Abstract
AbstractUsing the theory of cohomological invariants for algebraic stacks, we compute the Brauer group of the moduli stack of hyperelliptic curves${\mathcal {H}}_g$over any field of characteristic$0$. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.
Highlights
While Brauer groups of schemes have seen a lot of attention in modern algebraic geometry, computations of Brauer groups of moduli stacks over nonalgebraically closed fields only started appearing in recent years
The proofs of these results are based on standard tools in étale and flat cohomology coupled with a very delicate analysis of various presentations of the stack of elliptic curves, their relations, the stabiliser groups at various points and so on, which seem hard to apply to more complicated stacks
We extend a computation from [22] to compute the cohomological invariants with arbitrary coefficients of the stack of elliptic curves M1,1 and use it to compute its Brauer group, partially retrieving Antieau and Meier’s result [1]
Summary
Brauer groups of fields have long been an object of study in number theory, dating back to work of Noether and Brauer. They were later generalised by Grothendieck to schemes and more general objects, up to the vast generality of topoi. In 2019, Shin [26] showed that over an algebraically closed field of characteristic 2, the Brauer group of M1,1 is equal to Z/2Z The proofs of these results are based on standard tools in étale and flat cohomology coupled with a very delicate analysis of various presentations of the stack of elliptic curves, their relations, the stabiliser groups at various points and so on, which seem hard to apply to more complicated stacks
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