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 pp , the higher direct images of an FF -overconvergent isocrystal are FF -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.
Title: Around pp -adic Tate conjectures
Abstract: I shall explore Tate conjectures for smooth and proper varieties over pp -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 QQ -versus QpQp -structures on filtered (ϕ,N)(ϕ,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.
Title: On the construction of crystalline companions
Abstract: Let XX be a smooth irreducible variety over a finite field of characteristic pp . Fix a Weil cohomology theory with algebraically closed coefficients containing a prescribed algebraic closure of QQ (either ll -adic cohomology for some prime ll not equal to pp , or pp -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 “ll to pp “. We discuss the proof of this implication, i.e., the assertion that étale coefficient objects have crystalline companions.
Title: pp -adic variation of exponential sums on curves
Abstract: Understanding exponential sums over an algebraic variety in characteristic p>0p>0 is a fundamental problem in arithmetic geometry. One approach is to consider the LL -function associated an exponential sum. By Deligne’s work on the Weil conjectures, we know that this LL -function is rational and has roots that are ll -adic units whenever l≠pl≠p . It is natural to ask about the pp -adic properties of this LL -function, which are less well behaved. In this talk, we study the pp -adic variation of these LL -functions as our exponential sum varies over the pp -adic cyclotomic weight space. Generalizing work of Davis-Xiao-Wan, we prove that pp -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.
Title: Rank-22 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-22 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-22 graded stable Higgs bundle of degree −1−1 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 2626 complete solutions in the case of elliptic surfaces whose Kodaira-Spencer maps have zeros of torsion order 11 , 22 , 33 , 44 and 66 . We note that a similar phenomena appears in the work by Kontsevich on rank-22 ll -adic motivic local systems on the p rojec tive line with four punctured points. It looks quite mysterious, there should exist a relation between periodic Higgs bundles in the pp -adic world and the Hecke-eigenforms in the ll -adic world via Abe’s solution of Deligne conjecture on ll -to-pp companions.
More details avilable on: http://wwwf.imperial.ac.uk/~apal4/himr2019/workshop.html