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1. CJM 2013 (vol 65 pp. 1320)
Orbital $L$-functions for the Space of Binary Cubic Forms We introduce the notion of orbital $L$-functions
for the space of binary cubic forms
and investigate their analytic properties.
We study their functional equations and residue formulas in some detail.
Aside from their intrinsic interest,
the results from this paper are used to
prove the existence of secondary terms in counting
functions for cubic fields.
This is worked out in a companion paper.
Keywords:binary cubic forms, prehomogeneous vector spaces, Shintani zeta functions, $L$-functions, cubic rings and fields Categories:11M41, 11E76 |
2. 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 |
3. CJM 2007 (vol 59 pp. 1284)
On Effective Witt Decomposition and the Cartan--Dieudonn{Ã© Theorem Let $K$ be a number field, and let $F$ be a symmetric bilinear form in
$2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical
theorem of Witt states that the bilinear space $(Z,F)$ can be
decomposed into an orthogonal sum of hyperbolic planes and singular and
anisotropic components. We prove the existence of such a decomposition
of small height, where all bounds on height are explicit in terms of
heights of $F$ and $Z$. We also prove a special version of Siegel's
lemma for a bilinear space, which provides a small-height orthogonal
decomposition into one-dimensional subspaces. Finally, we prove an
effective version of the Cartan--Dieudonn{\'e} theorem. Namely, we show
that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can
be represented as a product of reflections of bounded heights with an
explicit bound on heights in terms of heights of $F$, $Z$, and
$\sigma$.
Keywords:quadratic form, heights Categories:11E12, 15A63, 11G50 |
4. CJM 2005 (vol 57 pp. 180)
On the Size of the Wild Set To every pair of algebraic number fields with isomorphic Witt rings
one can associate a number, called the {\it minimum number of wild
primes}. Earlier investigations have established lower bounds for this
number. In this paper an analysis is presented that expresses the
minimum number of wild primes in terms of the number of wild dyadic
primes. This formula not only gives immediate upper bounds, but can be
considered to be an exact formula for the minimum number of wild
primes.
Categories:11E12, 11E81, 19F15, 11R29 |
5. CJM 2001 (vol 53 pp. 434)
Values of the Dedekind Eta Function at Quadratic Irrationalities: Corrigendum Habib Muzaffar of Carleton University has pointed out to the authors
that in their paper [A] only the result
\[
\pi_{K,d}(x)+\pi_{K^{-1},d}(x)=\frac{1}{h(d)}\frac{x}{\log
x}+O_{K,d}\Bigl(\frac {x}{\log^2x}\Bigr)
\]
follows from the prime ideal theorem with remainder for ideal classes,
and not the stronger result
\[
\pi_{K,d}(x)=\frac{1}{2h(d)}\frac{x}{\log
x}+O_{K,d}\Bigl(\frac {x}{\log^2x}\Bigr)
\]
stated in Lemma~5.2. This necessitates changes in Sections~5 and 6 of
[A]. The main results of the paper are not affected by these changes.
It should also be noted that, starting on page 177 of [A], each and
every occurrence of $o(s-1)$ should be replaced by $o(1)$.
Sections~5 and 6 of [A] have been rewritten to incorporate the above
mentioned correction and are given below. They should replace the
original Sections~5 and 6 of [A].
Keywords:Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class group Categories:11F20, 11E45 |
6. CJM 2000 (vol 52 pp. 833)
W-Groups under Quadratic Extensions of Fields To each field $F$ of characteristic not $2$, one can associate a
certain Galois group $\G_F$, the so-called W-group of $F$, which
carries essentially the same information as the Witt ring $W(F)$ of
$F$. In this paper we investigate the connection between $\wg$ and
$\G_{F(\sqrt{a})}$, where $F(\sqrt{a})$ is a proper quadratic
extension of $F$. We obtain a precise description in the case when
$F$ is a pythagorean formally real field and $a = -1$, and show that
the W-group of a proper field extension $K/F$ is a subgroup of the
W-group of $F$ if and only if $F$ is a formally real pythagorean field
and $K = F(\sqrt{-1})$. This theorem can be viewed as an analogue of
the classical Artin-Schreier's theorem describing fields fixed by
finite subgroups of absolute Galois groups. We also obtain precise
results in the case when $a$ is a double-rigid element in $F$. Some
of these results carry over to the general setting.
Categories:11E81, 12D15 |
7. CJM 2000 (vol 52 pp. 613)
Small Solutions of $\phi_1 x_1^2 + \cdots + \phi_n x_n^2 = 0$ Let $\phi_1,\dots,\phi_n$ $(n\geq 2)$ be nonzero integers such that
the equation
$$
\sum_{i=1}^n \phi_i x_i^2 = 0
$$
is solvable in integers $x_1,\dots,x_n$ not all zero. It is shown
that there exists a solution satisfying
$$
0 < \sum_{i=1}^n |\phi_i| x_i^2 \leq 2 |\phi_1 \cdots \phi_n|,
$$
and that the constant 2 is best possible.
Keywords:small solutions, diagonal quadratic forms Category:11E25 |
8. CJM 1999 (vol 51 pp. 176)
Values of the Dedekind Eta Function at Quadratic Irrationalities Let $d$ be the discriminant of an imaginary quadratic field. Let
$a$, $b$, $c$ be integers such that
$$
b^2 - 4ac = d, \quad a > 0, \quad \gcd (a,b,c) = 1.
$$
The value of $\bigl|\eta \bigl( (b + \sqrt{d})/2a \bigr) \bigr|$ is
determined explicitly, where $\eta(z)$ is Dedekind's eta function
$$
\eta (z) = e^{\pi iz/12} \prod^\ty_{m=1} (1 - e^{2\pi imz})
\qquad \bigl( \im(z) > 0 \bigr). %\eqno({\rm im}(z)>0).
$$
Keywords:Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class group Categories:11F20, 11E45 |
9. CJM 1998 (vol 50 pp. 1323)
L'invariant de Hasse-Witt de la forme de Killing Nous montrons que l'invariant de Hasse-Witt de la forme de Killing
d'une alg{\`e}bre de Lie semi-simple $L$ s'exprime {\`a} l'aide de
l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de
$L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines
positives), et des invariants associ{\'e}s au groupe des
sym{\'e}tries du diagramme de Dynkin de $L$.
Categories:11E04, 11E72, 17B10, 17B20, 11E88, 15A66 |
10. CJM 1997 (vol 49 pp. 499)
Gorenstein Witt rings II The abstract Witt rings which are Gorenstein have been classified
when the dimension is one and the classification problem for those of
dimension zero has been reduced to the case of socle degree three. Here we
classifiy the Gorenstein Witt rings of fields with dimension zero and
socle degree three. They are of elementary type.
Categories:11E81, 13H10 |