Past Arithmetic Geometry Seminar

Let X be a smooth, complete geometrically connected curve defined over a one variable function field K over a finite field. Let G be a subgroup of the points of the Jacobian variety J of X defined over a separable closure of K with the property that G/p is finite, where p is the characteristic of K. Buium and Voloch, under the hypothesis that X is not defined over K^p, give an explicit bound for the number of points of X which lie in G (related to a conjecture of Lang, in the case of curves). In this joint work with Pazuki, we extend their result by requiring just that X is non isotrivial.

I will explain how to prove the exactness of the homotopy sequence of overconvergent p-adic fundamental groups for a smooth and projective morphism in characteristic p. We do so by first proving a corresponding result for rigid analytic varieties in characteristic 0, following dos Santos in the algebraic case. In characteristic p we proceed by a series of reductions to the case of a liftable family of curves, where we can apply the rigid analytic result. Joint work with Chris Lazda.

Let A be an abelian variety over the function field K of a curve over a finite field of characteristic p>0. We shall show that the group A(K^{p^{-\infty}}) is finitely generated, unless severe restrictions are put on the geometry of A. In particular, we shall show that if A is ordinary and has a point of bad reduction then A(K^{p^{-\infty}}) is finitely generated. This result can be used to give partial answers to questions of Scanlon, Ziegler, Esnault, Voloch and Poonen.

I will talk about a recent proof, joint with M. Gröchenig and D. Wyss, of a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The proof, inspired by an argument of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration. I will start with an introduction to Higgs bundles and their moduli spaces and then explain our argument.

For a smooth projective scheme Y over W(k) we consider an element in the motivic Chow group of the reduction Y_m over the truncated Witt ring W_m(k) and give a "Hodge" criterion - using the crystalline cycle class in relative crystalline cohomology - for the element to the lift to the continuous Chow group of Y. The result extends previous work of Bloch-Esnault-Kerz on the p-adic variational Hodge conjecture to a relative setting. In the course of the proof we derive two new results on the relative de Rham-Witt complex and its Nygaard filtration, and work with relative syntomic complexes to define relative motivic complexes for a smooth, formal lifting of Y_m over W(W_m(k)).

to come

Numbers like the special values of the gamma and the Bessel functions or the Euler-Mascheroni constant are not expected to be periods in the usual sense of algebraic geometry. However, they can be regarded as coefficients of the comparison isomorphism between two cohomology theories associated to pairs consisting of an algebraic variety and a regular function: the de Rham cohomology of a connection with irregular singularities, and the so-called “rapid decay cohomology”. Following ideas of Kontsevich and Nori, I will explain how this point of view allows one to construct a Tannakian category of exponential motives over a subfield of the complex numbers. The upshot is that one can attach to exponential periods a Galois group that conjecturally governs all algebraic relations between them. Classical results and conjectures in transcendence theory may be reinterpreted in this way. No prior knowledge of motives will be assumed, and I will focus on examples rather than on the more abstract aspects of the theory. This is a joint work with P. Jossen (ETH Zürich).

Beilinson has given a motivic construction of the Eisenstein cohomology on modular curves. This makes it possible to define Eisenstein classes in Deligne-Beilinson, syntomic, and ´etale cohomology. These Eisenstein classes can be computed in terms of real analytic and p-adic Eisenstein series or modular units. The resulting explicit expressions allow to prove results on special values of classical and p-adic L-functions and lead to explicit reciprocity laws. Harder has more generally defined and studied the Eisenstein cohomology for Hilbert modular varieties by analytic methods. In this talk we will explain a motivic and in particular algebraic construction of Harder’s Eisenstein cohomology classes, which generalizes Beilinson’s result. This opens the way to applications, similar as for modular curves, in the case of Hilbert modular varieties.

Following the spirit of Grothendieck’s Esquisse d’un Programme, the Ihara/Oda-Matsumoto conjecture predicted a combinatorial description of the absolute Galois group of Q based on its action on geometric fundamental groups of varieties. This conjecture was resolved in the 90’s by Pop using anabelian techniques. In this talk, I will discuss some satronger variants of this conjecture, focusing on the more recent solutions of its pro-ell and mod-ell two-step nilpotent variants.
 

The so-called Ax-Lindemann theorem asserts that the Zariski closure of a certain subset of a homogeneous variety (typically abelian or Shimura) is itself a homogeneous variety. This theorem has recently been proven in full generality by Klingler-Ullmo-Yafaev and Gao. This statement leads to a variety of questions about topological and Zariski closures of certain sets in  homogeneous varieties which can be approached by both ergodic and o-minimal techniques.  In a series of recent papers with E. Ullmo, we formulate conjectures and prove a certain number of results  of this type.  In this talk I will present these conjectures and results and explain the ideas of proofs
 

Continuation of the last talk.

This talk will give a description of the period ring B_dR of Fontaine, which uses de Rham algebra computations. 

This talk is part of the workshop on Beilinson's approach to p-adic Hodge  theory.

Following the notes and an article of B. Bhatt, we shall compute the de Rham algebra of the immersion of the zero-section of affine space over Z/p^nZ.

This talk is part of the workshop on Beilinson's approach to p-adic Hodge theory.

This talk will describe the basic properties of the de Rham algebra, which is a generalisation of the de Rham algebra over smooth schemes, which was introduced by L. Illusie in his monograph 'Complexe cotangent et déformations'.

This is the 4th talk of the study group on Beilinson's approach to p-adic Hodge theory, following the notes of Szamuley and Zabradi.

I shall finish the computation of the module of differentials of the ring of integers of the algebraic closure of Q_p and describe a universal thickening of C_p.

I shall also quickly introduce the derived de Rham algebra. Kevin McGerty will give a talk on the derived de Rham algebra in W5 or W6.

This is a survey (with some proofs) of chapter 2 of the notes http://renyi.mta.hu/~szamuely/beilintronew.pdf of T. Szamuely and G. Zabradi on Beilinson's approach to the p-adic Hodge decomposition theorem.

This is the first talk of the workshop organised by F. Brown, M. Kim and D. Rössler on Beilinson's approach to p-adic Hodge theory. 

In this talk, we shall give the definition and recall various properties of the cotangent complex, which was originally defined by L. Illusie in his monograph "Complexe cotangent et déformations" (Springer LNM 239, 1971).

This is an introduction to the article 

A. Beilinson, p-adic periods and derived de Rham cohomology, J. Amer. Math. Soc. 25 (2012), no. 3, 715--738.

Fang, Lu and Yoshikawa conjectured a few years ago that a certain string-theoretic invariant (originally introduced by the physicists M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa) of Calabi-Yau threefolds is a birational invariant. This conjecture can be viewed as a "secondary" analog (in dimension three) of the birational invariance of Hodge numbers of Calabi-Yau varieties established by Batyrev and Kontsevich. Using the arithmetic Riemann-Roch theorem, we prove a weak form of this conjecture.