In this paper, the theory of e-constants associated to tame finite group actions on arithmetic surfaces is used to define a Brauer group invariant µ(X ,G , V) associated to certain symplectic motives of weight one. The relationship between this invariant and w2(π) (the Galois-theoretic invariant associated to tame covers of surfaces defined by Cassou-Nogues, Erez and Taylor) is also discussed. In his paper (4), Deligne used elements in the Brauer group of Q and their relation- ship with certain e-constants to give a proof of the Frohlich-Queyrut theorem. In particular, he showed that certain global orthogonal root numbers are equal to one, by interpreting the associated local orthogonal root numbers as Stiefel-Whitney classes and then using the local root numbers to define an element of order two in the Brauer group of Q. This idea was furthered by Saito (in (13), for example) and others who defined Brauer group invariants associated to situations which can be interpreted as motives that are orthogonal and of even weight. In this paper, we define a Brauer group invariant associated to certain motives that are symplectic and have weight one. In order to construct the relevant motives, we first define X to be an arithmetic surface of dimension two which is flat, regular, and projective over Z. Throughout this paper, we assume that f : X− →Spec(Z) is the structure morphism. Let G be a finite group that acts tamely on X. In other words, for each closed point x ∈ X, the order of the inertia group of x is relatively prime to the residue characteristic of x .L etY be the quotient scheme X /G, which we assume is regular, and assume that for all finite places v, the fiber Yv =( Xv)/G = Y⊗ Z (Z/p(v)) has normal crossings and smooth irreducible components with multiplicities relatively prime to the residue characteristic of v. Finally, let V be a virtual representation of G over