Sebastian Pauli

Congruence Subgroups of PSL(2, Z) of Genus Less than or Equal to 24

In this paper, we report the computation and tabulation, using MAGMA, of all congruence subgroups of PSL(2, Z) of genus less than or equal to 24. We include full tables of the congruence groups of genus 0, 1, 2, and 3 and a summary of the remaining cases.

On the computation of all extensions of a p -ADIC field of a given degree

Let k be a p-adic field. It is well-known that k has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions K/k of a given degree and discriminant.

Factoring Polynomials Over Local Fields

We describe an efficient new algorithm for factoring a polynomial ?(x) over a fieldkthat is complete with respect to a discrete prime divisor. For every irreducible factor? (x) of ?(x) this algorithm returns an integral basis forkx/?(x)kx overk.

Computing the multiplicative group of residue class rings

Let k be a global field with maximal order 0k and let m0 be an ideal of 0k. We present algorithms for the computation of the multiplicative group (0k/m0)* of the residue class ring 0k/m0 and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order to obtain more efficient algorithms. They are applied to the computation of the ray class group Clkm modulo m = m0m∞, where m∞ denotes a formal product of real infinite places, and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields.

Algorithmic Number Theory

Invited Talks.- Computing Pro-P Galois Groups.- The Elliptic Curve Database for Conductors to 130000.- On the Computation of the Coefficients of a Modular Form.- Cohen-Lenstra Heuristics of Quadratic Number Fields.- Algebraic Number Theory.- An Algorithm for Computing p-Class Groups of Abelian Number Fields.- Computation of Locally Free Class Groups.- Numerical Results on Class Groups of Imaginary Quadratic Fields.- Cyclic Polynomials Arising from Kummer Theory of Norm Algebraic Tori.- The Totally Real Primitive Number Fields of Discriminant at Most 109.- A Modular Method for Computing the Splitting Field of a Polynomial.- Analytic and Elementary Number Theory.- On the Density of Sums of Three Cubes.- The Mertens Conjecture Revisited.- Fast Bounds on the Distribution of Smooth Numbers.- Use of Extended Euclidean Algorithm in Solving a System of Linear Diophantine Equations with Bounded Variables.- The Pseudosquares Prime Sieve.- Doubly-Focused Enumeration of Pseudosquares and Pseudocubes.- Lattices.- Practical Lattice Basis Sampling Reduction.- LLL on the Average.- On the Randomness of Bits Generated by Sufficiently Smooth Functions.- Curves and Varieties over Fields of Characteristic Zero.- Computing a Lower Bound for the Canonical Height on Elliptic Curves over ?.- Points of Low Height on Elliptic Curves and Surfaces I: Elliptic Surfaces over with Small d.- Shimura Curves for Level-3 Subgroups of the (2,3,7) Triangle Group, and Some Other Examples.- The Asymptotics of Points of Bounded Height on Diagonal Cubic and Quartic Threefolds.- Testing Equivalence of Ternary Cubics.- Classification of Genus 3 Curves in Special Strata of the Moduli Space.- Heegner Point Computations Via Numerical p-Adic Integration.- Symmetric Powers of Elliptic Curve L-Functions.- Determined Sequences, Continued Fractions, and Hyperelliptic Curves.- Computing CM Points on Shimura Curves Arising from Cocompact Arithmetic Triangle Groups.- Arithmetic of Generalized Jacobians.- Hidden Pairings and Trapdoor DDH Groups.- Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10.- Fast Bilinear Maps from the Tate-Lichtenbaum Pairing on Hyperelliptic Curves.- High Security Pairing-Based Cryptography Revisited.- Efficiently Computable Endomorphisms for Hyperelliptic Curves.- Construction of Rational Points on Elliptic Curves over Finite Fields.- 20 Years of ECM.- Discrete Logarithms.- An Index Calculus Algorithm for Plane Curves of Small Degree.- Signature Calculus and Discrete Logarithm Problems.- Spectral Analysis of Pollard Rho Collisions.- Hard Instances of the Constrained Discrete Logarithm Problem.

Factoring Polynomials over Local Fields II

We present an algorithm for factoring polynomials over local fields, in which the Montes algorithm is combined with elements from Zassenhaus Round Four algorithm. This algorithm avoids the computation of characteristic polynomials and the resulting precision problems that occur in the Round Four algorithm.

A new Algorithm for the Computation of logarithmic ℓ-Class Groups of Number Fields ∗

We present an algorithm for the computation of logarithmic l- class groups of number fields. Our principal motivation is the effective determination of the l-rank of the wild kernel in the K-theory of number fields.

RAMIFICATION POLYGONS, SPLITTING FIELDS, AND GALOIS GROUPS OF EISENSTEIN POLYNOMIALS

Let ϕ(x) be an Eisenstein polynomial of degree n over a local field and let α be a root of ϕ(x). The Newton polygon of ρ(x) = ϕ(αx+α)/αn is called the ramification polygon of ϕ(x). We attach an additional invariant, the segmental inertia degree, to each segment of the ramification polygon and use the slopes of the segments and their segmental inertia degrees to describe the splitting field of ϕ(x). Furthermore we present a method for determining the Galois group of ϕ(x) when the ramification polygon consists of one segment.

Enumerating extensions of (π)-adic fields with given invariants

We give an algorithm that constructs a minimal set of polynomials defining all extension of a

More Zeros of the Derivatives of the Riemann Zeta Function on the Left Half Plane

(\pi)