2014/02/28 by Takako Fukaya, Kazuya Kato, Romyar Sharifi · 1 citation
Mathematics · #math.AG #math.NT #msc:11F67 #msc:11Fxx #msc:11G16 #msc:11Gxx #msc:11R23 #msc:11R34 #msc:11R58 #msc:11Rxx
published as Iwasawa Theory 2012 - State of the Art and Recent Advances, Springer, 2014, 177-219 · 43 pages
arxiv created 2015/01/05 · arxiv updated 2015/01/07
This survey paper is focused on a connection between the geometry of GLd and the arithmetic of GLd-1 over global fields, for integers d ≥ 2. For d = 2 over ℚ, there is an explicit conjecture of the third author relating the geometry of modular curves and the arithmetic of cyclotomic fields, and it is proven in many instances by the work of the first two authors. The paper is divided into three parts: in the first, we explain the conjecture of the third author and the main result of the first two authors on it. In the second, we explain an analogous conjecture and result for d = 2 over \mathbbFq(t). In the third, we pose questions for general d over the rationals, imaginary quadratic fields, and global function fields.