Modular Forms, a Computational Approach

Stein, William

doi:10.1090/gsm/079

Cited by 264 publications

(299 citation statements)

References 144 publications

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“…The kernel routines run over small finite fields and are usually lifted over Z, Q or Z[X]. They are used in algebraic cryptanalysis [15,3], computational number theory [27], or integer linear programming [18] and they benefit from the experience in numerical linear algebra. In particular, a key point there is to embed the finite field elements in integers stored as floating point numbers, and then rely on the efficiency of the floating point matrix multiplication dgemm of the BLAS.…”

Section: Introductionmentioning

confidence: 99%

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Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

et al. 2016

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We present block algorithms and their implementation for the parallelization of sub-cubic Gaussian elimination on shared memory architectures. Contrarily to the classical cubic algorithms in parallel numerical linear algebra, we focus here on recursive algorithms and coarse grain parallelization. Indeed, sub-cubic matrix arithmetic can only be achieved through recursive algorithms making coarse grain block algorithms perform more efficiently than fine grain ones. This work is motivated by the design and implementation of dense linear algebra over a finite field, where fast matrix multiplication is used extensively and where costly modular reductions also advocate for coarse grain block decomposition. We incrementally build efficient kernels, for matrix multiplication first, then triangular system solving, on top of which a recursive PLUQ decomposition algorithm is built. We study the parallelization of these kernels using several algorithmic variants: either iterative or recursive and using different splitting strategies. Experiments show that recursive adaptive methods for matrix multiplication, hybrid recursive-iterative methods for triangular system solve and tile recursive versions of the PLUQ decomposition, together with various data mapping policies, provide the best performance on a 32 cores NUMA architecture. Overall, we show that the overhead of modular reductions is more than compensated by the fast linear algebra algorithms and that exact dense linear algebra matches the performance of full rank reference numerical software even in the presence of rank deficiencies.

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Section: Introductionmentioning

confidence: 99%

“…The latter is a key invariant used in many applications such as Gröbner basis computations [15] and computational number theory [27].…”

Section: Introductionmentioning

confidence: 99%

Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

et al. 2016

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“…we used the computational method of modular symbols outlined in [15], Chapter 8. The details are standard, and we omit them.…”

Section: Computing Special Valuesmentioning

confidence: 99%

Modular symbols, Eisenstein series, and congruences

Heumann¹,

Vatsal²

2014

J. Théor. Nombres Bordeaux

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“…By Lemma 9.1, the Galois representation on the p-torsion of G m , and of H m respectively arise from newforms at levels 100q and 128q. MAGMA allows us to list all the newforms at these levels (the MAGMA program for doing this is based on algorithms explained in [31]). Suppose the Galois representation on the p-torsion of G m and of H m arise from a particular given pair of newforms g and h, respectively.…”

Section: The Newforms Have Fourier Expansions Around the Cusp At Infimentioning

confidence: 99%

Almost powers in the Lucas sequence

Bugeaud¹,

Luca²,

Mignotte³

et al. 2008

J. Théor. Nombres Bordeaux

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A Henri Cohen,à l'occasion de son soixantième anniversaire Résumé. La liste complète des puissances pures qui apparaissent dans les suites de Fibonacci (F n ) n≥0 et de Lucas (L n ) n≥0 ne fut déterminée que tout récemment [10]. Les démonstrations combinent des techniques modulaires, issues de la preuve de Wiles du dernier théorème de Fermat, avec des méthodes classiques d'approximation diophantienne, dont la théorie de Baker. Dans le présent article, nous résolvons leséquations diophantiennes L n = q a y p , avec a > 0 et p ≥ 2, pour tous les nombres premiers q < 1087, et en fait pour tous les nombres premiers q < 10 6à l'exception de 13 d'entre eux. La stratégie suivie dans [10] s'avère inopérante en raison de la taille des bornes numériques données par les méthodes classiques et de la complexité deséquations de Thue qui apparaissent dans notreétude. La nouveauté mise en avant dans le présent article est l'utilisation simultanée de deux courbes de Frey afin d'aboutirà deséquations de Thue plus simples, et doncà de meilleures bornes numériques, qui contredisent les minorations que donne le crible modulaire.Abstract. The famous problem of determining all perfect powers in the Fibonacci sequence (F n ) n≥0 and in the Lucas sequence (L n ) n≥0 has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles' proof of Fermat's Last Theorem with classical techniques from Baker's theory and Diophantine approximation. In this paper we solve the Diophantine equations L n = q a y p , with a > 0 and p ≥ 2, for all primes q < 1087, and indeed for all but 13 primes q < 10 6 . Here the strategy of [10] is not sufficient due to the sizes of the bounds and complicated nature of the Thue equations involved. The novelty in the present paper is the use of the double-Frey approach to simplify the Thue equations and to cope with the large bounds obtained from Baker's theory.

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Modular Forms, a Computational Approach

Abstract: Abstract. This is a textbook about algorithms for computing with modular forms. It is nontraditional in that the primary focus is not on underlying theory; instead, it answers the question "how do you explicitly compute spaces of modular forms?" v To my grandmother, Annette Maurer.

Cited by 264 publications

References 144 publications

Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

Modular symbols, Eisenstein series, and congruences

Almost powers in the Lucas sequence

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