Function Field Analogues of Questions in Number Theory

点击翻译按钮获取中文摘要

My proposal is to formulate and prove analogues of several well-known conjectures in number theory in the function field setting. These analogues are both beautiful and natural, yet have been overlooked in the literature. The techniques created to attack such analogues are rich with unexpected applications in number theory. Very prominently, Deligne's work on the Weil conjectures and the subsequent results on exponential sums have led to major breakthroughs throughout number theory, and have even proven useful in combinatorics and ergodic theory.**Below I discuss two of my ongoing research projects which exemplify the above philosophy:*They also illustrate the principle that studying the function field analogue is often useful for making progress on the original problem, either directly as a step in the solution, or in a more subtle manner by providing intuition on how to proceed.**1) The Frey-Mazur conjecture states that for any prime p > 17, elliptic curves over the rationals can be classified up to isogeny simply by looking at their p-torsion as a Galois representation. This is a very deep conjecture which suggests a vast generalization of previous work of Mazur and others on torsion of elliptic curves. One can reformulate the Frey-Mazur conjecture as the statement that a certain family of moduli spaces M_p does not possess rational points. Together with Benjamin Bakker, we have been investigating this conjecture for elliptic curves defined over function fields (of any characteristic). The analogue is tantamount to the statement that M_p does not contain any low genus curves. Conditional on the conjecture of Bombieri-Lang, this would imply finiteness of rational points for the varieties M_p, providing a first step towards the original conjecture.**As is to be expected, the function field version of the conjecture involves some very interesting mathematics in and of itself: in particular, by combining methods from algebraic geometry, hyperbolic geometry, and diophantine approximation, Bakker and I have succeeded in proving the analogous conjecture for "fake elliptic curves", i.e. abelian surfaces admitting quaternionic multiplication. The original conjecture is as of yet elusive due to the spaces M_p being non-compact, but we are optimistic that the same methods can make further progress on the original problem and are investigating this further. **As our methods are also applicable to higher-dimensional moduli spaces related to abelian varieties, we hope that this work will be helpful in formulating a Frey-Mazur conjecture for abelian varieties, where the situation is further complicated by the group theory of the symplectic group of the Tate module. **2) There are many conjectures in number theory stating that various families of group orbits in homogeneous spaces become equidistributed. Methods to attack these questions generally split up into analytic methods (Duke, Iwaniec, ...) and ergodic theory methods (Lindenstrauss, Einsiedler, ...). One of the simplest unresolved cases is the so-called "mixing conjecture" of Venkatesh and Michel regarding pairs of Heegner points of growing discriminant. In recent work with Vivek Shende, we show that the function field analogue of these conjectures has a beautiful geometric description involving moduli spaces of vector bundles on curves of low gonality. In the case of the mixing conjecture, we show how the problem would follows from results on stabilization of cohomology of the Brill-Noether Loci of hyperelliptic curves. By establishing this, we prove the mixing conjecture in the function field setting (the result is currently conditional on an exponential bound for the sums of the Betti numbers of these spaces which we can only establish at present in characteristic 0; this appears t