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Fast Polynomial Factorization and Modular Composition
We obtain randomized algorithms for factoring degree $n$ univariate polynomials over $F_q$ requiring $O(n^{1.5 + o(1)} log^{1+o(1)} q+ n^{1 + o(1)}log^{2+o(1)} q)$ bit operations. When $log q <Expand
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Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology
We describe an algorithm for counting points on an arbitrary hyperelliptic curve over a finite field of odd characteristic, using Monsky-Washnitzer cohomology to compute a p-adic approximation to theExpand
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Relative P-adic Hodge Theory: Foundations
We describe a new approach to relative p-adic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry in the style of both Berkovich and Huber. We giveExpand
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A p-adic local monodromy theorem
We produce a canonical filtration for locally free sheaves on an open p-adic annulus equipped with a Frobenius structure. Using this filtration, we deduce a conjecture of Crew on p-adic differentialExpand
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Fast Modular Composition in any Characteristic
TLDR
We give an algorithm for modular composition of degree n univariate polynomials over a finite field F<sub>q</sub> requiring n <sup>1</sup><sup>+</sup>. Expand
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p-adic Differential Equations
Preface Introductory remarks Part I. Tools of p-adic Analysis: 1. Norms on algebraic structures 2. Newton polygons 3. Ramification theory 4. Matrix analysis Part II. Differential Algebra: 5.Expand
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The algebraic closure of the power series field in positive characteristic
For K an algebraically closed field, let K((t)) denote the quotient field of the power series ring over K. The “Newton-Puiseux theorem” states that if K has characteristic 0, the algebraic closure ofExpand
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Slope filtrations revisited.
We give a "second generation" exposition of the slope filtration theorem for modules with Frobenius action over the Robba ring, providing a number of simplifications in the arguments. Some of theseExpand
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Slope filtrations for relative Frobenius
The slope filtration theorem gives a partial analogue of the eigenspace decomposition of a linear transformation, for a Frobenius-semilinear endomorphism of a finite free module over the Robba ringExpand
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Solving constrained Pell equations
TLDR
We present a general elementary method for solving certain systems of Diophantine equations of the form x2 − ay2 = b, P (x, y) = z2, where P is a given integer polynomial. Expand
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