Section: Research Program

Algorithmic Number Theory

Concerning algorithmic number theory, the main problems we will be considering in the coming years are the following:

• Number fields. We will continue working on the problems of class groups and generators. In particular, the existence and accessibility of good defining polynomials for a fixed number field remain very largely open. The impact of better polynomials on the algorithmic performance is a very important parameter, which makes this problem essential.

• Lattice reduction. Despite a great amount of work in the past 35 years on the LLL algorithm and its successors, many open problems remain. We will continue the study of the use of interval arithmetic in this field and the analysis of variants of LLL along the lines of the Potential-LLL which provides improved reduction comparable to BKZ with a small block size but has better performance.

• Elliptic curves and Drinfeld modules. The study of elliptic curves is a very fruitful area of number theory with many applications in crypto and algorithms. Drinfeld modules are “cousins” of elliptic curves which have been less explored in the algorithm context. However, some recent advances  [74] have used them to provide some fast sophisticated factoring algorithms. As a consequence, it is natural to include these objects in our research directions.

Rigorous numerical computations

Some studies in this area will be driven by some other directions, for example, the rigorous evaluation of non algebraic functions on algebraic varieties might become central for some of our work on topology in small dimension (volumes of varieties, drawing of amoeba) or control theory (approximations of discriminant varieties) are our two main current sources of interesting problems. In the same spirit, the work on $L$-functions computations (extending the computation range, algorithmic tools for computing algebraic data from the $L$ function) will naturally follow.

On the other hand, another objective is to extend existing results on periods of algebraic curves to general curves and higher dimensional varieties is a general promising direction. This project aims at providing tools for integration on higher homology groups of algebraic curves, ie computing Gauss-Manin connections. It requires good understanding of their topology, and more algorithmic tools on differential equations.