On donne une formule exacte pour l’invariant de Rost $H^1\left(k,G\right)\to H^3\left(k\right)$ des groupes linéaires spéciaux.

We provide an exact formula for the Rost invariant $H^1\left(k,G\right)\to H^3\left(k\right)$ of special linear groups.

We study modular abelian varieties with odd congruence number by examining the cuspidal subgroup of $J_0\left(N\right)$. We show that the conductor of such abelian varieties must be of a special type. For example, if $N$ is the conductor of an absolutely simple modular abelian variety with odd congruence number, then $N$ has at most two prime divisors, and if $N$ is odd, then $N=p^{\alpha }$ or $N=pq$ for some primes $p$ and $q$. In the second half of the paper, we focus on modular elliptic curves with odd modular degree. Our results, combined with the work of Agashe, Ribet, and Stein for elliptic curves to have odd modular degree. In the process we prove Watkins’ conjecture for elliptic curves with odd modular degree and a nontrivial rational torsion point.

A ring extension $A\subseteq B$ is said to have depth one if $B$ is isomorphic to a direct summand of $A^n$ as an $\left(A,A\right)$-bimodule, for some positive integer $n$. We prove group-theoretic characterizations of this property in the case $kH\subseteq kG$, where $H$ is a subgroup of a finite group $G$ and $k$ is a field. We determine when the source algebra of a block of $kG$ with defect group $P$ is a depth-one extension of $kP$.

The variety of principal minors of $n\times n$ symmetric matrices, denoted $Z_n$, is invariant under the action of a group $G\subset GL\left(2^n\right)$ isomorphic to $SL{\left(2\right)}^{\times n}⋉{\mathfrak{S}}_n$. We describe an irreducible $G$-module of degree-four polynomials constructed from Cayley’s $2\times 2\times 2$ hyperdeterminant and show that it cuts out $Z_n$ set-theoretically. This solves the set-theoretic version of a conjecture of Holtz and Sturmfels. Standard techniques from representation theory and geometry are explored and developed for the proof of the conjecture and may be of use for studying similar $G$-varieties.

By analogy with the Riemann zeta function at positive integers, for each finite field ${\mathbb{F}}_{p^r}$ with fixed characteristic $p$, we consider Carlitz zeta values ${\zeta }_r\left(n\right)$ at positive integers $n$. Our theorem asserts that among the zeta values in the set ${\bigcup }_{r=1}^{\infty }\left\{{\zeta }_r\left(1\right),{\zeta }_r\left(2\right),{\zeta }_r\left(3\right),\dots \right\}$, all the algebraic relations are those relations within each individual family $\left\{{\zeta }_r\left(1\right),{\zeta }_r\left(2\right),{\zeta }_r\left(3\right),\dots \right\}$. These are the algebraic relations coming from the Euler–Carlitz and Frobenius relations. To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed.