Expand all Collapse all | Results 1 - 6 of 6 |
1. CJM 2011 (vol 63 pp. 992)
The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians
In this paper we study genus $2$ curves whose Jacobians admit a
polarized $(4,4)$-isogeny to a product of elliptic curves. We consider
base fields of characteristic different from $2$ and $3$, which we do
not assume to be algebraically closed.
We obtain a full classification of all principally polarized abelian
surfaces that can arise from gluing two elliptic curves along their
$4$-torsion, and we derive the relation their absolute invariants
satisfy.
As an intermediate step, we give a general description of Richelot
isogenies between Jacobians of genus $2$ curves, where previously only
Richelot isogenies with kernels that are pointwise defined over the
base field were considered.
Our main tool is a Galois theoretic characterization of genus $2$
curves admitting multiple Richelot isogenies.
Keywords:Genus 2 curves, isogenies, split Jacobians, elliptic curves Categories:11G30, 14H40 |
2. CJM 2010 (vol 62 pp. 1058)
On a Conjecture of S. Stahl
S. Stahl conjectured that the zeros of genus polynomials are real. In
this note, we disprove this conjecture.
Keywords:genus polynomial, zeros, real Category:05C10 |
3. CJM 2010 (vol 62 pp. 787)
An Explicit Treatment of Cubic Function Fields with Applications We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.
Keywords:cubic function field, discriminant, non-singularity, integral basis, genus, signature of a place, class number Categories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29 |
4. CJM 2008 (vol 60 pp. 1240)
Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links We define a family of formal Khovanov brackets
of a colored link depending on two parameters.
The isomorphism classes of these brackets are
invariants of framed colored links.
The Bar-Natan functors applied to these brackets
produce Khovanov and Lee homology theories categorifying the colored
Jones polynomial. Further,
we study conditions under which
framed colored link cobordisms induce chain transformations between
our formal brackets. We conjecture that
for special choice of parameters, Khovanov and Lee homology theories
of colored links are functorial (up to sign).
Finally, we extend the Rasmussen invariant to links and give examples
where this invariant is a stronger obstruction to sliceness
than the multivariable Levine--Tristram signature.
Keywords:Khovanov homology, colored Jones polynomial, slice genus, movie moves, framed cobordism Categories:57M25, 57M27, 18G60 |
5. CJM 2008 (vol 60 pp. 958)
A Note on a Conjecture of S. Stahl S. Stahl (Canad. J. Math. \textbf{49}(1997), no. 3, 617--640)
conjectured that the zeros of genus polynomial are real.
L. Liu and Y. Wang disproved this conjecture on the basis
of Example 6.7. In this note, it is pointed out
that there is an error in this example and a new generating matrix
and initial vector are provided.
Keywords:genus polynomial, zeros, real Categories:05C10, 05A15, 30C15, 26C10 |
6. CJM 1998 (vol 50 pp. 1209)
A lower bound for $K_X L$ of quasi-polarized surfaces $(X,L)$ with non-negative Kodaira dimension Let $X$ be a smooth projective surface over the complex
number field and let $L$ be a nef-big divisor on $X$. Here we consider
the following conjecture; If the Kodaira dimension $\kappa(X)\geq 0$,
then $K_{X}L\geq 2q(X)-4$, where $q(X)$ is the irregularity of $X$. In
this paper, we prove that this conjecture is true if (1) the case in which
$\kappa(X)=0$ or $1$, (2) the case in which $\kappa(X)=2$ and $h^{0}(L)\geq
2$, or (3) the case in which $\kappa(X)=2$, $X$ is minimal, $h^{0}(L)=1$,
and $L$ satisfies some conditions.
Keywords:Quasi-polarized surface, sectional genus Category:14C20 |