Die elliptischen Funktionen und ihre Anwendungen

Fricke, R.

doi:10.1007/978-3-642-19557-0

Cited by 17 publications

(33 citation statements)

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“…Fundamental domains for some of the groups Γ 0 (N ) and Γ 0 (N ) + have been determined already by Fricke [101][102][103][104]. The latter can be constructed as a quotient of the former if we can find a representation that is symmetric with respect to the Fricke involution F N .…”

Section: B Fundamental Domains For γ 0 (N ) +mentioning

confidence: 99%

Modular curves and the refined distance conjecture

Klaewer

2021

J. High Energ. Phys.

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We test the refined distance conjecture in the vector multiplet moduli space of 4D $$ \mathcal{N} $$ N = 2 compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations. This is reflected in the type IIA frame through the existence of a K3 fibration. We identify the degree d = 2N of the K3 fiber as a parameter that could potentially lead to large distances, which is substantiated by studying several explicit models. The moduli space geometry degenerates into the modular curve for the congruence subgroup Γ0(N)+. In order to probe the large N regime, we initiate the study of Calabi-Yau threefolds fibered by general degree d > 8 K3 surfaces by suggesting a construction as complete intersections in Grassmann bundles.

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Section: B Fundamental Domains For γ 0 (N ) +mentioning

confidence: 99%

Modular curves and the refined distance conjecture

Klaewer

2021

J. High Energ. Phys.

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“…Proof . By an old result, an elliptic curve with a rational point of order 5 will have its j -invariant of the form ( s 2 + 10 s + 5) 3 / s for some s∈Q [23]. Calculating the j -invariant of E α ,− n 2 , we must therefore have 256(α2+3n2false)3n4(α2+4n2)=(s2+10s+5false)3s.…”

Section: Torsion Pointsmentioning

confidence: 99%

Heron quadrilaterals via elliptic curves

Izadi

Khoshnam

Moody

2017

Rocky Mountain J. Math.

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A Heron quadrilateral is a cyclic quadrilateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form y2 = x3 + αx2 − n2x. This correspondence generalizes the notions of Goins and Maddox who established a similar connection between Heron triangles and elliptic curves. We further study this family of elliptic curves, looking at their torsion groups and ranks. We also explore their connection with congruent numbers, which are the α = 0 case. Congruent numbers are positive integers which are the area of a right triangle with rational side lengths.

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“…• If X 0 (p) is a curve of genus ≥ 1 and X 0 (p)(Q) is non-empty, then Mazur's theorem on isogenies of prime degree ( [24]) says that p is a prime in the list 11,17,19,37,43,67,163, but in all these cases X 0 (p)(Q) has only finitely many points (see for example Section 9 and…”

Section: Curves With Minimal Ramification At Pmentioning

confidence: 99%

Division fields of elliptic curves with minimal ramification

Lozano-Robledo¹

2015

Rev. Mat. Iberoam.

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Let E be an elliptic curve defined over Q, let p be a prime number, and let n ≥ 1. It is well-known that the p n-th division field Q(E[p n ]) of the elliptic curve E contains all the p n-th roots of unity. It follows that the Galois extension Q(E[p n ])/Q is ramified above p, and the ramification index e(p, Q(E[p n ])/Q) of any prime ℘ of Q(E[p n ]) lying above p is divisible by ϕ(p n). The goal of this article is to construct elliptic curves E/Q such that e(p, Q(E[p n ])/Q) is precisely ϕ(p n), and such that the Galois group of Q(E[p n ])/Q is as large as possible, i.e., isomorphic to GL(2, Z/p n Z).

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Die elliptischen Funktionen und ihre Anwendungen

Cited by 17 publications

References 0 publications

Modular curves and the refined distance conjecture

Modular curves and the refined distance conjecture

Heron quadrilaterals via elliptic curves

Division fields of elliptic curves with minimal ramification

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