Algebra and Number Theory Seminar: Poornima B

Title: Towards an Unrestricted Cut-by-Curves Criterion for Overconvergence of $F$-Isocrystals

Abstract:  The theory of $p$-adic differential equations first rose to prominence after Dwork used them to prove the rationality of zeta functions of a positive characteristic variety in 1960. Since then, there has been growing interest in the category of convergent $F$-isocrystals and the subcategory of overconvergent $F$-isocrystals due to this subcategory having good cohomology theory with finiteness properties. Recent work by Grubb, Kedlaya and Upton examines when a convergent $F$-isocrystal is overconvergent by restricting to smooth curves on the scheme under a mild tameness assumption (measured by the Swan conductor). In my talk, I will introduce the above categories and talk about work in progress about bounding the Swan conductor of an overconvergent $F$-isocrystal in terms of data associated to the corresponding convergent $F$-isocrystal.