24,441 research outputs found
Rational points over finite fields for regular models of algebraic varieties of Hodge type
Let be a discrete valuation ring of mixed characteristics , with
finite residue field and fraction field , let be a finite extension
of , and let be a regular, proper and flat -scheme, with generic
fibre and special fibre . Assume that is geometrically
connected and of Hodge type in positive degrees. Then we show that the
number of -rational points of satisfies the congruence mod . Thanks to \cite{BBE07}, we deduce such congruences from a
vanishing theorem for the Witt cohomology groups H^q(X_k, W\sO_{X_k,\Q}), for
. In our proof of this last result, a key step is the construction of a
trace morphism between the Witt cohomologies of the special fibres of two flat
regular -schemes and of the same dimension, defined by a surjective
projective morphism .Comment: 85 pages. Refereed version. Signs have been modified in some
definition
Resonance webs of hyperplane arrangements
Each irreducible component of the first resonance variety of a hyperplane
arrangement naturally determines a codimension one foliation on the ambient
space. The superposition of these foliations define what we call the resonance
web of the arrangement. In this paper we initiate the study of these objects
with emphasis on their spaces of abelian relations.Comment: (v2) Minor changes following suggestions of the referee. To appear in
the Proceedings of the 2nd MSJ-SI on Arrangements of Hyperplane
Galois invariant smoothness basis
This text answers a question raised by Joux and the second author about the
computation of discrete logarithms in the multiplicative group of finite
fields. Given a finite residue field \bK, one looks for a smoothness basis
for \bK^* that is left invariant by automorphisms of \bK. For a broad class
of finite fields, we manage to construct models that allow such a smoothness
basis. This work aims at accelerating discrete logarithm computations in such
fields. We treat the cases of codimension one (the linear sieve) and
codimension two (the function field sieve)
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