This article is essentially an announcement of the papers [7,8,9,10] of the authors, though some of the examples are not included in those papers. We consider what is called zeta and $L$-functions of root systems which can be regarded as a multi-variable version of Witten multiple zeta and $L$-functions. Furthermore, corresponding to these functions, Bernoulli polynomials of root systems are defined. First we state several analytic properties, such as analytic continuation and location of singularities of these functions. Secondly we generalize the Bernoulli polynomials and give some expressions of values of zeta and $L$-functions of root systems in terms of these polynomials. Finally we give some functional relations among them by our previous method. These relations include the known formulas for their special values formulated by Zagier based on Witten’s work.

For a few quadratic fields, the non-existence is proved of continuous irreducible mod 2 Galois representations of degree 2 unramified outside $\{2,\infty\}$.

This article is devoted to introduce a new class of nonassociative algebras with involution including the class of structurable algebras.