2:00 pm Thursday, April 12, 2012
Algebra, Number Theory, and Combinatorics : Arithmetic on a family of non-arithmetic curves by Pete Clark (University of Georgia) in RLM 9.166
Classical modular curves are a distinguished family of algebraic curves for several different reasons: (i) (modular) they parameterize elliptic curves with extra structure; (ii) (Hecke-theoretic) they are uniformized by Fuchsian groups which are, in a precise technical sense, arithmetic, the upshot of which is that they come with a "sufficiently large" Hecke algebra; (iii) (covering-theoretic) They are branched covers of the Riemann sphere of a very special type. All three of these perspectives give important insight into the arithmetic and geometry of modular curves. Since the 1960's there has been increasing attention given to a more general family of curves, the Shimura curves. Here one loses some traction on the moduli interpretation - a general Shimura curve over a totally real field is not a "PEL type Shimura variety" - but the Hecke-theoretic properties are still there. Indeed, by a theorem of Margulis, the Riemann surfaces which have sufficiently large Hecke algebras are precisely the (curves commensurable with) Shimura curves. The perspective of Fuchsian groups provides a third natural family: Riemann surfaces uniformized by torsionfree normal subgroups of hyperbolic triangle groups. Such surfaces were studied from a geometric-analytic perspective by J. Wolfart, and they are, in a certain precise sense, the "most distinguished" points on the moduli space over curves of genus g. However, there are only finitely many triangle groups which are arithmetic in the above technical sense, so for most of these curves, the Hecke theory is not available. This seems to have dissuaded experts from studying them. However, I maintain that algebraic curves uniformized by congruence subgroups of hyperbolic triangle groups are very interesting from an arithmetic geometric perspective...even though most of them are not "arithmetic" in the above technical sense! Nevertheless they can be studied via a variety of methods, including the (rapidly developing) arithmetic theory of branched covering maps, via the theory of Fuchsian groups and their finite quotients, and via embedding into higher dimensional quaternionic Shimura varieties. In this talk I will give a broad overview of these curves, including their classification in terms of PSL_2(F_q)-Galois Belyi maps, and the construction of models of these curves over the minimal possible number fields. All of this is joint work with John Voight...over a period of many years, but now almost finished. My strategy is to generate sufficient interest in this work to motivate us (especially, me) to finally finish it off.
Submitted by