776,312 research outputs found
Root numbers and ranks in positive characteristic
For a global field K and an elliptic curve E_eta over K(T), Silverman's
specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all
but finitely many t in P^1(K). If this inequality is strict for all but
finitely many t, the elliptic curve E_eta is said to have elevated rank. All
known examples of elevated rank for K=Q rest on the parity conjecture for
elliptic curves over Q, and the examples are all isotrivial.
Some additional standard conjectures over Q imply that there does not exist a
non-isotrivial elliptic curve over Q(T) with elevated rank. In positive
characteristic, an analogue of one of these additional conjectures is false.
Inspired by this, for the rational function field K = kappa(u) over any finite
field kappa with odd characteristic, we construct an explicit 2-parameter
family E_{c,d} of non-isotrivial elliptic curves over K(T) (depending on
arbitrary c, d in kappa^*) such that, under the parity conjecture, each E_{c,d}
has elevated rank.Comment: 40 pages; last version; to appear in Adv. Mat
On Chebyshev polynomials and torus knots
In this work we demonstrate that the q-numbers and their two-parameter
generalization, the q,p-numbers, can be used to obtain some polynomial
invariants for torus knots and links. First, we show that the q-numbers, which
are closely connected with the Chebyshev polynomials, can also be related with
the Alexander polynomials for the class T(s,2) of torus knots, s being an odd
integer, and used for finding the corresponding skein relation. Then, we
develop this procedure in order to obtain, with the help of q,p-numbers, the
generalized two-variable Alexander polynomials, and prove their direct
connection with the HOMFLY polynomials and the skein relation of the latter.Comment: 6 pages (two-column UJP style
Divisibility of characteristic numbers
We use homotopy theory to define certain rational coefficients characteristic
numbers with integral values, depending on a given prime number q and positive
integer t. We prove the first nontrivial degree formula and use it to show that
existence of morphisms between algebraic varieties for which these numbers are
not divisible by q give information on the degree of such morphisms or on zero
cycles of the target variety.Comment: This is the version published by Geometry & Topology Monographs on 29
January 200
Divisibility of the class numbers of imaginary quadratic fields
For a given odd integer , we provide some families of imaginary
quadratic number fields of the form whose ideal
class group has a subgroup isomorphic to .Comment: 10 pages, accepted for publication in Journal of Number Theory (2017
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