86,808 research outputs found
On Taking Square Roots without Quadratic Nonresidues over Finite Fields
We present a novel idea to compute square roots over finite fields, without
being given any quadratic nonresidue, and without assuming any unproven
hypothesis. The algorithm is deterministic and the proof is elementary. In some
cases, the square root algorithm runs in bit operations
over finite fields with elements. As an application, we construct a
deterministic primality proving algorithm, which runs in
for some integers .Comment: 14 page
Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties
We investigate the behaviour of Tamagawa numbers of semistable principally
polarised abelian varieties in extensions of local fields. In view of the
Raynaud parametrisation, this translates into a purely algebraic problem
concerning the number of -invariant points on a quotient of -lattices
for varying subgroups of and integers . In
particular, we give a simple formula for the change of Tamagawa numbers in
totally ramified extensions (corresponding to varying ) and one that
computes Tamagawa numbers up to rational squares in general extensions.
As an application, we extend some of the existing results on the -parity
conjecture for Selmer groups of abelian varieties by allowing more general
local behaviour. We also give a complete classification of the behaviour of
Tamagawa numbers for semistable 2-dimensional principally polarised abelian
varieties, that is similar to the well-known one for elliptic curves. The
appendix explains how to use this classification for Jacobians of genus 2
hyperelliptic curves given by equations of the form , under some
simplifying hypotheses.Comment: Two new lemmas are added. The first describes permutation
representations, and the second describes the dependence of the B-group on
the maximal fixpoint-free invariant sublattice. Contact details and
bibliographic details have been update
Number Fields in Fibers: the Geometrically Abelian Case with Rational Critical Values
Let X be an algebraic curve over Q and t a non-constant Q-rational function
on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a
point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the
number fields Q(P_1), ..., Q(P_N) there are at least cN distinct. We prove this
conjecture in the special case when t defines a geometrically abelian covering
of the projective line, and the critical values of t are all rational. This
implies, in particular, that our conjecture follows from a famous conjecture of
Schinzel.Comment: Some typos are corrected. The article is now accepted in Periodica
Math. Hungaric
Nonisomorphic curves that become isomorphic over extensions of coprime degrees
We show that one can find two nonisomorphic curves over a field K that become
isomorphic to one another over two finite extensions of K whose degrees over K
are coprime to one another.
More specifically, let K_0 be an arbitrary prime field and let r and s be
integers greater than 1 that are coprime to one another. We show that one can
find a finite extension K of K_0, a degree-r extension L of K, a degree-s
extension M of K, and two curves C and D over K such that C and D become
isomorphic to one another over L and over M, but not over any proper
subextensions of L/K or M/K.
We show that such C and D can never have genus 0, and that if K is finite, C
and D can have genus 1 if and only if {r,s} = {2,3} and K is an odd-degree
extension of F_3. On the other hand, when {r,s}={2,3} we show that genus-2
examples occur in every characteristic other than 3.
Our detailed analysis of the case {r,s} = {2,3} shows that over every finite
field K there exist nonisomorphic curves C and D that become isomorphic to one
another over the quadratic and cubic extensions of K.
Most of our proofs rely on Galois cohomology. Without using Galois
cohomology, we show that two nonisomorphic genus-0 curves over an arbitrary
field remain nonisomorphic over every odd-degree extension of the base field.Comment: LaTeX, 32 pages. Further references added to the discussion in
Section 1
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