6,201 research outputs found
Complex Multiplication Tests for Elliptic Curves
We consider the problem of checking whether an elliptic curve defined over a
given number field has complex multiplication. We study two polynomial time
algorithms for this problem, one randomized and the other deterministic. The
randomized algorithm can be adapted to yield the discriminant of the
endomorphism ring of the curve.Comment: 13 pages, 2 tables, 1 appendi
On the existence of distortion maps on ordinary elliptic curves
Distortion maps allow one to solve the Decision Diffie-Hellman problem on
subgroups of points on the elliptic curve. In the case of ordinary elliptic
curves over finite fields, it is known that in most cases there are no
distortion maps. In this article we characterize the existence of distortion
maps in the remaining cases.Comment: 3 Pages (Updated version corrects an error in the previous version
Triangulated categories of mixed motives
This book discusses the construction of triangulated categories of mixed
motives over a noetherian scheme of finite dimension, extending Voevodsky's
definition of motives over a field. In particular, it is shown that motives
with rational coefficients satisfy the formalism of the six operations of
Grothendieck. This is achieved by studying descent properties of motives, as
well as by comparing different presentations of these categories, following and
extending insights and constructions of Deligne, Beilinson, Bloch, Thomason,
Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, R\"ondigs, {\O}stv{\ae}r,
and others. In particular, the relation of motives with -theory is addressed
in full, and we prove the absolute purity theorem with rational coefficients,
using Quillen's localization theorem in algebraic -theory together with a
variation on the Grothendieck-Riemann-Roch theorem. Using resolution of
singularities via alterations of de Jong-Gabber, this leads to a version of
Grothendieck-Verdier duality for constructible motivic sheaves with rational
coefficients over rather general base schemes. We also study versions with
integral coefficients, constructed via sheaves with transfers, for which we
obtain partial results. Finally, we associate to any mixed Weil cohomology a
system of categories of coefficients and well behaved realization functors,
establishing a correspondence between mixed Weil cohomologies and suitable
systems of coefficients. The results of this book have already served as ground
reference in many subsequent works on motivic sheaves and their realizations,
and pointers to the most recent developments of the theory are given in the
introduction.Comment: This is the final version. To appear in the series Springer
Monographs in Mathematic
Dendroidal Segal spaces and infinity-operads
We introduce the dendroidal analogs of the notions of complete Segal space
and of Segal category, and construct two appropriate model categories for which
each of these notions corresponds to the property of being fibrant. We prove
that these two model categories are Quillen equivalent to each other, and to
the monoidal model category for infinity-operads which we constructed in an
earlier paper. By slicing over the monoidal unit objects in these model
categories, we derive as immediate corollaries the known comparison results
between Joyal's quasi-categories, Rezk's complete Segal spaces, and Segal
categories.Comment: We replaced a wrong technical lemma by a correct proposition at the
begining of Section 8. This does not affect the main results of this article
(in particular, the end of Section 8 is unchanged). To appear in J. Topo
Lefschetz and Hirzebruch-Riemann-Roch formulas via noncommutative motives
V. Lunts has recently established Lefschetz fixed point theorems for
Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov
introduced the noncommutative analogue of the Hirzebruch-Riemann-Roch theorem.
In this short article, we see how these constructions and computations formally
stem from their motivic counterparts.Comment: 16 pages; revised versio
Mixed Weil cohomologies
We define, for a regular scheme and a given field of characteristic zero
\KK, the notion of \KK-linear mixed Weil cohomology on smooth -schemes
by a simple set of properties, mainly: Nisnevich descent, homotopy invariance,
stability (which means that the cohomology of \GG_{m} behaves correctly), and
K\"unneth formula. We prove that any mixed Weil cohomology defined on smooth
-schemes induces a symmetric monoidal realization of some suitable
triangulated category of motives over to the derived category of the field
\KK. This implies a finiteness theorem and a Poincar\'e duality theorem for
such a cohomology with respect to smooth and projective -schemes (which can
be extended to smooth -schemes when is the spectrum of a perfect field).
This formalism also provides a convenient tool to understand the comparison of
such cohomology theories. Our main examples are algebraic de Rham cohomology
and rigid cohomology, and the Berthelot-Ogus isomorphism relating them.Comment: update references; hopefully improve the expositio
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