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Search: MSC category 11E ( Forms and linear algebraic groups [See also 19Gxx] {For quadratic forms in linear algebra, see 15A63} )

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1. CJM Online first

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 Cartan--Dieudonn{é 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

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 $o(s-1)$ 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

7. CJM 2000 (vol 52 pp. 833)

Mináč, Ján; Smith, Tara L.
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

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

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 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

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

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