Rational torsion points on Jacobians of modular curves

Abstract We consider the generalized Jacobian {\widetilde{J}} of the modular curve {X_{0}(N)} of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the {\mathbb{Q}} -rational torsion points on {\widetilde{J}} up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on {\widetilde{J}} and its Eisenstein property.

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A MODULI INTERPRETATION FOR THE NON-SPLIT CARTAN MODULAR CURVE

Glasgow Mathematical Journal ◽

10.1017/s0017089517000180 ◽

2017 ◽

Vol 60 (2) ◽

pp. 411-434 ◽

Cited By ~ 2

Author(s):

MARUSIA REBOLLEDO ◽

CHRISTIAN WUTHRICH

Keyword(s):

Elliptic Curves ◽

Half Plane ◽

London Math ◽

Number Fields ◽

Arithmetic Geometry ◽

Modular Curves ◽

Torsion Points ◽

Modular Curve ◽

Upper Half Plane ◽

Cartan Subgroups

AbstractModular curves likeX0(N) andX1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here byXnsp(p) andXnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures ofp-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc.(3)77(1) (1998), 1–38;J. Algebra231(1) (2000), 414–448).

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Torsion points on modular curves

Inventiones mathematicae ◽

10.1007/s002220050370 ◽

2000 ◽

Vol 140 (3) ◽

pp. 487-509 ◽

Cited By ~ 4

Author(s):

Matthew H. Baker

Keyword(s):

Modular Curves ◽

Torsion Points

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Optimized CSIDH Implementation Using a 2-Torsion Point

Cryptography ◽

10.3390/cryptography4030020 ◽

2020 ◽

Vol 4 (3) ◽

pp. 20 ◽

Cited By ~ 1

Author(s):

Donghoe Heo ◽

Suhri Kim ◽

Kisoon Yoon ◽

Young-Ho Park ◽

Seokhie Hong

Keyword(s):

Elliptic Curve ◽

Group Action ◽

Large Degree ◽

Transitive Group ◽

Optimization Method ◽

Torsion Point ◽

Torsion Points ◽

Diffie Hellman ◽

Odd Degree ◽

Montgomery Curve

The implementation of isogeny-based cryptography mainly use Montgomery curves, as they offer fast elliptic curve arithmetic and isogeny computation. However, although Montgomery curves have efficient 3- and 4-isogeny formula, it becomes inefficient when recovering the coefficient of the image curve for large degree isogenies. Because the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) requires odd-degree isogenies up to at least 587, this inefficiency is the main bottleneck of using a Montgomery curve for CSIDH. In this paper, we present a new optimization method for faster CSIDH protocols entirely on Montgomery curves. To this end, we present a new parameter for CSIDH, in which the three rational two-torsion points exist. By using the proposed parameters, the CSIDH moves around the surface. The curve coefficient of the image curve can be recovered by a two-torsion point. We also proved that the CSIDH while using the proposed parameter guarantees a free and transitive group action. Additionally, we present the implementation result using our method. We demonstrated that our method is 6.4% faster than the original CSIDH. Our works show that quite higher performance of CSIDH is achieved while only using Montgomery curves.

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Mazur’s rational torsion result for pointless genus one curves: examples

Research in Number Theory ◽

10.1007/s40993-020-00231-z ◽

2021 ◽

Vol 7 (1) ◽

Author(s):

Arjan Dwarshuis ◽

Majken Roelfszema ◽

Jaap Top

Keyword(s):

Algebraic Geometry ◽

Elliptic Curves ◽

Rational Point ◽

Torsion Points ◽

Arbitrary Genus ◽

Mazur’S Theorem ◽

Genus One

AbstractThis note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

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