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denote the largest product of distinct primes whose sum does not exceed
. The main result of this paper is that the property for all
, we have
denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis.
Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of
-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over
, ordered by height. We describe databases of elliptic curves over
, ordered by height, in which we compute ranks and
-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.
We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows a Jacobi eigenform to be generated directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms, it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.
In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in
with the number field sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special-
strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various polynomial selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special-
and various enumeration strategies.
We propose an algorithm to verify the
-part of the class number for a number field
is totally real and an abelian extension of the rational field
is any prime. On fields of degree 4 or higher, this algorithm has been shown heuristically to be faster than classical algorithms that compute the entire class number, with improvement increasing with larger field degrees.
We study elliptic curves over quadratic fields with isogenies of certain degrees. Let
be a positive integer such that the modular curve
is hyperelliptic of genus
and such that its Jacobian has rank
. We determine all points of
defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to
-isomorphism, every elliptic curve over a quadratic field
-isogenous, for some
, to the twist of its Galois conjugate by a quadratic extension
. We determine
explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to
-isomorphism, all elliptic curves with
-isogenies over quadratic fields are in fact
We construct explicit bases for spaces of overconvergent
-adic modular forms when
and study their interaction with the Atkin operator. This results in an extension of Lauder’s algorithms for overconvergent modular forms. We illustrate these algorithms with computations of slope sequences of some
-adic eigencurves and the construction of Chow–Heegner points on elliptic curves via special values of Rankin triple product L-functions.
We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1–13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.
The problem of solving polynomial equations over finite fields has many applications in cryptography and coding theory. In this paper, we consider polynomial equations over a ‘large’ finite field with a ‘small’ characteristic. We introduce a new algorithm for solving this type of equations, called the successive resultants algorithm (SRA). SRA is radically different from previous algorithms for this problem, yet it is conceptually simple. A straightforward implementation using Magma was able to beat the built-in Roots function for some parameters. These preliminary results encourage a more detailed study of SRA and its applications. Moreover, we point out that an extension of SRA to the multivariate case would have an important impact on the practical security of the elliptic curve discrete logarithm problem in the small characteristic case.
We develop algorithms to turn quotients of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and their behavior under certain quotients. We illustrate the power of our ideas in a new modular normal form algorithm for modules over rings of integers, vastly outperforming classical algorithms.
We describe how the Hardy–Ramanujan–Rademacher formula can be implemented to allow the partition function p(n) to be computed with softly optimal complexity O(n1/2+o(1)) and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate p(1019), an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of p(n), where our implementation of the Hardy–Ramanujan–Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver’s tabulation of 76 065 congruences. Supplementary materials are available with this article.
We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we rigorously prove the formula for 16714 of the 16725 such curves of conductor less than 5000.
We present p-adic algorithms for computing Hecke polynomials and Hecke eigenforms associated to spaces of classical modular forms, using the theory of overconvergent modular forms. The algorithms have a running time which grows linearly with the logarithm of the weight and are well suited to investigating the dimension variation of certain p-adically defined spaces of classical modular forms.
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