7,233 research outputs found
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
Solving discrete logarithms on a 170-bit MNT curve by pairing reduction
Pairing based cryptography is in a dangerous position following the
breakthroughs on discrete logarithms computations in finite fields of small
characteristic. Remaining instances are built over finite fields of large
characteristic and their security relies on the fact that the embedding field
of the underlying curve is relatively large. How large is debatable. The aim of
our work is to sustain the claim that the combination of degree 3 embedding and
too small finite fields obviously does not provide enough security. As a
computational example, we solve the DLP on a 170-bit MNT curve, by exploiting
the pairing embedding to a 508-bit, degree-3 extension of the base field.Comment: to appear in the Lecture Notes in Computer Science (LNCS
Discretisation for odd quadratic twists
The discretisation problem for even quadratic twists is almost understood,
with the main question now being how the arithmetic Delaunay heuristic
interacts with the analytic random matrix theory prediction. The situation for
odd quadratic twists is much more mysterious, as the height of a point enters
the picture, which does not necessarily take integral values (as does the order
of the Shafarevich-Tate group). We discuss a couple of models and present data
on this question.Comment: To appear in the Proceedings of the INI Workshop on Random Matrix
Theory and Elliptic Curve
A database of genus 2 curves over the rational numbers
We describe the construction of a database of genus 2 curves of small
discriminant that includes geometric and arithmetic invariants of each curve,
its Jacobian, and the associated L-function. This data has been incorporated
into the L-Functions and Modular Forms Database (LMFDB).Comment: 15 pages, 7 tables; bibliography formatting and typos fixe
Elliptic curves of large rank and small conductor
For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the
smallest conductor known, improving on the previous records by factors ranging
from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search
methods, and tabulate, for each r=5,6,...,11, the five curves of lowest
conductor, and (except for r=11) also the five of lowest absolute discriminant,
that we found.Comment: 16 pages, including tables and one .eps figure; to appear in the
Proceedings of ANTS-6 (June 2004, Burlington, VT). Revised somewhat after
comments by J.Silverman on the previous draft, and again to get the correct
page break
A table of elliptic curves over the cubic field of discriminant -23
Let F be the cubic field of discriminant -23 and O its ring of integers. Let
Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let
Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of
us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the
action of the Hecke operators. The goal of that paper was to test the
modularity of elliptic curves over F. In the present paper, we complement and
extend this prior work in two ways. First, we tabulate more elliptic curves
than were found in our prior work by using various heuristics ("old and new"
cohomology classes, dimensions of Eisenstein subspaces) to predict the
existence of elliptic curves of various conductors, and then by using more
sophisticated search techniques (for instance, torsion subgroups, twisting, and
the Cremona-Lingham algorithm) to find them. We then compute further invariants
of these curves, such as their rank and representatives of all isogeny classes.
Our enumeration includes conjecturally the first elliptic curves of ranks 1 and
2 over this field, which occur at levels of norm 719 and 9173 respectively
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Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer
Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here we extend the original empirical approach, to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine-learning and topological data analysis. Key quantities such as rank, Weierstrass coefficients, period, conductor, Tamagawa number, regulator and order of the Tate-Shafarevich group give rise to a high-dimensional point-cloud whose statistical properties we investigate. We reveal patterns and distributions in the rank versus Weierstrass coefficients, as well as the Beta distribution of the BSD ratio of the quantities. Via gradient boosted trees, machine learning is applied in finding inter-correlation amongst the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and more in general in pure Mathematics
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