What is the Rank of the Demjanenko Matrix?

Folz, H. G.; Zimmer, Horst G.

doi:10.1016/s0747-7171(87)80053-6

Cited by 6 publications

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“…The matrices M m and M m which appear above would seem to have some relation to the Demjanenko matrices used by Folz and Zimmer [6] in studying torsion points on elliptic curves and used by other authors ( [7], [16], [17]) to investigate the class number of (abelian) number fields.…”

Section: Description Of the Algorithmmentioning

confidence: 91%

“…In this article we will explain how to computeĥ(P ) without factoring ∆, essentially by grouping together terms in (3) whoseλ p 's have the same form. Our method is not completely factorization free, since it does require the prime factorization of the quantity gcd(c 4 , c 6 ); but in practice, it is usually feasible to factor gcd(c 4 , c 6 ) even when ∆ is far too large to be effectively factored.…”

Section: P|∆ P Dλ P (P ) (3)mentioning

confidence: 99%

“…Compute Associated Quantities. Compute the quantities b 2 , b 4 , b 6 , b 8 , c 4 , c 6 , ∆ listed in (4).…”

Section: Description Of the Algorithmmentioning

confidence: 99%

“…Additive Reduction Contributions. For primes p ≥ 5, so in particular for p ≥ p 0 , the primes of additive reduction are precisely the primes dividing both c 4 and c 6 . We already found these primes in step (2) when we factored gcd(c 4 , c 6 ), so we can use the algorithm [20] to compute theλ p (P )'s and add them onto the total, while also removing them from ∆:…”

Section: Description Of the Algorithmmentioning

confidence: 99%

“…. , a 6 are large. We note that this is not a mere intellectual exercise, since curves with huge integer coefficients have already made their appearance in the search for curves whose Mordell-Weil group E(Q) has large rank [5], [11], [12], [13], [14], and the standard tool for proving that a set of points P 1 , .…”

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Computing canonical heights with little (or no) factorization

Silverman

1997

Math. Comp.

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Abstract. Let E/Q be an elliptic curve with discriminant ∆, and let P ∈ E(Q). The standard method for computing the canonical heightĥ(P ) is as a sum of local heightsĥ(P ) =λ∞(P) + pλ p(P ). There are well-known series for computing the archimedean heightλ∞(P ), and the non-archimedean heightsλp(P ) are easily computed as soon as all prime factors of ∆ have been determined. However, for curves with large coefficients it may be difficult or impossible to factor ∆. In this note we give a method for computing the nonarchimedean contribution toĥ(P ) which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm.Let E be an elliptic curve defined over a number field K, say given by a Weierstrass equationThe canonical height on E is a quadratic formThe canonical height is an extremely important theoretical and computational tool in the arithmetic study of elliptic curves. See [18, Chapter VIII, Section 9] for the definition and basic properties ofĥ, and [20], [21], and [23] for some discussion of how to computeĥ in practice. In this paper, which may be considered as a continuation of our earlier note [20], we will discuss the computation of the canonical height for curves E whose coefficients a 1 , . . . , a 6 are large. We note that this is not a mere intellectual exercise, since curves with huge integer coefficients have already made their appearance in the search for curves whose Mordell-Weil group E(Q) has large rank [5], [11], [12], [13], [14], and the standard tool for proving that a set of points P 1 , . . . , P r ∈ E(Q) is linearly independent is to check the non-vanishing of the height regulator matrix det P i , P j . Here the height pairing · , · is defined (up to a normalizing factor) by the formulaTate's definitionĥ(P ) = lim n→∞ 4 −n h x(2 n P ) of the canonical height is not practical for numerical computations. Instead, one uses the Néron-Tate decomposition of the canonical height into a sum of local heights, one for each distinct

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Section: Description Of the Algorithmmentioning

confidence: 91%

Section: P|∆ P Dλ P (P ) (3)mentioning

confidence: 99%

“…Compute Associated Quantities. Compute the quantities b 2 , b 4 , b 6 , b 8 , c 4 , c 6 , ∆ listed in (4).…”

Section: Description Of the Algorithmmentioning

confidence: 99%

Section: Description Of the Algorithmmentioning

confidence: 99%

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Computing canonical heights with little (or no) factorization

Silverman

1997

Math. Comp.

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On a Special Ideal Contained in the Stickelberger Ideal

Skula

1996

Journal of Number Theory

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In this paper an ideal B of the group ring R=Z[G ] generated by a special Kummer element is studied (G means a cyclic group of order l&1 and l and odd prime). The group index [R*: B] is expressed by means of the relative class number of the l th cyclotomic field (R* a subring of R). The ideal B is investigated also mod l. The connection with the (modified) Demjanenko and the Benneton system of congruences is mentioned.

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Demjanenko matrix and 2-divisibility of class numbers

Schwarz

1993

Arch. Math

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What is the Rank of the Demjanenko Matrix?

Cited by 6 publications

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Computing canonical heights with little (or no) factorization

Computing canonical heights with little (or no) factorization

On a Special Ideal Contained in the Stickelberger Ideal

Demjanenko matrix and 2-divisibility of class numbers

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