Pub Date : 2014-03-26DOI: 10.1112/S1461157015000054
U. Thiel
We present a computer algebra package based on Magma for performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded $G$ -module structure of the simple modules, and the Calogero–Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino’s conjecture for several exceptional complex reflection groups. Supplementary materials are available with this article.
{"title":"Champ : a Cherednik algebra Magma package","authors":"U. Thiel","doi":"10.1112/S1461157015000054","DOIUrl":"https://doi.org/10.1112/S1461157015000054","url":null,"abstract":"We present a computer algebra package based on Magma for performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded $G$\u0000 -module structure of the simple modules, and the Calogero–Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino’s conjecture for several exceptional complex reflection groups. Supplementary materials are available with this article.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"33 1","pages":"266-307"},"PeriodicalIF":0.0,"publicationDate":"2014-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000054","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-02-20DOI: 10.1112/S1461157014000436
D. Boyd, G. Martin, Mark Thom
The discriminant of a trinomial of the form x n x m 1 has the form n n (n m) n m m m if n and m are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, whenn is congruent to 2 (mod 6) we have that ((n 2 n+1)=3) 2 always divides n n (n 1) n 1 . In addition, we discover many other square factors of these discriminants that do not t into these parametric families. The set of primes whose squares can divide these sporadic values asn varies seems to be independent ofm, and this set can be seen as a generalization of the Wieferich primes, those primes p such that 2 p is congruent to 2 (mod p 2 ). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants.
如果n和m是相对素数，那么x n x m1的三叉式的判别式是n n (n m) n m m m。我们研究这些判别式何时具有非平凡的平方因子。我们解释了这些判别值的平方因子的各种看似不太可能的参数族:例如，当n等于2(对6取模)时，我们得到(n 2 n+1)=3) 2总是能除nn (n 1) n 1。此外，我们还发现了这些判别式的许多其他平方因子，它们不属于这些参数族。素数的平方可以除以这些零星值的集合似乎与m无关，这个集合可以看作是维费里希素数的推广，这些素数使得2p等于2(对p2取模)。我们提供了这些零星素数的密度和这些三叉判别式的无平方值的密度的启发式。
{"title":"Squarefree values of trinomial discriminants","authors":"D. Boyd, G. Martin, Mark Thom","doi":"10.1112/S1461157014000436","DOIUrl":"https://doi.org/10.1112/S1461157014000436","url":null,"abstract":"The discriminant of a trinomial of the form x n x m 1 has the form n n (n m) n m m m if n and m are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, whenn is congruent to 2 (mod 6) we have that ((n 2 n+1)=3) 2 always divides n n (n 1) n 1 . In addition, we discover many other square factors of these discriminants that do not t into these parametric families. The set of primes whose squares can divide these sporadic values asn varies seems to be independent ofm, and this set can be seen as a generalization of the Wieferich primes, those primes p such that 2 p is congruent to 2 (mod p 2 ). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"26 1","pages":"148-169"},"PeriodicalIF":0.0,"publicationDate":"2014-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000436","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-02-19DOI: 10.1112/S1461157014000199
A. Elsenhans, J. Jahnel
We construct explicit $K3$ surfaces over $mathbb{Q}$ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces constructed.
{"title":"Examples of surfaces with real multiplication","authors":"A. Elsenhans, J. Jahnel","doi":"10.1112/S1461157014000199","DOIUrl":"https://doi.org/10.1112/S1461157014000199","url":null,"abstract":"We construct explicit $K3$\u0000 surfaces over $mathbb{Q}$\u0000 having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces constructed.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"14-35"},"PeriodicalIF":0.0,"publicationDate":"2014-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000199","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present an algorithm that computes the Hasse-Witt matrix of given hyperelliptic curve over Q at all primes of good reduction up to a given bound N. It is simpler and faster than the previous algorithm developed by the authors.
{"title":"Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time","authors":"David Harvey, Andrew V. Sutherland","doi":"10.1090/conm/663/13352","DOIUrl":"https://doi.org/10.1090/conm/663/13352","url":null,"abstract":"We present an algorithm that computes the Hasse-Witt matrix of given hyperelliptic curve over Q at all primes of good reduction up to a given bound N. It is simpler and faster than the previous algorithm developed by the authors.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"257-273"},"PeriodicalIF":0.0,"publicationDate":"2014-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60550141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-01DOI: 10.1112/S1461157014000205
Jinxiang Zeng
Let f ∈ S 2 (Γ 0 ( N )) be a normalized newform such that the abelian variety A f attached by Shimura to f is the Jacobian of a genus-two curve. We give an efficient algorithm for computing Galois representations associated to such newforms.
