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1. CJM Online first
Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$ |
Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$ We exhibit a set of minimal generators of the defining ideal of the
Rees Algebra associated with the ideal of three bivariate homogeneous
polynomials parametrizing a proper rational curve in projective plane,
having a minimal syzygy of degree 2.
Keywords:Rees Algebras, rational plane curves, minimal generators Categories:13A30, 14H50 |
2. CJM 2011 (vol 64 pp. 81)
Pseudoprime Reductions of Elliptic Curves
Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication,
and for each prime
$p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer
$b$, we consider elliptic pseudoprimes to the base
$b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq
x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E,
b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such
that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called
elliptic curve pseudoprimes). Motivated by cryptography applications,
we address the problem of finding upper bounds for
$Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$,
generalising some of the literature for the classical pseudoprimes
to this new setting.
Keywords:Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes Categories:11N36, 14H52 |
3. 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 |
4. CJM 2010 (vol 63 pp. 86)
On Vojta's $1+\varepsilon$ Conjecture We give another proof of Vojta's $1+\varepsilon$ conjecture.
Keywords:Vojta, 1+epsilon Categories:14G40, 14H15 |
5. CJM 2010 (vol 62 pp. 1131)
Moduli Spaces of Reflexive Sheaves of Rank 2
Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of
dimension at least two and let $X \subset Y$ be the zero set of a section
$\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the
functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform
$\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two
forgetful maps between the functors, we prove that the scheme structure of
\emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$
at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves
on $Y$ become closely related. Using this relationship, we get criteria for the
dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $
{\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose
deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $
(\emph{e.g.,} of diameter at most 2),
we get necessary and sufficient conditions of unobstructedness that coincide
in the diameter one case. The conditions are further equivalent to the
vanishing of certain graded Betti numbers of the free graded minimal
resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible
component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter
one is reduced (generically smooth) and we compute its dimension. We also
determine a good lower bound for the dimension of any component of ${\rm
M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small''
deficiency module $M$.
Keywords:moduli space, reflexive sheaf, Hilbert scheme, space curve, Buchsbaum sheaf, unobstructedness, cup product, graded Betti numbers.xdvi Categories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07 |
6. 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 |
7. CJM 2009 (vol 61 pp. 109)
The Ample Cone of the Kontsevich Moduli Space We produce ample (resp.\ NEF, eventually free) divisors in the
Kontsevich space $\Kgnb{0,n} (\mathbb P^r, d)$ of $n$-pointed,
genus $0$, stable maps to $\mathbb P^r$, given such divisors in
$\Kgnb{0,n+d}$. We prove that this produces all ample (resp.\ NEF,
eventually free) divisors in $\Kgnb{0,n}(\mathbb P^r,d)$.
As a consequence, we construct a contraction of the boundary
$\bigcup_{k=1}^{\lfloor d/2 \rfloor} \Delta_{k,d-k}$ in
$\Kgnb{0,0}(\mathbb P^r,d)$, analogous to a contraction of
the boundary $\bigcup_{k=3}^{\lfloor n/2 \rfloor}
\tilde{\Delta}_{k,n-k}$ in $\kgnb{0,n}$ first constructed by Keel
and McKernan.
Categories:14D20, 14E99, 14H10 |
8. CJM 2009 (vol 61 pp. 3)
Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers Let $X$ be a smooth projective geometrically connected curve over
a finite field with function field $K$. Let $\G$ be a connected semisimple group
scheme over $X$. Under certain hypotheses we prove the equality of
two numbers associated with $\G$.
The first is an arithmetic invariant, its Tamagawa number. The second
is a geometric invariant, the number of connected components of the moduli
stack of $\G$-torsors on $X$. Our results are most useful for studying
connected components as much is known about Tamagawa numbers.
Categories:11E, 11R, 14D, 14H |
9. CJM 2008 (vol 60 pp. 297)
Transitive Factorizations in the Hyperoctahedral Group The classical Hurwitz enumeration problem has a presentation in terms of
transitive factorizations in the symmetric group. This presentation suggests
a generalization from type~$A$ to other
finite reflection groups and, in particular, to type~$B$.
We study this generalization both from a combinatorial and a geometric
point of view, with the prospect of providing a means of understanding more
of the structure of the moduli spaces of maps with an $\gS_2$-symmetry.
The type~$A$ case has been well studied and connects Hurwitz numbers
to the moduli space of curves. We conjecture an analogous setting for the
type~$B$ case that is studied here.
Categories:05A15, 14H10, 58D29 |
10. CJM 2006 (vol 58 pp. 1000)
On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of $\operatorname{SL}_n$ We compute some Hodge and Betti numbers of the moduli space of stable rank $r$,
degree $d$ vector bundles on a smooth projective curve. We
do not assume $r$ and $d$ are coprime.
In the process we equip the cohomology of an arbitrary algebraic stack with a
functorial mixed Hodge structure. This Hodge structure is
computed in the case of the moduli stack of rank $r$, degree
$d$ vector bundles on a curve. Our methods also yield a formula
for the Poincar\'e
polynomial of the moduli stack that is valid over any
ground field. In the last section we use the previous sections
to give a proof that the Tamagawa number of $\sln$ is one.
