論文

Abstract We construct a family of affinoids in the Lubin–Tate perfectoid space and their formal models such that the middle cohomology of their reductions realizes the local Langlands correspondence and the local Jacquet–Langlands correspondence for the simple supercuspidal representations. The reductions of the formal models are isomorphic to the perfections of some Artin–Schreier varieties, whose cohomology realizes primitive Galois representations. We show also the Tate conjecture for Artin–Schreier varieties associated to quadratic forms.

Abstract We study Van der Geer–Van der Vlugt curves in a ramification‐theoretic view point. We give explicit formulae on the ‐polynomials of these curves. As a result, we show that these curves are supersingular and give sufficient conditions for these curves to be maximal or minimal.

Abstract For an additive polynomial and a positive integer, we define an irreducible smooth representation of a Weil group of a non-archimedean local field. We study several invariants of this representation. We obtain a necessary and sufficient condition for it to be primitive.

Abstract The van der Geer–van der Vlugt curves form a class of Artin–Schreier coverings of the projective line over finite fields. We provide an explicit formula for their ‐polynomials in characteristic 2, expressed in terms of characters of maximal abelian subgroups of the associated Heisenberg groups. For this purpose, we develop new methods specific to characteristic 2 that exploit the structure of the Heisenberg groups and the geometry of Lang torsors for . As an application, we construct examples of curves in this family that attain the Hasse–Weil bound.

We study the mod Weil representation of a finite unitary group and related Deligne–Lusztig inductions. In particular, we study their decomposition as representations of a symplectic group, and give a construction of a mod Howe correspondence for including the case where .

Abstract We introduce a certain Artin–Schreier scheme over a finite field associated to a pair of coprime integers {(m,n)} with {n\geq 3} divisible by the characteristic of the base field, and study the middle étale cohomology group of it. If m is even, the variety admits actions of some finite Heisenberg groups. We study the middle cohomology as representations of the Heisenberg groups. If m is odd, we compute the Frobenius eigenvalues of it concretely. This affine scheme comes from the reduction of a certain affinoid in a Lubin–Tate space.

We construct a stable formal model of a Lubin–Tate curve with level three, and study the action of a Weil group and a division algebra on its stable reduction. Further, we study a structure of cohomology of the Lubin–Tate curve. Our study is purely local and includes the case where the characteristic of the residue field of a local field is two.