{"title":"Computing Galois representations of modular abelian surfaces","authors":"Jinxiang Zeng","doi":"10.1112/S1461157014000205","DOIUrl":"https://doi.org/10.1112/S1461157014000205","url":null,"abstract":"Let f ∈ S 2 (Γ 0 ( N )) be a normalized newform such that the abelian variety A f attached by Shimura to f is the Jacobian of a genus-two curve. We give an efficient algorithm for computing Galois representations associated to such newforms.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"36-48"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000205","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-01DOI: 10.1112/S1461157014000047
B. Eick, M. Kirschmer, C. Leedham-Green
We exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank $2$ in $mathrm{PSL}_2(mathbb{R})$ or $mathrm{SL}_2(mathbb{R})$ . This algorithm, together with methods for checking whether a two-generator subgroup of $mathrm{PSL}_2(mathbb{R})$ or $mathrm{SL}_2(mathbb{R})$ is discrete and free, have been implemented in Magma for groups defined over real algebraic number fields. Supplementary materials are available with this article.
{"title":"The constructive membership problem for discrete free subgroups of rank 2 of","authors":"B. Eick, M. Kirschmer, C. Leedham-Green","doi":"10.1112/S1461157014000047","DOIUrl":"https://doi.org/10.1112/S1461157014000047","url":null,"abstract":"We exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank $2$ in $mathrm{PSL}_2(mathbb{R})$ or $mathrm{SL}_2(mathbb{R})$ . This algorithm, together with methods for checking whether a two-generator subgroup of $mathrm{PSL}_2(mathbb{R})$ or $mathrm{SL}_2(mathbb{R})$ is discrete and free, have been implemented in Magma for groups defined over real algebraic number fields. Supplementary materials are available with this article.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"345-359"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000047","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-01DOI: 10.1112/s146115701400014x
T. Fukuda, K. Komatsu
We propose a fast method of calculating the p-part of the class numbers in certain non-cyclotomic Zp-extensions of an imaginary quadratic field using elliptic units constructed by Siegel functions. We carried out practical calculations for p = 3 and determined λ-invariants of such Z3-extensions which were not known in our previous paper.
{"title":"Class number calculation using Siegel functions","authors":"T. Fukuda, K. Komatsu","doi":"10.1112/s146115701400014x","DOIUrl":"https://doi.org/10.1112/s146115701400014x","url":null,"abstract":"We propose a fast method of calculating the p-part of the class numbers in certain non-cyclotomic Zp-extensions of an imaginary quadratic field using elliptic units constructed by Siegel functions. We carried out practical calculations for p = 3 and determined λ-invariants of such Z3-extensions which were not known in our previous paper.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"295-302"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/s146115701400014x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-01DOI: 10.1112/S1461157013000363
H. Bez, N. Bez
{"title":"A note on magnitude bounds for the mask coefficients of the interpolatory Dubuc–Deslauriers subdivision scheme","authors":"H. Bez, N. Bez","doi":"10.1112/S1461157013000363","DOIUrl":"https://doi.org/10.1112/S1461157013000363","url":null,"abstract":"","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"226-232"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000363","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-01DOI: 10.1112/S1461157014000023
A. Dujella, J. C. Peral
We construct an elliptic curve over the field of rational functions with torsion group Z/2Z × Z/4Z and rank equal to four, and an elliptic curve over Q with the same torsion group and rank nine. Both results improve previous records for ranks of curves of this torsion group. They are obtained by considering elliptic curves induced by Diophantine triples.
{"title":"High rank elliptic curves with torsion Z/2Z × Z/4Z induced by Diophantine triples","authors":"A. Dujella, J. C. Peral","doi":"10.1112/S1461157014000023","DOIUrl":"https://doi.org/10.1112/S1461157014000023","url":null,"abstract":"We construct an elliptic curve over the field of rational functions with torsion group Z/2Z × Z/4Z and rank equal to four, and an elliptic curve over Q with the same torsion group and rank nine. Both results improve previous records for ranks of curves of this torsion group. They are obtained by considering elliptic curves induced by Diophantine triples.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"282-288"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-01DOI: 10.1112/S1461157014000059
T. Fisher
We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n = 7, 11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, that is, pairs of non-isogenous elliptic curves over Q whose n-torsion subgroups are isomorphic as Galois modules. We also find a non-trivial pair of 11-congruent elliptic curves over Q(T ), and hence give an explicit infinite family of non-trivial pairs of 11-congruent elliptic curves over Q. Supplementary materials are available with this article.
{"title":"On families of 7- and 11-congruent elliptic curves","authors":"T. Fisher","doi":"10.1112/S1461157014000059","DOIUrl":"https://doi.org/10.1112/S1461157014000059","url":null,"abstract":"We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n = 7, 11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, that is, pairs of non-isogenous elliptic curves over Q whose n-torsion subgroups are isomorphic as Galois modules. We also find a non-trivial pair of 11-congruent elliptic curves over Q(T ), and hence give an explicit infinite family of non-trivial pairs of 11-congruent elliptic curves over Q. Supplementary materials are available with this article.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"536-564"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000059","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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