1. CJM 2016 (vol 68 pp. 395)
 Garibaldi, Skip; Nakano, Daniel K.

Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
The representation theory of semisimple algebraic groups over
the complex numbers (equivalently, semisimple complex Lie algebras
or Lie groups, or real compact Lie groups) and the question of
whether a
given complex representation is symplectic or orthogonal has
been solved since at least the 1950s. Similar results for Weyl
modules of split reductive groups over fields of characteristic
different from 2 hold by
using similar proofs. This paper considers analogues of these
results for simple, induced and tilting modules of split reductive
groups over fields of prime characteristic as well as a complete
answer for Weyl modules over fields of characteristic 2.
Keywords:orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups Categories:20G05, 11E39, 11E88, 15A63, 20G15 

2. CJM 2013 (vol 65 pp. 1320)
 Taniguchi, Takashi; Thorne, Frank

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 

3. CJM 2009 (vol 61 pp. 3)
 Behrend, Kai; Dhillon, Ajneet

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 

4. CJM 2007 (vol 59 pp. 1284)
 Fukshansky, Lenny

On Effective Witt Decomposition and the CartanDieudonn{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 smallheight orthogonal
decomposition into onedimensional subspaces. Finally, we prove an
effective version of the CartanDieudonn{\'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 

5. CJM 2005 (vol 57 pp. 180)
 Somodi, Marius

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 

6. CJM 2001 (vol 53 pp. 434)
 van der Poorten, Alfred J.; Williams, Kenneth S.

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 $o(s1)$ 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 

7. CJM 2000 (vol 52 pp. 833)
 Mináč, Ján; Smith, Tara L.

WGroups under Quadratic Extensions of Fields
To each field $F$ of characteristic not $2$, one can associate a
certain Galois group $\G_F$, the socalled Wgroup 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 Wgroup of a proper field extension $K/F$ is a subgroup of the
Wgroup 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 ArtinSchreier's theorem describing fields fixed by
finite subgroups of absolute Galois groups. We also obtain precise
results in the case when $a$ is a doublerigid element in $F$. Some
of these results carry over to the general setting.
Categories:11E81, 12D15 

8. CJM 2000 (vol 52 pp. 613)
 Ou, Zhiming M.; Williams, Kenneth S.

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 

9. CJM 1999 (vol 51 pp. 176)
 van der Poorten, Alfred; Williams, Kenneth S.

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 

10. CJM 1998 (vol 50 pp. 1323)
 Morales, Jorge

L'invariant de HasseWitt de la forme de Killing
Nous montrons que l'invariant de HasseWitt de la forme de Killing
d'une alg{\`e}bre de Lie semisimple $L$ 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 

11. CJM 1997 (vol 49 pp. 499)
 Fitzgerald, Robert W.

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 
