737 research outputs found
Linear relations in families of powers of elliptic curves
Motivated by recent work of Masser and Zannier on simultaneous torsion on the
Legendre elliptic curve of equation , we
prove that, given linearly independent points on with coordinates in
, there are at most finitely many complex numbers
such that the points satisfy
two independent relations on . This is a special case of
conjectures about Unlikely Intersections on families of abelian varieties
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
Bounded gaps between primes with a given primitive root, II
Let be a natural number, and let be a set containing at
least primes. We show that one can find infinitely many strings of
consecutive primes each of which has some as a primitive
root, all lying in an interval of length . This is
a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a
result on an elliptic analogue of Artin's conjecture. Let be an
elliptic curve with an irrational -torsion point. Assume GRH. Then for every
, there are infinitely many strings of consecutive primes for which
is cyclic, all lying an interval of length . If has CM, then the GRH assumption can be removed. Here , ,
and are absolute constants
Snyder's Model -- de Sitter Special Relativity Duality and de Sitter Gravity
Between Snyder's quantized space-time model in de Sitter space of momenta and
the \dS special relativity on \dS-spacetime of radius with Beltrami
coordinates, there is a one-to-one dual correspondence supported by a minimum
uncertainty-like argument. Together with Planck length , should be a fundamental constant. They lead to a
dimensionless constant . These indicate that physics at these two scales should be dual to
each other and there is in-between gravity of local \dS-invariance
characterized by . A simple model of \dS-gravity with a gauge-like action on
umbilical manifolds may show these characters. It can pass the observation
tests and support the duality.Comment: 32 page
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