We present a new method to propagate p-adic precision in computations, which also applies to other ultrametric fields. We illustrate it with some examples and give a toy application to the stable computation of the SOMOS 4 sequence.

This paper generalizes work of Buzzard and Kilford to the case p = 3, giving an explicit bound for the overconvergence of the quotient Eκ/V (Eκ) and using this bound to prove that the eigencurve is a union of countably many annuli over the boundary of weight space.

Using the differential precision methods developed previously by the same authors, we study the p-adic stability of standard operations on matrices and vector spaces. We demonstrate that lattice-based methods surpass naive methods in many applications, such as matrix multiplication and sums and intersections of subspaces. We also analyze determinants , characteristic polynomials and LU factorization using these differential methods. We supplement our observations with numerical experiments.Comment: ISSAC 2015, Jul 2015, Bath, United Kingdom. 201

Abstract. We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme G over a finite field k and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on G and show that it is an extension of the group of characters of G(k) by a cohomology group determined by the component group scheme of G. We also classify all morphisms in the category character sheaves on G. As an application, we study character sheaves on Greenberg transforms of locally finite type Néron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of p-adic tori.

We analyze the precision of the characteristic polynomial χ(A) of an n × n p-adic matrix A using differential precision methods developed previously. When A is integral with precision O(p N ), we give a criterion (checkable in time O˜(n ω )) for χ(A) to have precision exactly O(p N ). We also give a O˜(n 3 ) algorithm for determining the optimal precision when the criterion is not satisfied, and give examples when the precision is larger than O(p N ).