Introduction to Quadratic Forms over Fields

Convergence of the Motivic Adams Spectral Sequence

“…Therefore, the completion at p is η-complete if and only if (7) W ∧ p = 0. Referring to [11], if the field F has a real ordering (which is equivalent to −1 not being a sum of squares), then there exists an epimorphism W → Z, which means that (7) cannot hold. Therefore, completion at η is necessary for convergence of the Adams spectral sequence at p > 2 for any such field.…”

“…Since we already remarked that attaching cells can be interpreted as an operation in the motivic stable homotopy category, we may assume (without changing the dimensions of the cells constituting X) that the attaching map of e is actually 0. This gives us a map (11) S m+ α → X.…”

“…Let X 1 be the homotopy cofiber of the map (11). Note carefully that X 1 can be constructed by following the construction of X while "omitting" the cell e, i.e.…”

On commensurable hyperbolic Coxeter groups

“…When n is even, the isomorphism class of B is a complete invariant of the commensurability class of (see [21,Theorem 7.2]). Since two quaternion algebras over a number field are isomorphic if and only if their ramification sets are the same (see [20] When n is odd and greater than 3, the situation is more complicated since the quaternion algebra B is not a complete invariant. In fact, a complete invariant consists of the quaternion algebra B, the discriminant (or signed determinant) δ of the quadratic form q V and a set of prime ideals [21,Corollary 7.5]).…”

Sums of squares in certain quaternion and octonion algebras

“…Our notation coincides with that employed in [3], aside from our usage of n×ϕ to denote the orthogonal sum of n ∈ N copies of a quadratic form ϕ. Moreover, for a, b ∈ F × , we will let k(a) (respectively k(a, b)) denote the least n ∈ N such that n × 1, −a (respectively n × 1, −a, −b, ab ) is isotropic (over F , unless stated otherwise).…”