# Workshop on p-adic coefficients and geometry

## 26th-28th June, 2019 Imperial College London

Talks will be at 180 Queen's Gate, Huxley Building, Room 140. Coffee breaks will be at 180 Queen's Gate, Huxley Building, Room 549.

 Wed 26 9:30-10:00 Registration 10:00-11:00 Joe Kramer-Miller $$p$$-adic variation of exponential sums on curves 11:00-11:30 Coffee 11:30-12:30 Kiran Kedlaya On the construction of crystalline companions I 12:30-14:30 Lunch

 Thu 27 10:00-11:00 Kang Zuo Rank-$$2$$ motivic local systems over punctured projective line via an arithmetic Simpson correspondence 11:00-11:30 Coffee 11:30-12:30 Kiran Kedlaya On the construction of crystalline companions II 12:30-14:30 Lunch

 Fri 28 10:00-11:00 Valentina di Proietto A crystalline incarnation of Berthelot's conjecture and Künneth formula for isocrystals 11:00-11:30 Coffee 11:30-12:30 Oli Gregory Around $$p$$-adic Tate conjectures 12:30-14:30 Lunch

Speaker: Valentina di Proietto

Title: A crystalline incarnation of Berthelot's conjecture and Künneth formula for isocrystals

Abstract: Berthelot's conjecture predicts that under a proper and smooth morphism of varieties in characteristic $$p$$, the higher direct images of an $$F$$-overconvergent isocrystal are $$F$$-overconvergent isocrystals. In a joint work with Fabio Tonini and Lei Zhang we prove that this is true for crystals up to isogeny. As an application we prove a Künneth formula for the crystalline fundamental group.

Speaker: Oli Gregory

Title: Around $$p$$-adic Tate conjectures

Abstract: I shall explore Tate conjectures for smooth and proper varieties over $$p$$-adic fields, especially a conjecture of Raskind which concerns varieties with totally degenerate reduction. After first motivating the conjecture and discussing some evidence, I will reformulate Raskind's conjecture into a subtle question about $$\mathbb Q$$-versus $$\mathbb Q_p$$-structures on filtered $$(\phi,N)$$-modules. I will then use this reformulation to show that Raskind's conjecture can fail even for abelian surfaces. This is joint work with Christian Liedtke.

Speaker: Kiran Kedlaya

Title: On the construction of crystalline companions

Abstract: Let $$X$$ be a smooth irreducible variety over a finite field of characteristic $$p$$. Fix a Weil cohomology theory with algebraically closed coefficients containing a prescribed algebraic closure of $$\mathbb Q$$ (either $$l$$-adic cohomology for some prime $$l$$ not equal to $$p$$, or $$p$$-adic rigid cohomology). Deligne's "companion conjecture" then asserts that if one takes every irreducible lisse coefficient with finite determinant and associates to it the tuple of Frobenius characteristic polynomials at all closed points, then the resulting collection of tuples does not depend on the choice of the Weil cohomology theory. By previous work of Deligne, Drinfeld, Abe-Esnault, and the speaker, this is known in all cases except "$$l$$ to $$p$$". We discuss the proof of this implication, i.e., the assertion that étale coefficient objects have crystalline companions.

Speaker: Joe Kramer-Miller

Title: $$p$$-adic variation of exponential sums on curves

Abstract: Understanding exponential sums over an algebraic variety in characteristic $$p>0$$ is a fundamental problem in arithmetic geometry. One approach is to consider the $$L$$-function associated an exponential sum. By Deligne's work on the Weil conjectures, we know that this $$L$$-function is rational and has roots that are $$l$$-adic units whenever $$l\neq p$$. It is natural to ask about the $$p$$-adic properties of this $$L$$-function, which are less well behaved. In this talk, we study the $$p$$-adic variation of these $$L$$-functions as our exponential sum varies over the $$p$$-adic cyclotomic weight space. Generalizing work of Davis-Xiao-Wan, we prove that $$p$$-adic families of exponential sums over certain curves satisfy properties analogous to Coleman's spectral halo conjecture. Time permitting, we will explain applications to the Newton stratum of Artin-Schreier moduli spaces. This is joint work with James Upton.

Speaker: Kang Zuo

Title: Rank-$$2$$ motivic local systems over punctured projective line via an arithmetic Simpson correspondence

Abstract: We propose an arithmetic Simpson correspondence for Higgs bundles over arithmetic scheme and speculate a relation between the arithmetic dynamic system over the projective line with four punctured points arising from rank-$$2$$ Higgs-de Rham flow and the multiplication map on the associated elliptic curve as the double cover of the projective line ramified at the four points. It predicts that a rank-$$2$$ graded stable Higgs bundle of degree $$-1$$ over the projective line with logarithmic singularities at four punctured points corresponds to the local system arising from an abelian scheme endowed with a real multiplication over the projective line with bad reductions at those punctured points if and only if the zero of the Higgs field is the image of a torsion point on the associated elliptic curve. We have already constructed $$26$$ complete solutions in the case of elliptic surfaces whose Kodaira-Spencer maps have zeros of torsion order $$1$$, $$2$$, $$3$$, $$4$$ and $$6$$. We note that a similar phenomena appears in the work by Kontsevich on rank-$$2$$ $$l$$-adic motivic local systems on the projective line with four punctured points. It looks quite mysterious, there should exist a relation between periodic Higgs bundles in the $$p$$-adic world and the Hecke-eigenforms in the $$l$$-adic world via Abe's solution of Deligne conjecture on $$l$$-to-$$p$$ companions.