3,096 research outputs found
Quadratic Chabauty for (bi)elliptic curves and Kim's conjecture
We explore a number of problems related to the quadratic Chabauty method for
determining integral points on hyperbolic curves. We remove the assumption of
semistability in the description of the quadratic Chabauty sets
containing the integral points
of an elliptic curve of rank at most . Motivated
by a conjecture of Kim, we then investigate theoretically and computationally
the set-theoretic difference . We also consider some algorithmic questions arising
from Balakrishnan--Dogra's explicit quadratic Chabauty for the rational points
of a genus-two bielliptic curve. As an example, we provide a new solution to a
problem of Diophantus which was first solved by Wetherell. Computationally, the
main difference from the previous approach to quadratic Chabauty is the use of
the -adic sigma function in place of a double Coleman integral.Comment: Replaced Conjecture 4.12 with Theorem 1.8; rewrote the introduction
and fixed minor issues according to the referee's and PhD examiners'
suggestions; 42 page
Towards Spinfoam Cosmology
We compute the transition amplitude between coherent quantum-states of
geometry peaked on homogeneous isotropic metrics. We use the holomorphic
representations of loop quantum gravity and the
Kaminski-Kisielowski-Lewandowski generalization of the new vertex, and work at
first order in the vertex expansion, second order in the graph (multipole)
expansion, and first order in 1/volume. We show that the resulting amplitude is
in the kernel of a differential operator whose classical limit is the canonical
hamiltonian of a Friedmann-Robertson-Walker cosmology. This result is an
indication that the dynamics of loop quantum gravity defined by the new vertex
yields the Friedmann equation in the appropriate limit.Comment: 8 page
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4
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