Exhibiting SHA[2] on hyperelliptic Jacobians
نویسندگان
چکیده
منابع مشابه
Exhibiting Sha[2] on Hyperelliptic Jacobians
We discuss approaches to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of gen...
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In his previous papers [25, 26, 28] the author proved that in characteristic 6= 2 the jacobian J(C) of a hyperelliptic curve C : y = f(x) has only trivial endomorphisms over an algebraic closure Ka of the ground field K if the Galois group Gal(f) of the irreducible polynomial f(x) ∈ K[x] is either the symmetric group Sn or the alternating group An. Here n ≥ 9 is the degree of f . The goal of th...
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Consider the Jacobian of a hyperelliptic genus two curve de ned over a nite eld. Under certain restrictions on the endomorphism ring of the Jacobian, we give an explicit description of all non-degenerate, bilinear, anti-symmetric and Galois-invariant pairings on the Jacobian. From this description it follows that no such pairing can be computed more e ciently than the Weil pairing. To establish...
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In [17] the author proved that in characteristic 0 the jacobian J(C) = J(Cf ) of a hyperelliptic curve C = Cf : y 2 = f(x) has only trivial endomorphisms over an algebraic closure Ka of the ground field K if the Galois group Gal(f) of the irreducible polynomial f ∈ K[x] is “very big”. Namely, if n = deg(f) ≥ 5 and Gal(f) is either the symmetric group Sn or the alternating group An then the ring...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2006
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2005.10.007