bibitem{orton} L. Orton, An elementary proof of a weak exceptional zero conjecture. Canad. J. Math. 56 (2004), no. 2, 373--405.
\bibitem{JTT} J. T. Teitelbaum, Values of $p$-adic $L$-functions and a $p$-adic Poisson kernel , Invent. Math. {\bf 101} (1990), no.~2, 395--410.
The goal of the project is to explicitly compute as many as possible of the following objects as $\mathbf{\Gamma}$ runs through the groups given above: