Computing rational points  on rank 1 elliptic curves  via $L$-series and canonical heights

Silverman, Joseph H.

doi:10.1090/s0025-5718-99-01068-6

Cited by 10 publications

(8 citation statements)

References 14 publications

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“…Since E is modular [23], if we can show that L$(E, 1){0, then work of Kolyvagin [8,9] and Gross and Zagier [5] tells us that E(Q) has rank 1. Accocrding to ( [19], Proposition 4.1) we calculate that the error in our estimation is off by no more than 1.25 and so in particular, L$(E, 1){0 and we have that , (EÂQ)$(ZÂ5Z) 5 . If we sum the first 10, 000 terms of this series (using PARI to find the a n and E 1 (x)) we get L$(E, 1)r3.76.…”

Section: The Local Pairingssupporting

confidence: 55%

5-Torsion in the Shafarevich–Tate Group of a Family of Elliptic Curves

Beaver

2000

Journal of Number Theory

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We compute the ,-Selmer group for a family of elliptic curves, where , is an isogeny of degree 5, then find a practical formula for the Cassels Tate pairing on the ,-Selmer groups and use it to show that a particular family of elliptic curves have non-trivial 5-torsion in their Shafarevich Tate group.2000 Academic Press INTRODUCTIONLet K be a number field, EÂK an elliptic curve defined over K, and E(K) the Mordell Weil group of points on E with coordinates in K. Denote by (EÂK) the Shafarevich Tate group of EÂK. The Shafarevich Tate group arises frequently in the study of elliptic curves although much about its size and structure remains mysterious. The group is believed to be finite and the conjecture of Birch and Swinnerton-Dyer asserts that its cardinality is a factor in the leading term of the L-series of the elliptic curve at s=1. Kolyvagin [8] proved that E(Q) and (EÂQ) are both finite for a certain class of modular curves. Rubin [16] showed that if E is an elliptic curve defined over an imaginary quadratic field, K, with complex multiplication by K, then if L(E ÂK , 1){0, (EÂK) is finite. Although finiteness of the Shafarevich Tate group has been proven in some cases, a general result is far from being achieved and we instead try to gather information about its cardinality. For example how large is the m-torsion subgroup? In the past, the 2-and 3-torsion in (EÂK) have been studied with the most success. Kramer [10] used a descent procedure to produce a family of elliptic curves with arbitrarily large 2-torsion in (EÂK); and Cassels [2] showed that the 3-torsion can be arbitrarily large by constructing certain quadratic twists of E. Here we will give a method for finding non-trivial 5-torsion

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Section: The Local Pairingssupporting

confidence: 55%

5-Torsion in the Shafarevich–Tate Group of a Family of Elliptic Curves

Beaver

2000

Journal of Number Theory

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“…Finding rational non-torsion points on curves defined over Q is certainly non-trivial, and seems impossibly hard unless the point on the lifted curve has small height [Sil99]. There does not seem to be any obvious way to find a lift with rational points of small height (even though they certainly exist).…”

Section: Bcm On Elliptic Curves Modulo Compositesmentioning

confidence: 99%

Spreading Alerts Quietly and the Subgroup Escape Problem

et al. 2014

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We introduce a new cryptographic primitive called the blind coupon mechanism (BCM). In effect, the BCM is an authenticated bit commitment scheme, which is AND-homomorphic. It has not been known how to construct such commitments before. We show that the BCM has natural and important applications. In particular, we use it to construct a mechanism for transmitting alerts undetectably in a messagepassing system of n nodes. Our algorithms allow an alert to quickly propagate to all nodes without its source or existence being detected by an adversary, who controls all message traffic. Our proofs of security are based on a new subgroup escape problem, which seems hard on certain groups with bilinear pairings and on elliptic curves over the ring Z n .

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“…Remark. In his work [6], Silverman proposed a method to find rational points on rank one elliptic curves. Remark.…”

Section: B Finding a Good Liftingmentioning

confidence: 99%