Section: New Results

Algorithmic number theory

Participant : Henri Cohen.

The book [18] by Henri Cohen on Modular Forms: A Classical Approach has been published. The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete subject in itself and abounds with an amazing number of surprising identities. This comprehensive textbook, gives a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. Content include: elliptic functions and theta functions, the modular group, its subgroups, and general aspects of holomorphic and nonholomorphic modular forms, with an emphasis on explicit examples. The heart of the book is the classical theory developed by Hecke and continued up to the Atkin–Lehner–Li theory of newforms and including the theory of Eisenstein series, Rankin–Selberg theory, and a more general theory of theta series including the Weil representation. The final chapter also explores in some detail more general types of modular forms such as half-integral weight, Hilbert, Jacobi, Maass, and Siegel modular forms.

The article by Bill Allombert, Jean-Paul Allouche and Michel Mendès France on Euler's divergent series and an elementary model in Statistical Physics has been published in Statistical Physics Ars Mathematica Contemporanea. This article study the multiple integral of a multivariate exponential taken with respect either to the Lebesgue measure or to the discrete uniform Bernoulli measure. In the first case the integral is linked to Euler's everywhere divergent power series and its generalizations, while in the second case the integral is linked to a one-dimensional model of spin systems as encountered in physics.

Bill Allombert has worked with Nicolas Brisebarre and Alain Lasjaunias on a two-valued sequence and related continued fractions in power series fields. They explicitly describe a noteworthy transcendental continued fraction in the field of power series over $\mathbb{Q}$, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set $\{1,2\}$.

In the Pari software, K. Belabas and H. Cohen have added an extensive new package mf for modular forms. This package allows to build spaces of classical modular form $M_k({\Gamma }_0\left(N\right),\chi )$ where $2k\in \mathbb{Z}$ and perform standard tasks like finding bases, splitting the space using Hecke operators and the computation of eigenforms. It also solves important difficult problems: the computation of forms of weight 1, the realization of Shimura lifts as an explicit isomorphism between Kohnen's $+$-space $S_k^{+}(\Gamma 0\left(4N\right),\chi )$ and $S_{2k-1}({\Gamma }_0\left(N\right),{\chi }^2)$ and the Fourier expansion of $f{\mid }_k\gamma$ for arbitrary $f$ and arbitrary $\gamma \in {\mathrm{GL}}_2{\left(\mathbb{Q}\right)}^{+}$, which includes as a special case the expansion of $f$ at all cusps (where other modular form packages usualy deal with the expansion at infinity and the cusps reachable via Atkin-Lehner operators, e.g. all cusps in squarefree levels). The latter is especially important as it allows an explicit description of Atkin-Lehner operators, the evaluation of $f$ arbitrary points in the upper-half plane, the computation of period polynomials and Pettersson products, etc.