We analyse the Higgs branch of 4d $\mathcal{N}=2$ SQCD gauge theories with non-connected groups $\widetilde{\mathrm{SU}}(N) = \mathrm{SU}(N) \rtimes_{I,II} \mathbb{Z}_2$ whose study was initiated in arXiv:1804.01108. derive Hasse diagrams corresponding to mechanism using adapted characters for representations groups. propose 3d $\mathcal{N}=4$ magnetic quivers branches type $I$ discrete gauging...

Let f(X1, . . . , Xm) be a quadratic form in m variables X1, . . . , Xm with integer coefficients. Then it is well-known that the Diophantine equation f(X1, . . . , Xm) = 0 has a nontrivial solution in integers if and only if the equation has a nontrivial solution in real numbers and the congruence f(X1, . . . , Xm) ≡ 0 (mod N) has a nontrivial solution for every integer N > 1. Such a principle...

Let k be a complete discrete valuation field of equal characteristic p > 0 with possibly imperfect residue field. Let Gk be its Galois group. We will prove that the conductors computed by the arithmetic ramification filtrations on Gk defined in [2] coincide with the differential Artin conductors and Swan conductors of Galois representations of Gk defined in [15]. As a consequence, we give a Has...

The Hasse principle is said to hold for a class of varieties over a number field K if for any variety X in the class, the set of rational points X(K) is non-empty whenever the set of adelic points X(AK) is non-empty. Manin [Man] observed that the failure of the Hasse principle can often be explained in terms of the Brauer group of X, Br(X). The product rule implies that X(K) must be contained i...

Let k be a complete discretely valued field of equal characteristic p > 0 with possibly imperfect residue field and let Gk be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on Gk defined in [3] coincide with the differential Artin conductors and Swan conductors of Galois representations of Gk defined in [17]. As a consequence, we give a Hasse-...

We discuss the geometry and arithmetic of higher-dimensional analogues of Châtelet surfaces; namely, we describe the structure of their Brauer and Picard groups and show that they can violate the Hasse principle. In addition, we use these varieties to give straightforward generalizations of two recent results of Poonen. Specifically, we prove that, assuming Schinzel’s hypothesis, the non-m powe...

We describe the classical Hasse principle for the existence of nontrivial zeros for quadratic forms over number fields, namely, local zeros over all completions at places of the number field imply nontrivial zeros over the number field itself. We then go on to explain more general questions related to the Hasse principle for nontrivial zeros of quadratic forms over function fields, with referen...