137 research outputs found
Shalika germs for tamely ramified elements in
Degenerating the action of the elliptic Hall algebra on the Fock space, we
give a combinatorial formula for the Shalika germs of tamely ramified regular
semisimple elements of over a nonarchimedean local field. As a
byproduct, we compute the weight polynomials of affine Springer fibers in type
A and orbital integrals of tamely ramified regular semisimple elements. We
conjecture that the Shalika germs of correspond to residues of torus
localization weights of a certain quasi-coherent sheaf on
the Hilbert scheme of points on , thereby finding a geometric
interpretation for them. As corollaries, we obtain the polynomiality in of
point-counts of compactified Jacobians of planar curves, as well as a virtual
version of the Cherednik-Danilenko conjecture on their Betti numbers. Our
results also provide further evidence for the ORS conjecture relating
compactified Jacobians and HOMFLY-PT invariants of algebraic knots.Comment: 47 pages, added clarifications on the unramified case and an
application to components of affine Springer fibers, fixed typos and
reference
Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve
It has been conjectured that for sufficiently large, there are no
quadratic polynomials in with rational periodic points of period
. Morton proved there were none with , by showing that the genus~
algebraic curve that classifies periodic points of period~4 is birational to
, whose rational points had been previously computed. We prove there
are none with . Here the relevant curve has genus~, but it has a
genus~ quotient, whose rational points we compute by performing
a~-descent on its Jacobian and applying a refinement of the method of
Chabauty and Coleman. We hope that our computation will serve as a model for
others who need to compute rational points on hyperelliptic curves. We also
describe the three possible Gal-stable -cycles, and show that
there exist Gal-stable -cycles for infinitely many .
Furthermore, we answer a question of Morton by showing that the genus~
curve and its quotient are not modular. Finally, we mention some partial
results for
Simultaneous -orderings and minimising volumes in number fields
In the paper "On the interpolation of integer-valued polynomials" (Journal of
Number Theory 133 (2013), pp. 4224--4232.) V. Volkov and F. Petrov consider the
problem of existence of the so-called -universal sets (related to
simultaneous -orderings of Bhargava) in the ring of Gaussian integers. We
extend their results to arbitrary imaginary quadratic number fields and prove
an existence theorem that provides a strong counterexample to a conjecture of
Volkov-Petrov on minimal cardinality of -universal sets. Along the way, we
discover a link with Euler-Kronecker constants and prove a lower bound on
Euler-Kronecker constants which is of the same order of magnitude as the one
obtained by Ihara.Comment: new version, substantial corrections in section 6, will appear in
Journal of Number Theor
Nilpoten Conjugacy Classes of Reductive p-adic Lie Algebras and Definability in Pas's Language
We will study the following question: Are nilpotent conjugacy classes of reductive Lie algebras over p-adic fields definable by a formula in Pas's language. We answer in the affirmative in three cases: special orthogonal Lie algebras so(n) for n odd, special linear Lie algebra sl(3) and the exceptional Lie algebra G2 over p-adic fields. The nilpotent conjugacy classes in these three cases have been parameterized by Waldspurger (so(n)) and S. DeBacker(sl(3), G2). For sl (3) and G2 we are required to extend Pas's language by a finite number of symbols
Tropical totally positive matrices
We investigate the tropical analogues of totally positive and totally
nonnegative matrices. These arise when considering the images by the
nonarchimedean valuation of the corresponding classes of matrices over a real
nonarchimedean valued field, like the field of real Puiseux series. We show
that the nonarchimedean valuation sends the totally positive matrices precisely
to the Monge matrices. This leads to explicit polyhedral representations of the
tropical analogues of totally positive and totally nonnegative matrices. We
also show that tropical totally nonnegative matrices with a finite permanent
can be factorized in terms of elementary matrices. We finally determine the
eigenvalues of tropical totally nonnegative matrices, and relate them with the
eigenvalues of totally nonnegative matrices over nonarchimedean fields.Comment: The first author has been partially supported by the PGMO Program of
FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author
is sported by the French Chateaubriand grant and INRIA postdoctoral
fellowshi
Complex Multiplication of Exactly Solvable Calabi-Yau Varieties
We propose a conceptual framework that leads to an abstract characterization
for the exact solvability of Calabi-Yau varieties in terms of abelian varieties
with complex multiplication. The abelian manifolds are derived from the
cohomology of the Calabi-Yau manifold, and the conformal field theoretic
quantities of the underlying string emerge from the number theoretic structure
induced on the varieties by the complex multiplication symmetry. The geometric
structure that provides a conceptual interpretation of the relation between
geometry and the conformal field theory is discrete, and turns out to be given
by the torsion points on the abelian varieties.Comment: 44 page
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