The Magma Algebra System I: The User Language

Bosma, Wieb; Cannon, John; Playoust, Catherine

doi:10.1006/jsco.1996.0125

Cited by 6,079 publications

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References 7 publications

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“…Condition ( (1,4), (2,5), or (3,6) Proof. Let J be an abelian surface over F q whose Weil polynomial P J = X 4 + aX 3 + bX 2 + aqX + q 2 satisfies one of (1.1)-(1.5).…”

Section: Moreover the Surface J Is Simple If And Only If Eithermentioning

confidence: 99%

“…13 (9, 42) 9 (6, 20) 7 (4,16) 5 (3,6) or (8,26) 4 (2,5), (4, 11), or (6, 17) 3 (1,4), (3,5), or (4, 10) 2 (0, 3), (1,0), (1,4), (2,5), or (3,6) The special form required of the Frobenius endomorphism in [4] has an immediate consequence for the shape of its characteristic polynomial, and by inspection the above polynomials do not have the required shape. Thus the main result of [4] follows from the above result.…”

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confidence: 99%

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$p^k$-torsion of genus two curves over $\mathbb {F}_{p^m}$

Zieve

2010

Math. Comp.

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Abstract. We determine the isogeny classes of abelian surfaces over F q whose group of F q -rational points has order divisible by q 2 . We also solve the same problem for Jacobians of genus-2 curves.In a recent paper [4], Ravnshøj proved: if C is a genus-2 curve over a prime field F p , and if one assumes that the endomorphism ring of the Jacobian J of C is the ring of integers in a primitive quartic CM-field, and that the Frobenius endomorphism of J has a certain special form, then p 2 #J(F p ). Our purpose here is to deduce this conclusion under less restrictive hypotheses. We write q = p m , where p is prime, and for any abelian variety J over F q we let P J denote the Weil polynomial of J, namely the characteristic polynomial of the Frobenius endomorphism π J of J. As shown by Tate [6, Thm. 1], two abelian varieties over F q are isogenous if and only if their Weil polynomials are identical. Thus, the following result describes the isogeny classes of abelian surfaces J over F q for which q 2 | #J(F q ). The special form required of the Frobenius endomorphism in [4] has an immediate consequence for the shape of its characteristic polynomial, and by inspection the above polynomials do not have the required shape. Thus the main result of [4] follows from the above result.

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“…Condition ( (1,4), (2,5), or (3,6) Proof. Let J be an abelian surface over F q whose Weil polynomial P J = X 4 + aX 3 + bX 2 + aqX + q 2 satisfies one of (1.1)-(1.5).…”

Section: Moreover the Surface J Is Simple If And Only If Eithermentioning

confidence: 99%

mentioning

confidence: 99%

$p^k$-torsion of genus two curves over $\mathbb {F}_{p^m}$

Zieve

2010

Math. Comp.

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“…Suppose Γ = A, B is a finite-volume Kleinian group such that the six conditions of the theorem hold. Then Condition (4) ensures that kΓ is a number field with exactly one complex place, Conditions (1) and (6) imply that trΓ consists of algebraic integers by Lemma 3.3, and Conditions (4) and (5) guarantee that AΓ is ramified at all real places of kΓ, so Γ is arithmetic by Theorem 5.2. Finally, Conditions (1) and (3) Proof.…”

Section: Theorem 54 a Finite-covolume Kleinian Group γ Is An Arithmmentioning

confidence: 99%

Jørgensen number and arithmeticity

Callahan

2009

Conform. Geom. Dyn.

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Abstract. The Jørgensen number of a rank-two non-elementary Kleinian group Γ isJørgensen's Inequality guarantees J(Γ) ≥ 1, and Γ is a Jørgensen group if J(Γ) = 1. This paper shows that the only torsion-free Jørgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic Jørgensen groups, and establishes a characterization of cocompact arithmetic Jørgensen groups. The paper concludes with computations of J(Γ) for several noncocompact Kleinian groups including some two-bridge knot and link groups.

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“…Algorithms exist to find all solutions to S-unit equations; see, for example, [15]. In the case S = ∅ this is implemented in Magma [2], so when f is monic with f (0) = ±1 we may find all solutions this way. Note, however, that the algorithm requires us to find the unit group of the number field K, which can be time-consuming when K has large degree or discriminant.…”

Section: Lemma 2 B(fg N) = B(f N)b(g N)mentioning

confidence: 99%

Unimodular integer circulants

Cremona¹

2008

Math. Comp.

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Abstract. We study families of integer circulant matrices and methods for determining which are unimodular. This problem arises in the study of cyclically presented groups, and leads to the following problem concerning polynomials with integer coefficients: given a polynomial f (x) ∈ Z[x], determine all those n ∈ N such that Res(f (x), x n − 1) = ±1. In this paper we describe methods for resolving this problem, including a method based on the use of Strassman's Theorem on p-adic power series, which are effective in many cases. The methods are illustrated with examples arising in the study of cyclically presented groups and further examples which illustrate the strengths and weaknesses of the methods for polynomials of higher degree.

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The Magma Algebra System I: The User Language

Abstract: In the first of two papers on Magma, a new system for computational algebra, we present the Magma language, outline the design principles and theoretical background, and indicate its scope and use. Particular attention is given to the constructors for structures, maps, and sets.

Cited by 6,079 publications

References 7 publications

$p^k$-torsion of genus two curves over $\mathbb {F}_{p^m}$

$p^k$-torsion of genus two curves over $\mathbb {F}_{p^m}$

Jørgensen number and arithmeticity

Unimodular integer circulants

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