Kiran S. Kedlaya

Good formal structures for flat meromorphic connections, I: Surfaces

We give a criterion under which one can obtain a good decomposition (in the sense of Malgrange) of a formal flat connection on a complex analytic or algebraic variety of arbitrary dimension. The criterion is stated in terms of the spectral behavior of differential operators, and generalizes Robbas construction of the Hukuhara-Levelt-Turrittin decomposition in the one-dimensional case. As an application, we prove the existence of good formal structures for flat meromorphic connections on surfaces after suitable blowing up; this verifies a conjecture of Sabbah, and extends a result of Mochizuki for algebraic connections. Our proof uses a finiteness argument on the valuative tree associated to a point on a surface, in order to verify the numerical criterion.

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 of K((t)) is the union of the fields K((t1/n)) over n ∈ N. We answer a question of Abhyankar by constructing an algebraic closure of K((t)) for any field K of positive characteristic explicitly in terms of certain generalized power series.

Fast Modular Composition in any Characteristic

We give an algorithm for modular composition of degree n univariate polynomials over a finite field F_q requiring n ¹ ⁺ ^o(1) log¹ ⁺ ^o(1) q bit operations; this had earlier been achieved in characteristic n^o(1) by Umans (2008). As an application, we obtain a randomized algorithm for factoring degree n polynomials over F_q requiring (n^1.5 ⁺ ^o(1) + n ¹ ⁺ ^o(1) log q) log¹ ⁺ ^o(1) q bit operations, improving upon the methods of von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998). Our results also imply algorithms for irreducibility testing and computing minimal polynomials whose running times are best-possible, up to lower order terms.As in Umans (2008), we reduce modular composition to certain instances of multipoint evaluation of multivariate polynomials. We then give an algorithm that solves this problem optimally (up to lower order terms), in arbitrary characteristic. The main idea is to lift to characteristic 0, apply a small number of rounds of multimodular reduction, and finish with a small number of multidimensional FFTs. The final evaluations are then reconstructed using the Chinese Remainder Theorem. As a bonus, we obtain a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithm uses techniques which are commonly employed in practice, so it may be competitive for real problem sizes. This contrasts with previous asymptotically fast methods relying on fast matrix multiplication.

Semistable reduction for overconvergent

Let

F

X

-isocrystals I: Unipotence and logarithmic extensions

be a smooth variety over a field

Sato-Tate distributions and Galois endomorphism modules in genus 2

k

Good formal structures for flat meromorphic connections, II: Excellent schemes

of characteristic

Semistable reduction for overconvergent F -isocrystals, IV: local semistable reduction at nonmonomial valuations

p>0

Finiteness of rigid cohomology with coefficients

, and let