Regularity of quotients of Drinfeld modular schemes

Let $A$ be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal $I \subset A$, Drinfeld defined the notion of structure of level $I$ on a Drinfeld module. We extend this to that of level $N$, where $N$ is a finitely generated torsion $A$-module. The case where $N=(I^{-1}/A)^d$, where $d$ is the rank of the Drinfeld module,coincides with the structure of level $I$. The moduli functor is representable by a regular affine scheme. The automorphism group $\mathrm{Aut}_{A}(N)$ acts on the moduli space. Our theorem gives a class of subgroups for which the quotient of the moduli scheme is regular. Examples include generalizations of $\Gamma_0$ and of $\Gamma_1$. We also show that parabolic subgroups appearing in the definition of Hecke correspondences are such subgroups

Local newforms for the general linear groups over a non-archimedean local field

In [12], Jacquet--Piatetskii-Shapiro--Shalika defined a family of compact open subgroups of $p$-adic general linear groups indexed by non-negative integers, and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of non-negative integers. For the proof, we introduce the Rankin--Selberg integrals for Speh representations.Comment: 60 page

First and second KK-groups of an elliptic curve over a global field of positive characteristic

In this paper, we show that the maximal divisible subgroup of groups $K_1$ and $K_2$ of an elliptic curve $E$ over a function field is uniquely divisible. Further those $K$-groups modulo this uniquely divisible subgroup are explicitly computed. We also calculate the motivic cohomology groups of the minimal regular model of $E$, which is an elliptic surface over a finite field.Comment: This paper will appear at Annales de l'Institut Fourie

The role of fibers in the femoral attachment of the anterior cruciate ligament in resisting tibial displacement

Purpose The purpose was to clarify the load-bearing functions of the fibers of the femoral anterior cruciate ligament (ACL) attachment in resisting tibial anterior drawer and rotation. Methods A sequential cutting study was performed on 8 fresh-frozen human knees. The femoral attachment of the ACL was divided into a central area that had dense fibers inserting directly into the femur and anterior and posterior fan-like extension areas. The ACL fibers were cut sequentially from the bone: the posterior fan-like area in 2 stages, the central dense area in 4 stages, and then the anterior fan-like area in 2 stages. Each knee was mounted in a robotic joint testing system that applied tibial anteroposterior 6-mm translations and 10° or 15° of internal rotation at 0° to 90° of flexion. The reduction of restraining force or moment was measured after each cut. Results The central area resisted 82% to 90% of the anterior drawer force; the anterior fan-like area, 2% to 3%; and the posterior fan-like area, 11% to 15%. Among the 4 central areas, most load was carried close to the roof of the intercondylar notch: the anteromedial bundle resisted 66% to 84% of the force and the posterolateral bundle resisted 16% to 9% from 0° to 90° of flexion. There was no clear pattern for tibial internal rotation, with the load shared among the posterodistal and central areas near extension and mostly the central areas in flexion. Conclusions Under the experimental conditions described, 66% to 84% of the resistance to tibial anterior drawer arose from the ACL fibers at the central-proximal area of the femoral attachment, corresponding to the anteromedial bundle; the fan-like extension fibers contributed very little. This work did not support moving a single-bundle ACL graft to the side wall of the notch or attempting to cover the whole attachment area if the intention was to mimic how the natural ACL resists tibial displacements. Clinical Relevance There is ongoing debate about how best to reconstruct the ACL to restore normal knee function, including where is the best place for ACL graft tunnels. This study found that the most important area on the femur, in terms of resisting displacement of the tibia, was in the central-anterior part of the femoral ACL attachment, near the roof of the intercondylar notch. The testing protocol did not lead to data that would support using a large ACL graft tunnel that attempts to cover the whole natural femoral attachment area