1,274 research outputs found
Second p descents on elliptic curves
Let p be a prime and let C be a genus one curve over a number field k
representing an element of order dividing p in the Shafarevich-Tate group of
its Jacobian. We describe an algorithm which computes the set of D in the
Shafarevich-Tate group such that pD = C and obtains explicit models for these D
as curves in projective space. This leads to a practical algorithm for
performing 9-descents on elliptic curves over the rationals.Comment: 45 page
On the nonexistence of certain curves of genus two
We prove that if q is a power of an odd prime then there is no genus-2 curve
over F_q whose Jacobian has characteristic polynomial of Frobenius equal to x^4
+ (2-2q)x^2 + q^2. Our proof uses the Brauer relations in a biquadratic
extension of Q to show that every principally polarized abelian surface over
F_q with the given characteristic polynomial splits over F_{q^2} as a product
of polarized elliptic curves.Comment: LaTeX, 13 page
Discrete logarithms in curves over finite fields
A survey on algorithms for computing discrete logarithms in Jacobians of
curves over finite fields
Solving discrete logarithms on a 170-bit MNT curve by pairing reduction
Pairing based cryptography is in a dangerous position following the
breakthroughs on discrete logarithms computations in finite fields of small
characteristic. Remaining instances are built over finite fields of large
characteristic and their security relies on the fact that the embedding field
of the underlying curve is relatively large. How large is debatable. The aim of
our work is to sustain the claim that the combination of degree 3 embedding and
too small finite fields obviously does not provide enough security. As a
computational example, we solve the DLP on a 170-bit MNT curve, by exploiting
the pairing embedding to a 508-bit, degree-3 extension of the base field.Comment: to appear in the Lecture Notes in Computer Science (LNCS
Twists of X(7) and primitive solutions to x^2+y^3=z^7
We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent
argument involving the simple group of order 168 reduces the problem to the
determination of the set of rational points on a finite set of twists of the
Klein quartic curve X. To restrict the set of relevant twists, we exploit the
isomorphism between X and the modular curve X(7), and use modularity of
elliptic curves and level lowering. This leaves 10 genus-3 curves, whose
rational points are found by a combination of methods.Comment: 47 page
Potential Sha for abelian varieties
We show that the p-torsion in the Tate-Shafarevich group of any principally
polarized abelian variety over a number field is unbounded as one ranges over
extensions of degree O(p), the implied constant depending only on the dimension
of the abelian variety.Comment: Version 2: improved exposition and corrected various small error
The Cassels-Tate pairing on polarized abelian varieties
Let (A,\lambda) be a principally polarized abelian variety defined over a
global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd
denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and
Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd
\rightarrow \Q/\Z. If A is an elliptic curve, then by a result of Cassels the
pairing is alternating. But in general it is only antisymmetric.
Using some new but equivalent definitions of the pairing, we derive general
criteria deciding whether it is alternating and whether there exists some
alternating nondegenerate pairing on \Sha(A)_\nd. These criteria are expressed
in terms of an element c \in \Sha(A)_\nd that is canonically associated to the
polarization \lambda. In the case that A is the Jacobian of some curve, a
down-to-earth version of the result allows us to determine effectively whether
\#\Sha(A) (if finite) is a square or twice a square. We then apply this to
prove that a positive proportion (in some precise sense) of all hyperelliptic
curves of even genus g \ge 2 over \Q have a Jacobian with nonsquare \#\Sha (if
finite). For example, it appears that this density is about 13% for curves of
genus 2. The proof makes use of a general result relating global and local
densities; this result can be applied in other situations.Comment: 41 pages, published versio
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