Schedule for: 17w5118 - $p$-adic Cohomology and Arithmetic Applications

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Let $K$ be a finitely generated infinite field over the finite prime field $\mathbb{F}_p$ with separable closure $K^s$ and let $X$ be a smooth projective variety over $K$. By Deligne the cohomology groups $H^i_{\mathrm{\acute{e}t}}(X_{K^s},\mathbb{Q}_\ell)$ for varying primes $\ell\neq p$ form a ($\mathbb{Q}$-rational) compatible system of Galois representations of $\mathrm{Gal}(K^s/K)$ and its restriction to the geometric Galois group $G^{\mathrm{geo}}_K=\mathrm{Gal}(K^s/K\mathbb{F}_p^s)$ is semisimple. Using mainly algebraic geometry, representation theory and Bruhat-Tits theory, Cadoret, Hui and Tamagawa showed recently that also the family of reductions $H^i_{\mathrm{\acute{e}t}}(X_{K^s},\mathbb{F}_\ell)$ is semisimple as a representation of $G^{\mathrm{geo}}_K$ for almost all $\ell$, the key case being that of a global function field $K$. This has important consequence for the image of $G^{\mathrm{geo}}_K$ for its action on the adelic module~$H^i_{\mathrm{\acute{e}t}}(X_{K^s},\mathbb{A}_{\mathbb{Q}})$. In joint work with W. Gajda and S. Petersen, using automorphic methods as a main tool, we prove the analog of the above result for any $E$-rational compatible system of Galois representations of a global function field. In the talk I shall explain the context, indicate the applications and sketch how automorphic methods come to bear on the problem.

We prove that the monodromy of a cohomologically rigid integrable connection $(E,\nabla)$ on a smooth complex projective variety $X$ is integral. This answers positively a special case of a conjecture by Carlos Simpson. To this aim, we prove that the mod $p$ reduction of a rigid integrable connection $(E,\nabla) $ has the structure of an isocrystal with Frobenius structure. We also prove that rigid integrable connections with vanishing $p$- curvatures are unitary. This allows one to prove new cases of Grothendieck’s $p$-curvature conjecture. Joint with Michael Groechenig

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

In the first part of this talk we are going to define certain integral structures, depending on a congruence level, for the sheaves of differential operators on a formal scheme which is a blow-up of a formal scheme which itself is formally smooth over a complete discrete valuation ring of mixed characteristic. When one takes the projective limit over all blow-ups, one obtains the sheaf of differential operators on the associated rigid space, introduced independently by K. Ardakov and S. Wadsley. In the second part we will explain what it means that formal models of flag varieties are D-affine (this concept is analogous to that of Beilinson-Bernstein and Brylinski-Kashiwara in the algebraic context). If time permits, we will explain an example which illustrates that methods and results from rigid cohomology can be used in connection with those sheaves to analyze locally analytic representations of p-adic groups. This is joint work with C. Huyghe, D. Patel, and T. Schmidt.

I will talk on a recent result and possible applications concerning the comparison between integral Monsky-Washnitzer cohomology and overconvergent de Rham-Witt cohomology for smooth and affine schemes. This extends work of Davis and Zureick-Brown. Joint with Johannes Sprang.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk concerns a twenty-thousand-year old mistake: the natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to the cohomological interpretation of the Hasse-Weil zeta function by regularized determinants envisioned by Deninger.

de Jong conjecture predicts that any isocrystal on a geometrically simply connected projective smooth variety over a perfect field of characteristic $p>0$ would be constant. I will report some results related to this conjecture. Joint work in progress with Hélène Esnault.

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 describe an extension of Berthelot's theory of arithmetic differential operators to a class of morphisms of adic formal schemes that are not necessarily of finite type, or even adic. If time permits I will explain how one can use this to give generalizations of the theory of convergent and overconvergent isocrystals.

A central ingredient in Kim's work on integral points of hyperbolic curves is the "unipotent Kummer map" which goes from integral points to certain torsors for the prounipotent completion of the fundamental group, and which, roughly speaking, sends an integral point to the torsor of homotopy classes of paths connecting it to a fixed base-point. In joint work with Tomer Schlank, we introduce a space $\Omega$ of rational motivic loops, and we construct a double factorization of the unipotent Kummer map which may be summarized schematically as points $\rightarrow$ rational motivic points $\rightarrow$ $\Omega$-torsors $\rightarrow$ $\pi_1$-torsors. Our "connectedness theorem" says that any two motivic points are connected by a non-empty torsor. Our "concentration theorem" says that for an affine curve, $\Omega$ is actually equal to $\pi_1$.

We report on joint work-in-progress with Ambrus Pal on the following conjecture. Conjecture: Let $X$ be a smooth variety over a finite field $k$ and let $E$ be an overconvergent F-isocrystal on $X$ that has rank 2 and is absolutely irreducible. Suppose further that $E$ has "infinite monodromy at a divisor at infinity." Then $E$ comes from a family of abelian varieties.

I will first describe two problems in the arithmetic of elliptic curves over function fields, then my work in progress on these conjectures using tools from $p$-adic cohomology. The first ingredient is a pro-representability theorem of what could be called arithmetic deformations of crystals and Dieudonne crystals. The second ingredient is the application of the Taylor-Wiles method in this setting.

I will report on my work in progress done jointly with Frits Beukers. In 1990s V. Batyrev used Dwork modules to describe the mixed Hodge structure on the middle cohomology of affine hypersurfaces in algebraic tori. Dwork modules are crystals where the Frobenius map and connection can be described explicitly. We use these crystals to show several p-adic congruences for the coefficients of powers of a Laurent polynomial. The congruence mentioned in the title involves the L-function of the toric exponential sums and yields p-adic unit-root formulas.

Let k be a finite field of characteristic p, and let l be a prime not equal to p. An old conjecture of Deligne states that on any smooth variety over k, any lisse l-adic sheaf which is irreducible of finite determinant admits a "crystalline companion" with matching Frobenius characteristic polynomials at all closed points. We report on progress towards constructing crystalline companions in the category of overconvergent F-isocrystals, building on work of Drinfeld, Deligne, and Abe-Esnault.

We study the formal neighbourhood of a point in µ-ordinary locus of an integral model of a Hodge type Shimura variety, and prove the existence of a structure analogous to the Serre-Tate structure on the deformation space of an ordinary abelian variety.This is joint work with Rong Zhou.

A Simpson correspondence establishes an equivalence between a category of modules equipped with a connexion and a category of Higgs bundles. In a joint work with Michel Gros and Adolfo Quirós, we describe such an equivalence in the case of modules equipped with a q-connexion when q is a p-th root of unity. This is modeled on the characteristic p case and relies on the notion of quantum divided powers that we also discuss.

We report on an ongoing joint project with Kiran Kedlaya and David Harvey on the computation of zeta functions of nondegenerate toric hypersurfaces over finite fields using $p$-adic cohomology

We introduce a notion of $F$-isocrystals with logarithmic decay and give a conjecture relating this notion to slope filtrations. When the unit-root subcrystal has rank one we prove this conjecture. Combining this with a monodromy theorem we give a new proof of the Drinfeld-Kedlaya theorem. We also prove a generalized version of Wan's conjecture on genus stability for towers of curves coming from geometry.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.