1,540 research outputs found
Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces
We discover suprising connections between three seemingly different problems:
finding right triangles with rational sides in a non-Euclidean geometry,
finding three integers such that the difference of the squares of any two is a
square, and the problem of finding rational points on an algebraic surface in
algebraic geometry. We will also reinterpret Euler's work on the second problem
with a modern point of view.Comment: 11 pages, 1 figur
Explicit Selmer groups for cyclic covers of P^1
For any abelian variety J over a global field k and an isogeny phi: J -> J,
the Selmer group Sel^phi(J,k) is a subgroup of the Galois cohomology group
H^1(Gal(ksep/k), J[phi]), defined in terms of local data. When J is the
Jacobian of a cyclic cover of P^1 of prime degree p, the Selmer group has a
quotient by a subgroup of order at most p that is isomorphic to the `fake
Selmer group', whose definition is more amenable to explicit computations. In
this paper we define in the same setting the `explicit Selmer group', which is
isomorphic to the Selmer group itself and just as amenable to explicit
computations as the fake Selmer group. This is useful for describing the
associated covering spaces explicitly and may thus help in developing methods
for second descents on the Jacobians considered.Comment: 12 pages. Mostly expository changes and a new title (previously
`unfaking the fake Selmer group'
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