Spring 2012

Intercity Number Theory Seminar

11:45-12:45	Anneroos Everts (Groningen), From finite automata to power series and back again Abstract. Christol's theorem links algebra in an unexpected way with a concept from computer sciences: a power series over a finite field is algebraic if and only if its coefficients are generated by a finite automaton. We examined the proof of Christol's theorem to find answers to the following two questions: Given a finite automaton with m states, what can we say about the algebraic degree of the corresponding power series? Conversely, given an algebraic power series of algebraic degree d and bounded coefficients, can we find a bound on the number of states of an automaton that generates the power series? In this talk I will explain Christol's theorem and the concept of finite automata, and give answers to the questions above.
13:30-14:30	Paul Helminck (Groningen), Tropical elliptic curves and j-invariants Abstract. Any elliptic curve over the field of complex Puiseux series has a "tropicalization": an associated tropical curve. We construct, for any elliptic curve over the field of Puiseux series that has a j-invariant with negative valuation, a model such that its tropification is a tropical elliptic curve. Moreover, we show that the tropical j-invariant of this tropical curve is minus the valuation of the j-invariant. Special cases of this were already proven by Markwig, and similar results were obtained quite recently by M. Baker and by Sturmfels and Chan.
14:45-15:45	Wilke Trei (Carl von Ossietzky University Oldenburg), Elliptic Curve Arithmetic on Vectorized Hardware Platforms Abstract. Parallelization of computational intensive algorithms has always been an important task in computational number theory. Modern hardware requires a high on-chip parallelization for gaining maximum possible performance. We discuss several mathematical concepts to implement modular arithmetic with a focus on elliptic curve scalar multiplication on graphic cards and present a new performance record for Lenstra's elliptic curve factoring algorithm on an ordinary personal computer. The resulting implementation is of high cryptographic interest, for instance it can easily be modified to speed up intermediate factorization in the number field sieve algorithm.
16:00-17:00	Arthemy Kiselev (Groningen), The deformation quantisation problem for multiplicative structures on noncommutative jet spaces. Abstract. We outline the basic notions and concepts from the differential calculus --up to the Schouten bracket-- on a class of noncommutative jet spaces, and then we pose the deformation quantisation problems for the non-associative but commutative multiplications in the two spaces of differential functions (i.e., the noncommutative fields) and integral functionals (i.e, the Hamiltonians), aiming to restore the associative but not commutative star-products. During the entire talk, the constructions and reasonings will appeal to the profound properties of a pair of pants borrowed from the topological closed string theory.

Intercity Number Theory Seminar

Belgian-Dutch algebraic geometry day

14:00-15:00	Ted Chinburg, Small generators for S-arithmetic groups Abstract. A surprising discovery of H. W. Lenstra, Jr., was that one can find generators of small height for groups of S-units of number fields once S is moderately large. I will discuss joint work with Matt Stover on generalizing Lenstra's results from S-units to the S-integral points of linear algebraic groups. This has applications to finding presentations for such groups.
15:30-16:30	TBA,
15:30-16:30	Mathieu Romagny, Models of groups schemes of roots of unity Abstract. I will explain the construction of a family of models over OK of the group scheme μpn,K of pn-th roots of unity over a p-adic field K. This construction is inspired by work of Sekiguchi and Suwa in the late nineties. The contemplation of these models in light of the recent classification of finite flat group schemes by Breuil and Kisin leads to conjecture that they exhaust all possible models of μpn,K. This is joint work with A. Mézard and D. Tossici.
16:45-17:45	Mircea Mustaţă, Adjoint line bundles in positive characteristic Abstract. In characteristic zero, adjoint line bundles enjoy many positivity properties that all go back to Kodaira's Vanishing Theorem. I will explain how certain positivity properties can be recovered in positive characteristic by making use of the Frobenius morphism.