The L-functions and modular forms database project

The Langlands Programme, formulated by Robert Langlands in the 1960s and since much developed and refined, is a web of interrelated theory and conjectures concerning many objects in number theory, their interconnections, and connections to other fields. At the heart of the Langlands Programme is the concept of an L-function. The most famous L-function is the Riemann zeta-function, and as well as being ubiquitous in number theory itself, L-functions have applications in mathematical physics and cryptography. Two of the seven Clay Mathematics Million Dollar Millennium Problems, the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture, deal with their properties. Many different mathematical objects are connected in various ways to L-functions, but the study of those objects is highly specialized, and most mathematicians have only a vague idea of the objects outside their specialty and how everything is related. Helping mathematicians to understand these connections was the motivation for the L-functions and Modular Forms Database (LMFDB) project. Its mission is to chart the landscape of L-functions and modular forms in a systematic, comprehensive and concrete fashion. This involves developing their theory, creating and improving algorithms for computing and classifying them, and hence discovering new properties of these functions, and testing fundamental conjectures. In the lecture I gave a very brief introduction to L-functions for non-experts, and explained and demonstrated how the large collection of data in the LMFDB is organized and displayed, showing the interrelations between linked objects, through our website www.lmfdb.org. I also showed how this has been created by a world-wide open source collaboration, which we hope may become a model for others.Comment: 14 pages with one illustration. Based on a plenary lecture given at FoCM'14, December 2014, Montevideo, Urugua

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+y2m=zpx^{2\ell}+y^{2m}=z^p

Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of conductor $n<100$, with $5 \nmid n$ and $n \ne 29$, $87$, $89$, then every semistable elliptic curve $E$ over $K$ is modular. Let $\ell$, $m$, $p$ be prime, with $\ell$, $m \ge 5$ and $p \ge 3$.To a putative non-trivial primitive solution of the generalized Fermat $x^{2\ell}+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\mathbb{Q}(\zeta_p)^+$, and study its mod $\ell$ representation with the help of level lowering and our modularity result. We deduce the non-existence of non-trivial primitive solutions if $p \le 11$, or if $p=13$ and $\ell$, $m \ne 7$.Comment: Introduction rewritten to emphasise the new modularity theorem. Paper revised in the light of referees' comment

ON USING SAGE TO SOLVE CONSTRAINED OPTIMIZATION PROBLEMS APPLYING THE LAGRANGE MULTIPLIERS METHOD

This paper combines Calculus and Programming to solve constrained optimization problems common in many areas, notably in Economics. It uses Lagrange multipliers, a well-known technique for maximizing (or minimizing) functions, and the free open-source mathematics software system Sage to compute the maximum (minimum) automatically. Moreover, Sage can be used interactively to work out the solution and to graphically interpret the results, which we find a valuable and practical approach in teaching such techniques to the undergraduate level. In this paper we carry out an exercise describing how these three interdisciplinary areas can work together

Cryptanalysis of McEliece Cryptosystem Based on Algebraic Geometry Codes and their subcodes

We give polynomial time attacks on the McEliece public key cryptosystem based either on algebraic geometry (AG) codes or on small codimensional subcodes of AG codes. These attacks consist in the blind reconstruction either of an Error Correcting Pair (ECP), or an Error Correcting Array (ECA) from the single data of an arbitrary generator matrix of a code. An ECP provides a decoding algorithm that corrects up to $\frac{d^*-1-g}{2}$ errors, where $d^*$ denotes the designed distance and $g$ denotes the genus of the corresponding curve, while with an ECA the decoding algorithm corrects up to $\frac{d^*-1}{2}$ errors. Roughly speaking, for a public code of length $n$ over $\mathbb F_q$, these attacks run in $O(n^4\log (n))$ operations in $\mathbb F_q$ for the reconstruction of an ECP and $O(n^5)$ operations for the reconstruction of an ECA. A probabilistic shortcut allows to reduce the complexities respectively to $O(n^{3+\varepsilon} \log (n))$ and $O(n^{4+\varepsilon})$. Compared to the previous known attack due to Faure and Minder, our attack is efficient on codes from curves of arbitrary genus. Furthermore, we investigate how far these methods apply to subcodes of AG codes.Comment: A part of the material of this article has been published at the conferences ISIT 2014 with title "A polynomial time attack against AG code based PKC" and 4ICMCTA with title "Crypt. of PKC that use subcodes of AG codes". This long version includes detailed proofs and new results: the proceedings articles only considered the reconstruction of ECP while we discuss here the reconstruction of EC