ρ — Adic Analogues of Ramanujan Type Formulas for 1/π
Following Ramanujan’s work on modular equations and approximations of π, there are formulas for 1/π of the form [PLEASE CHECK FORMULA IN THE PDF] for d = 2, 3, 4, 6, where λd are singular values that correspond to elliptic curves with complex multiplication, and α, δ are explicit algebraic numbers....
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MDPI AG
2013-03-01
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| Series: | Mathematics |
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| Online Access: | http://www.mdpi.com/2227-7390/1/1/9 |
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doaj-d61958a99faa4d8c860195096a2ac9882020-11-25T01:10:16ZengMDPI AGMathematics2227-73902013-03-011193010.3390/math1010009ρ — Adic Analogues of Ramanujan Type Formulas for 1/πSarah ChisholmAlyson DeinesLing LongGabriele NebeHolly SwisherFollowing Ramanujan’s work on modular equations and approximations of π, there are formulas for 1/π of the form [PLEASE CHECK FORMULA IN THE PDF] for d = 2, 3, 4, 6, where λd are singular values that correspond to elliptic curves with complex multiplication, and α, δ are explicit algebraic numbers. In this paper we prove a ρ-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.http://www.mdpi.com/2227-7390/1/1/9Ramanujan type supercongruencesAtkin and Swinnerton-Dyer congruenceshypergeometric serieselliptic curvescomplex multiplicationperiodsmodular formsPicard–Fuchs equation |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Sarah Chisholm Alyson Deines Ling Long Gabriele Nebe Holly Swisher |
| spellingShingle |
Sarah Chisholm Alyson Deines Ling Long Gabriele Nebe Holly Swisher ρ — Adic Analogues of Ramanujan Type Formulas for 1/π Mathematics Ramanujan type supercongruences Atkin and Swinnerton-Dyer congruences hypergeometric series elliptic curves complex multiplication periods modular forms Picard–Fuchs equation |
| author_facet |
Sarah Chisholm Alyson Deines Ling Long Gabriele Nebe Holly Swisher |
| author_sort |
Sarah Chisholm |
| title |
ρ — Adic Analogues of Ramanujan Type Formulas for 1/π |
| title_short |
ρ — Adic Analogues of Ramanujan Type Formulas for 1/π |
| title_full |
ρ — Adic Analogues of Ramanujan Type Formulas for 1/π |
| title_fullStr |
ρ — Adic Analogues of Ramanujan Type Formulas for 1/π |
| title_full_unstemmed |
ρ — Adic Analogues of Ramanujan Type Formulas for 1/π |
| title_sort |
ρ — adic analogues of ramanujan type formulas for 1/π |
| publisher |
MDPI AG |
| series |
Mathematics |
| issn |
2227-7390 |
| publishDate |
2013-03-01 |
| description |
Following Ramanujan’s work on modular equations and approximations of π, there are formulas for 1/π of the form [PLEASE CHECK FORMULA IN THE PDF] for d = 2, 3, 4, 6, where λd are singular values that correspond to elliptic curves with complex multiplication, and α, δ are explicit algebraic numbers. In this paper we prove a ρ-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication. |
| topic |
Ramanujan type supercongruences Atkin and Swinnerton-Dyer congruences hypergeometric series elliptic curves complex multiplication periods modular forms Picard–Fuchs equation |
| url |
http://www.mdpi.com/2227-7390/1/1/9 |
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AT sarahchisholm radicanaloguesoframanujantypeformulasfor1p AT alysondeines radicanaloguesoframanujantypeformulasfor1p AT linglong radicanaloguesoframanujantypeformulasfor1p AT gabrielenebe radicanaloguesoframanujantypeformulasfor1p AT hollyswisher radicanaloguesoframanujantypeformulasfor1p |
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