Rigorous computation of the endomorphism ring of a Jacobian

Costa, Edgar; Mascot, Nicolas; Sijsling, Jeroen; Voight, John

doi:10.1090/mcom/3373

Cited by 55 publications

(73 citation statements)

References 29 publications

(26 reference statements)

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“…Both of the curves D 1 and D 2 have endomorphism ring Z over Q, as can be verified by the -adic methods in [10]. Similar considerations show that the Jacobians of D 1 and D 2 are not isogenous.…”

supporting

confidence: 60%

“…Numerical computation shows that End(Jac(C)) = Z and that End(Jac(F )) = Z, which can be confirmed by the -adic methods in [10]. It follows that Hom(Jac(F ), Jac(C)) = 0.…”

Section: Examplesmentioning

confidence: 54%

“…While this is worthwhile and part of future work, rigorous error bounds would make little difference for our applications, since we have no a priori height bound on the rational numbers we are trying to recognize from floating point approximations. One of our objectives is to provide input for the rigorous verification procedures described in [10]. Our main application is to find decompositions of Jac(C) via its endomorphism ring End k (J) = Hom k (J, J).…”

Section: Curvementioning

confidence: 99%

“…A computation on homology again shows that this morphism cannot come from a morphism of curves C → D 3 ; if it did, the degree of such a morphism would have to be 6, which is impossible in light of the Riemann-Hurwitz formula. An explicit correspondence between C and D 3 can in principle be found by using the methods in [10]; however, this will still be a rather involved calculation, which we have therefore not performed yet. of the Macbeath curve from [16], which is due to Bradley Brock.…”

Section: Examplesmentioning

confidence: 99%

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Numerical computation of endomorphism rings of Jacobians

2019

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We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as the decomposition of their Jacobians up to isogeny, can be calculated heuristically. Particular applications include the determination of (generically) non-Galois morphisms between curves and the identification of Prym varieties.

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supporting

confidence: 60%

“…Numerical computation shows that End(Jac(C)) = Z and that End(Jac(F )) = Z, which can be confirmed by the -adic methods in [10]. It follows that Hom(Jac(F ), Jac(C)) = 0.…”

Section: Examplesmentioning

confidence: 54%

Section: Curvementioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

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Numerical computation of endomorphism rings of Jacobians

2019

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“…The fact that a ℓ is effective follows from the fact that one can effectively compute a parameter of maximal growth for the ℓ-adic torsion representation (Remark 3.7), an upper bound for the value of d (Remark 4.14), and the endomorphism ring End K (E) ([1], [8], [16]).…”

Section: 3mentioning

confidence: 99%

An explicit open image theorem for products of elliptic curves

Lombardo

2016

Journal of Number Theory

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Let E be an elliptic curve defined over a number field K, let α ∈ E(K) be a point of infinite order, and let N −1 α be the set of N -division points of α in E(K). We prove strong effective and uniform results for the degrees of the Kummer extensions [K(E[N ], N −1 α) : K(E[N ])]. When K = Q, and under a minimal assumption on α, we show that the inequality [Q(E[N ], N −1 α) : Q(E[N ])] cN 2 holds with a constant c independent of both E and α.We shall often use divisibility conditions involving the symbols ℓ ∞ (where ℓ is a prime) and ∞. Our convention is that every power of ℓ divides ℓ ∞ , every positive integer divides ∞, and ℓ ∞ divides ∞.Recall from the Introduction that we denote by K M the field K(E[M ]) generated by the coordinates of the M -torsion points of E, and by K M,N (for N | M ) the field K(E[M ], N −1 α). We extend this notation by setting K ℓ ∞ = n K ℓ n , K ∞ = M K M , and more generally, forIf H is a subgroup of GL 2 (Z ℓ ), we denote by Z ℓ [H] the sub-Z ℓ -algebra of Mat 2 (Z ℓ ) generated by the elements of H.Let G be a (profinite) group. We write G ′ for its derived subgroup, namely, the subgroup of G (topologically) generated by commutators, and G ab = G/G ′ for its abelianisation, namely, its largest abelian (profinite) quotient. We say that a finite simple group S occurs in a profinite group G if there are closed subgroups H 1 , H 2 of G, with H 1 ⊳ H 2 , such that H 2 /H 1 is isomorphic to S. Finally, we denote by exp G the exponent of a finite group G, namely, the smallest integer e 1 such that g e = 1 for every g ∈ G.2.2. The ℓ-adic and adelic failures. We start by observing that it is enough to restrict our attention to the case N = M :Remark 2.1. Suppose that there is a constant C 1 satisfying M 2 [K M,M : K M ] divides C for all positive integers M . Then for any N | M , since [K M,M : K M,N ] divides (M/N ) 2 , we have that

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Chabauty–Coleman Experiments for Genus 3 Hyperelliptic Curves

Balakrishnan

Bianchi

Cantoral-Farfán

et al. 2019

Association for Women in Mathematics Series

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We describe a computation of rational points on genus 3 hyperelliptic curves C defined over Q whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in C(Q).

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Rigorous computation of the endomorphism ring of a Jacobian

Abstract: We describe several improvements and generalizations to algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field.

Cited by 55 publications

References 29 publications

Numerical computation of endomorphism rings of Jacobians

Numerical computation of endomorphism rings of Jacobians

An explicit open image theorem for products of elliptic curves

Chabauty–Coleman Experiments for Genus 3 Hyperelliptic Curves

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