Function Field Analogues of Questions in Number Theory

数论问题的函数域类似物

• 批准号: RGPIN-2014-05784
• 负责人: Tsimerman, Jacob
• 金额: $ 2.84万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611843
• 关键词: Function; Field; Analogues; Questions; Number

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

我的建议是制定并证明几种知名的被命名的类似物。在组合学和千古理论中。 下面我讨论了我正在进行的两个研究项目，这些项目体现了上述理念： 他们说，通过提供有关如何继续进行的攻击，研究田间田间方面的问题，以使进度问题成为解决方案的一步，或者以更单一的方式进行。 1）弗雷·马祖尔（Frey-Mazur）的猜想是，作为galois代表的任何prime p> 17。我们一直在调查在任何特征上定义的Eliptic曲线（模拟）。对于品种M_P，提供了迈向原始猜想的第一步。 正如预期的。我们乐观地认为，相同的方法可以在原始问题上取得进一步的进展，并正在进一步研究。 由于我们的方法也适用于更高的Ating Afrey-Mazur猜想，适用于泰特模块的Abelian品种。 2）在均质空间中，有许多猜想的构造量通常分布在分析方法上SimpleSolved Case是所谓的“ Venkatesh和Michel关于公开的所谓”。在与Vivek Shende的最新工作中，我们表明这些猜想的田野模拟具有美丽的几何震动模量模量，矢量巨头庞然大物。该问题是从高椭圆形曲线的Brill-Noether基因座的结果中遵循的