9 research outputs found
Kakeya sets over non-archimedean local rings
In a recent paper of Ellenberg, Oberlin, and Tao, the authors asked whether
there are Besicovitch phenomena in F_q[[t]]^n. In this paper, we answer their
question in the affirmative by explicitly constructing a Kakeya set in
F_q[[t]]^n of measure 0. Furthermore, we prove that any Kakeya set in
F_q[[t]]^2 or Z_p^2 is of Minkowski dimension 2.Comment: 10 page
Explicit computations of Hida families via overconvergent modular symbols
In [Pollack-Stevens 2011], efficient algorithms are given to compute with
overconvergent modular symbols. These algorithms then allow for the fast
computation of -adic -functions and have further been applied to compute
rational points on elliptic curves (e.g. [Darmon-Pollack 2006, Trifkovi\'c
2006]). In this paper, we generalize these algorithms to the case of families
of overconvergent modular symbols. As a consequence, we can compute -adic
families of Hecke-eigenvalues, two-variable -adic -functions,
-invariants, as well as the shape and structure of ordinary Hida-Hecke
algebras.Comment: 51 pages. To appear in Research in Number Theory. This version has
added some comments and clarifications, a new example, and further
explanations of the previous example