83 research outputs found
Rational divisors in rational divisor classes
We discuss the situation where a curve C, defined over a number field K, has a known K-rational divisor class of degree 1, and consider whether this class contains an actual K-rational divisor. When C has points everywhere locally, the local to global principle of the Brauer group gives the existence of such a divisor. In this situation, we give an alternative, more down to earth, approach, which indicates how to compute this divisor in certain situations. We also discuss examples where C does not have points everywhere locally, and where no such K-rational divisor is contained in the K-rational divisor class
Selmer Groups in Twist Families of Elliptic Curves
The aim of this article is to give some numerical data related to the order
of the Selmer groups in twist families of elliptic curves. To do this we assume
the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated
theorem of Waldspurger to get a fast algorithm to compute . Having
an extensive amount of data we compare the distribution of the order of the
Selmer groups by functions of type with small. We discuss how the
"best choice" of is depending on the conductor of the chosen elliptic
curves and the congruence classes of twist factors.Comment: to appear in Quaestiones Mathematicae. 16 page
On sums of squares and on elliptic curves over function fields
It has long been known that every positive semidefinite function of R(x, y) is the sum of four squares. This paper gives the first example of such a function which is not expressible as the sum of three squares. The proof depends on the determination of the points on a certain elliptic curve defined over C(x). The 2-component of the Tate-Safarevic group of this curve is nontrivial and infinitely divisible.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/33657/1/0000167.pd
Testing Hardy nonlocality proof with genuine energy-time entanglement
We show two experimental realizations of Hardy ladder test of quantum
nonlocality using energy-time correlated photons, following the scheme proposed
by A. Cabello \emph{et al.} [Phys. Rev. Lett. \textbf{102}, 040401 (2009)].
Unlike, previous energy-time Bell experiments, these tests require precise
tailored nonmaximally entangled states. One of them is equivalent to the
two-setting two-outcome Bell test requiring a minimum detection efficiency. The
reported experiments are still affected by the locality and detection
loopholes, but are free of the post-selection loophole of previous energy-time
and time-bin Bell tests.Comment: 5 pages, revtex4, 6 figure
Ranks of twists of elliptic curves and Hilbert's Tenth Problem
In this paper we investigate the 2-Selmer rank in families of quadratic
twists of elliptic curves over arbitrary number fields. We give sufficient
conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer
rank, and we give lower bounds for the number of twists (with bounded
conductor) that have a given 2-Selmer rank. As a consequence, under appropriate
hypotheses we can find many twists with trivial Mordell-Weil group, and
(assuming the Shafarevich-Tate conjecture) many others with infinite cyclic
Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our
results that if the Shafarevich-Tate conjecture holds, then Hilbert's Tenth
Problem has a negative answer over the ring of integers of every number field.Comment: Minor changes. To appear in Inventiones mathematica
Can a Lattice String Have a Vanishing Cosmological Constant?
We prove that a class of one-loop partition functions found by Dienes, giving
rise to a vanishing cosmological constant to one-loop, cannot be realized by a
consistent lattice string. The construction of non-supersymmetric string with a
vanishing cosmological constant therefore remains as elusive as ever. We also
discuss a new test that any one-loop partition function for a lattice string
must satisfy.Comment: 14 page
Topological Strings and (Almost) Modular Forms
The B-model topological string theory on a Calabi-Yau threefold X has a
symmetry group Gamma, generated by monodromies of the periods of X. This acts
on the topological string wave function in a natural way, governed by the
quantum mechanics of the phase space H^3(X). We show that, depending on the
choice of polarization, the genus g topological string amplitude is either a
holomorphic quasi-modular form or an almost holomorphic modular form of weight
0 under Gamma. Moreover, at each genus, certain combinations of genus g
amplitudes are both modular and holomorphic. We illustrate this for the local
Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four
dimensions and local P_2 and P_1 x P_1. As a byproduct, we also obtain a simple
way of relating the topological string amplitudes near different points in the
moduli space, which we use to give predictions for Gromov-Witten invariants of
the orbifold C^3/Z_3.Comment: 62 pages, 1 figure; v2: minor correction
Cyclotomic integers, fusion categories, and subfactors
Dimensions of objects in fusion categories are cyclotomic integers, hence
number theoretic results have implications in the study of fusion categories
and finite depth subfactors. We give two such applications. The first
application is determining a complete list of numbers in the interval (2,
76/33) which can occur as the Frobenius-Perron dimension of an object in a
fusion category. The smallest number on this list is realized in a new fusion
category which is constructed in the appendix written by V. Ostrik, while the
others are all realized by known examples. The second application proves that
in any family of graphs obtained by adding a 2-valent tree to a fixed graph,
either only finitely many graphs are principal graphs of subfactors or the
family consists of the A_n or D_n Dynkin diagrams. This result is effective,
and we apply it to several families arising in the classification of subfactors
of index less then 5.Comment: 47 pages, with an appendix by Victor Ostri
Generalised Elliptic Functions
We consider multiply periodic functions, sometimes called Abelian functions,
defined with respect to the period matrices associated with classes of
algebraic curves. We realise them as generalisations of the Weierstras
P-function using two different approaches. These functions arise naturally as
solutions to some of the important equations of mathematical physics and their
differential equations, addition formulae, and applications have all been
recent topics of study.
The first approach discussed sees the functions defined as logarithmic
derivatives of the sigma-function, a modified Riemann theta-function. We can
make use of known properties of the sigma function to derive power series
expansions and in turn the properties mentioned above. This approach has been
extended to a wide range of non hyperelliptic and higher genus curves and an
overview of recent results is given.
The second approach defines the functions algebraically, after first
modifying the curve into its equivariant form. This approach allows the use of
representation theory to derive a range of results at lower computational cost.
We discuss the development of this theory for hyperelliptic curves and how it
may be extended in the future.Comment: 16 page
- …