Categories:14H, 14L |
11. CJM 2005 (vol 57 pp. 338)
Certain Exponential Sums and Random Walks on Elliptic Curves For a given elliptic curve $\E$, we obtain an upper bound
on the discrepancy of sets of
multiples $z_sG$ where $z_s$ runs through a sequence
$\cZ=\(z_1, \dots, z_T\)$
such that $k z_1,\dots, kz_T $ is a permutation of
$z_1, \dots, z_T$, both sequences taken modulo $t$, for
sufficiently many distinct values of $k$ modulo $t$.
We apply this result to studying an analogue of the power generator
over an elliptic curve. These results are elliptic curve analogues
of those obtained for multiplicative groups of finite fields and
residue rings.
Categories:11L07, 11T23, 11T71, 14H52, 94A60 |
12. CJM 2004 (vol 56 pp. 612)
Solvable Points on Projective Algebraic Curves We examine the problem of finding rational points defined over
solvable extensions on algebraic curves defined over general fields.
We construct non-singular, geometrically irreducible projective curves
without solvable points of genus $g$, when $g$ is at least $40$, over
fields of arbitrary characteristic. We prove that every smooth,
geometrically irreducible projective curve of genus $0$, $2$, $3$ or
$4$ defined over any field has a solvable point. Finally we prove
that every genus $1$ curve defined over a local field of
characteristic zero with residue field of characteristic $p$ has a
divisor of degree prime to $6p$ defined over a solvable extension.
Categories:14H25, 11D88 |
13. CJM 2003 (vol 55 pp. 609)
Integrable Systems Associated to a Hopf Surface A Hopf surface is the quotient of the complex surface $\mathbb{C}^2
\setminus \{0\}$ by an infinite cyclic group of dilations of
$\mathbb{C}^2$. In this paper, we study the moduli spaces
$\mathcal{M}^n$ of stable $\SL (2,\mathbb{C})$-bundles on a Hopf
surface $\mathcal{H}$, from the point of view of symplectic geometry.
An important point is that the surface $\mathcal{H}$ is an elliptic
fibration, which implies that a vector bundle on $\mathcal{H}$ can be
considered as a family of vector bundles over an elliptic curve. We
define a map $G \colon \mathcal{M}^n \rightarrow \mathbb{P}^{2n+1}$
that associates to every bundle on $\mathcal{H}$ a divisor, called the
graph of the bundle, which encodes the isomorphism class of the bundle
over each elliptic curve. We then prove that the map $G$ is an
algebraically completely integrable Hamiltonian system, with respect
to a given Poisson structure on $\mathcal{M}^n$. We also give an
explicit description of the fibres of the integrable system. This
example is interesting for several reasons; in particular, since the
Hopf surface is not K\"ahler, it is an elliptic fibration that does
not admit a section.
Categories:14J60, 14D21, 14H70, 14J27 |
14. CJM 2003 (vol 55 pp. 248)
A Generalized Torelli Theorem Given a smooth projective curve $C$ of positive genus $g$, Torelli's
theorem asserts that the pair $\bigl( J(C),W^{g-1} \bigr)$ determines
$C$. We show that the theorem is true with $W^{g-1}$ replaced by
$W^d$ for each $d$ in the range $1\le d\le g-1$.
Category:14H99 |
15. CJM 2003 (vol 55 pp. 331)
The Maximum Number of Points on a Curve of Genus $4$ over $\mathbb{F}_8$ is $25$ We prove that the maximum number of rational points on a smooth,
geometrically irreducible genus 4 curve over the field of 8 elements
is 25. The body of the paper shows that 27 points is not possible by
combining techniques from algebraic geometry with a computer
verification. The appendix shows that 26 points is not possible by
examining the zeta functions.
Categories:11G20, 14H25 |
16. CJM 1999 (vol 51 pp. 936)
Galois Representations with Non-Surjective Traces Let $E$ be an elliptic curve over $\q$, and let $r$ be an integer.
According to the Lang-Trotter conjecture, the number of primes $p$
such that $a_p(E) = r$ is either finite, or is asymptotic to
$C_{E,r} {\sqrt{x}} / {\log{x}}$ where $C_{E,r}$ is a non-zero
constant. A typical example of the former is the case of rational
$\ell$-torsion, where $a_p(E) = r$ is impossible if $r \equiv 1
\pmod{\ell}$. We prove in this paper that, when $E$ has a rational
$\ell$-isogeny and $\ell \neq 11$, the number of primes $p$ such
that $a_p(E) \equiv r \pmod{\ell}$ is finite (for some $r$ modulo
$\ell$) if and only if $E$ has rational $\ell$-torsion over the
cyclotomic field $\q(\zeta_\ell)$. The case $\ell=11$ is special,
and is also treated in the paper. We also classify all those
occurences.
Category:14H52 |
17. CJM 1998 (vol 50 pp. 1253)
Integral representation of $p$-class groups in ${\Bbb Z}_p$-extensions and the Jacobian variety For an arbitrary finite Galois $p$-extension $L/K$ of
$\zp$-cyclotomic number fields of $\CM$-type with Galois group $G =
\Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $ \mu_L^-$
are zero, we obtain unconditionally and explicitly the Galois
module structure of $\clases$, the minus part of the $p$-subgroup
of the class group of $L$. For an arbitrary finite Galois
$p$-extension $L/K$ of algebraic function fields of one variable
over an algebraically closed field $k$ of characteristic $p$ as its
exact field of constants with Galois group $G = \Gal(L/K)$ we
obtain unconditionally and explicitly the Galois module structure
of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated
to $L/k$.
Keywords:${\Bbb Z}_p$-extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structure Categories:11R33, 11R23, 11R58, 14H40 |