Incremental ascent of a modular tower via branch cycle designs | | Posted on:2003-11-27 | Degree:Ph.D | Type:Dissertation | | University:University of California, Irvine | Candidate:Bailey, Paul Lawrence | Full Text:PDF | | GTID:1462390011485929 | Subject:Mathematics | | Abstract/Summary: | | | Let G be a finite group and let C be an r-tuple of conjugacy classes from G which generate G. The reduced Hurwitz space (G, C)in,rd parameterizes weak equivalence classes of ramified covers of the Riemann sphere with ramification in C. If the rank r is four, the reduced space is a Riemann surface. The modular curves Y1(n) are such spaces, with G = Dn and C four conjugacy classes of involutions.; We ask three modest questions regarding reduced rank four Hurwitz spaces: (1) How many components are there? (2) What are the genera of the components? (3) What are the fields of definition of the components?; Let p be a prime which divides the order of G. The universal elementary p-Frattini cover is versal for Frattini covers of G with elementary p-group kernel. Inductively define . If p does not divide the orders of the elements in C, these conjugacy classes lift uniquely to , producing a sequence of Riemann surfaces &ldots;H&parl0; k+1p G&d5;,C&parr0;in, rd→H&parl0; kpG&d5; ,C&parr0;in,rd→&cdots;→ H&parl0;G,C&parr0;in,rd→ J4, which is called a Modular Tower, and is denoted by MTp(G, C); this generalizes towers of modular curves. Understanding a Modular Tower requires combining knowledge of the base space and techniques of lifting information.; Certain configurations of the branch points give Harbater-Mumford covers, which are necessarily defined over , producing real points on the Hurwitz space. If p = 2, these are the only points which lie in projective systems of real points up the tower, and lay at the center of computations.; Given a ramified cover, we develop its Nielsen graph, which dictates which covers can factor through the given one. Classical generators for the base space of the cover lift to an embedded realization of the graph in the covering space; this is a branch cycle design, and it produces classical generators for the covering space. Using branch cycle designs as platforms and real points as ladders, we ascend to the first level of the Modular Tower MT2(A4 , ), and answer some of the questions posed above. | | Keywords/Search Tags: | Modulartower, Bold, Hspsp, Branchcycle | | Related items |
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