529 research outputs found
The Infrastructure of a Global Field of Arbitrary Unit Rank
In this paper, we show a general way to interpret the infrastructure of a
global field of arbitrary unit rank. This interpretation generalizes the prior
concepts of the giant step operation and f-representations, and makes it
possible to relate the infrastructure to the (Arakelov) divisor class group of
the global field. In the case of global function fields, we present results
that establish that effective implementation of the presented methods is indeed
possible, and we show how Shanks' baby-step giant-step method can be
generalized to this situation.Comment: Revised version. Accepted for publication in Math. Com
Sato-Tate groups of genus 2 curves
We describe the analogue of the Sato-Tate conjecture for an abelian variety
over a number field; this predicts that the zeta functions of the reductions
over various finite fields, when properly normalized, have a limiting
distribution predicted by a certain group-theoretic construction related to
Hodge theory, Galois images, and endomorphisms. After making precise the
definition of the "Sato-Tate group" appearing in this conjecture, we describe
the classification of Sato-Tate groups of abelian surfaces due to
Fite-Kedlaya-Rotger-Sutherland. (These are notes from a three-lecture series
presented at the NATO Advanced Study Institute "Arithmetic of Hyperelliptic
Curves" held in Ohrid (Macedonia) August 25-September 5, 2014, and are expected
to appear in a proceedings volume.)Comment: 20 pages; includes custom class file; v2: formula of Birch correcte
Explicit root numbers of abelian varieties
The Birch and Swinnerton-Dyer conjecture predicts that the parity of the
algebraic rank of an abelian variety over a global field should be controlled
by the expected sign of the functional equation of its -function, known as
the global root number. In this paper, we give explicit formulae for the local
root numbers as a product of Jacobi symbols. This enables one to compute the
global root number, generalising work of Rohrlich who studies the case of
elliptic curves. We provide similar formulae for the root numbers after
twisting the abelian variety by a self-dual Artin representation. As an
application, we find a rational genus two hyperelliptic curve with a simple
Jacobian whose root number is invariant under quadratic twist.Comment: Corrected formulation of Theorem 7.4; to appear in Transactions of
the AM
On the automorphisms of the non-split Cartan modular curves of prime level
We study the automorphisms of the non-split Cartan modular curves
of prime level . We prove that if all the automorphisms preserve
the cusps. Furthermore, if and , the
automorphism group is generated by the modular involution given by the
normalizer of a non-split Cartan subgroup of . We
also prove that for every such that has a CM rational
point, the existence of an exceptional rational automorphism would give rise to
an exceptional rational point on the modular curve associated to
the normalizer of a non-split Cartan subgroup of
Groups from Cyclic Infrastructures and Pohlig-Hellman in Certain Infrastructures
In discrete logarithm based cryptography, a method by Pohlig and Hellman
allows solving the discrete logarithm problem efficiently if the group order is
known and has no large prime factors. The consequence is that such groups are
avoided. In the past, there have been proposals for cryptography based on
cyclic infrastructures. We will show that the Pohlig-Hellman method can be
adapted to certain cyclic infrastructures, which similarly implies that certain
infrastructures should not be used for cryptography. This generalizes a result
by M\"uller, Vanstone and Zuccherato for infrastructures obtained from
hyperelliptic function fields.
We recall the Pohlig-Hellman method, define the concept of a cyclic
infrastructure and briefly describe how to obtain such infrastructures from
certain function fields of unit rank one. Then, we describe how to obtain
cyclic groups from discrete cyclic infrastructures and how to apply the
Pohlig-Hellman method to compute absolute distances, which is in general a
computationally hard problem for cyclic infrastructures. Moreover, we give an
algorithm which allows to test whether an infrastructure satisfies certain
requirements needed for applying the Pohlig-Hellman method, and discuss whether
the Pohlig-Hellman method is applicable in infrastructures obtained from number
fields. Finally, we discuss how this influences cryptography based on cyclic
infrastructures.Comment: 14 page
On the Sato-Tate conjecture for non-generic abelian surfaces
We prove many non-generic cases of the Sato-Tate conjecture for abelian
surfaces as formulated by Fite, Kedlaya, Rotger and Sutherland, using the
potential automorphy theorems of Barnet-Lamb, Gee, Geraghty and Taylor.Comment: 21 pages. Minor changes and corrections. With an appendix by Francesc
Fit\'e. Essentially final version, to appear in Transactions of the AM
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