1,686,163 research outputs found
Elliptic curves, modular forms, and sums of Hurwitz class numbers
Let H(N) denote the Hurwitz class number. It is known that if is a prime,
then {equation*} \sum_{|r|<2\sqrt p}H(4p-r^2) = 2p. {equation*} In this paper,
we investigate the behavior of this sum with the additional condition . Three different methods will be explored for determining the values
of such sums. First, we will count isomorphism classes of elliptic curves over
finite fields. Second, we will express the sums as coefficients of modular
forms. Third, we will manipulate the Eichler-Selberg trace for ula for Hecke
operators to obtain Hurwitz class number relations. The cases and 4 are
treated in full. Partial results, as well as several conjectures, are given for
and 7.Comment: Preprint of an old pape
Donaldson-Thomas invariants, torus knots, and lattice paths
In this paper we find and explore the correspondence between quivers, torus
knots, and combinatorics of counting paths. Our first result pertains to quiver
representation theory -- we find explicit formulae for classical generating
functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We
then focus on quivers corresponding to torus knots and show that their
classical generating functions, in the extremal limit and framing , are
generating functions of lattice paths under the line of the slope .
Generating functions of such paths satisfy extremal A-polynomial equations,
which immediately follows after representing them in terms of the Duchon
grammar. Moreover, these extremal A-polynomial equations encode
Donaldson-Thomas invariants, which provides an interesting example of
algebraicity of generating functions of these invariants. We also find a
quantum generalization of these statements, i.e. a relation between motivic
quiver generating functions, quantum extremal knot invariants, and -weighted
path counting. Finally, in the case of the unknot, we generalize this
correspondence to the full HOMFLY-PT invariants and counting of Schr\"oder
paths.Comment: 45 pages. Corrected typos in new versio
Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties
We investigate the behaviour of Tamagawa numbers of semistable principally
polarised abelian varieties in extensions of local fields. In view of the
Raynaud parametrisation, this translates into a purely algebraic problem
concerning the number of -invariant points on a quotient of -lattices
for varying subgroups of and integers . In
particular, we give a simple formula for the change of Tamagawa numbers in
totally ramified extensions (corresponding to varying ) and one that
computes Tamagawa numbers up to rational squares in general extensions.
As an application, we extend some of the existing results on the -parity
conjecture for Selmer groups of abelian varieties by allowing more general
local behaviour. We also give a complete classification of the behaviour of
Tamagawa numbers for semistable 2-dimensional principally polarised abelian
varieties, that is similar to the well-known one for elliptic curves. The
appendix explains how to use this classification for Jacobians of genus 2
hyperelliptic curves given by equations of the form , under some
simplifying hypotheses.Comment: Two new lemmas are added. The first describes permutation
representations, and the second describes the dependence of the B-group on
the maximal fixpoint-free invariant sublattice. Contact details and
bibliographic details have been update
Unexpected distribution phenomenon resulting from Cantor series expansions
We explore in depth the number theoretic and statistical properties of
certain sets of numbers arising from their Cantor series expansions. As a
direct consequence of our main theorem we deduce numerous new results as well
as strengthen known ones.Comment: 32 page
The spectral curve and the Schroedinger equation of double Hurwitz numbers and higher spin structures
We derive the spectral curves for -part double Hurwitz numbers, -spin
simple Hurwitz numbers, and arbitrary combinations of these cases, from the
analysis of the unstable (0,1)-geometry. We quantize this family of spectral
curves and obtain the Schroedinger equations for the partition function of the
corresponding Hurwitz problems. We thus confirm the conjecture for the
existence of quantum curves in these generalized Hurwitz number cases.Comment: 15 pages, journal publication versio
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