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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=667233
• 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

我的建议是制定和证明类似的几个著名的代数数论中的功能领域的设置。这些类似物既美丽又自然，但在文献中被忽视了。攻击这种类似物的技术在数论中有着意想不到的应用。非常突出的是，德利涅在魏尔定理上的工作以及随后关于指数和的结果导致了整个数论的重大突破，甚至在组合数学和遍历理论中也被证明是有用的。下面我将讨论我正在进行的两个研究项目，它们体现了上述哲学：* 它们还说明了这样一个原则，即研究函数场模拟通常有助于在原始问题上取得进展，要么直接作为解决方案的一个步骤，要么以更微妙的方式提供如何进行的直觉。1)Frey-Mazur猜想指出，对于任何p > 17的素数，有理数上的椭圆曲线可以简单地通过将它们的p-挠视为伽罗瓦表示来分类到Isaac。这是一个非常深刻的猜想，这表明了广泛的推广以前的工作马祖尔和其他扭转的椭圆曲线。我们可以将Frey-Mazur猜想重新表述为这样一个命题：某类模空间M_p不具有有理点。我们与Benjamin Bakker一起研究了定义在函数域（任何特征）上的椭圆曲线的这个猜想。这种类比相当于M_p不包含任何低亏格曲线的陈述。在Besili-Lang猜想的条件下，这将意味着簇M_p的有理点的有限性，提供了通向原始猜想的第一步。正如所料，功能领域版本的猜想涉及一些非常有趣的数学本身：特别是，通过结合方法从代数几何，双曲几何，丢番图近似，巴克和我已经成功地证明了类似的猜想“假椭圆曲线”，即阿贝尔表面承认四元数乘法。由于空间M_p是非紧的，原始猜想至今仍是难以捉摸的，但我们乐观地认为，同样的方法可以在原始问题上取得进一步的进展，并正在进一步研究这一问题。** 由于我们的方法也适用于与阿贝尔簇相关的高维模空间，我们希望这项工作有助于为阿贝尔簇建立Frey-Mazur猜想，其中Tate模的辛群的群论使情况进一步复杂化。 **2）在数论中有许多定理指出，齐次空间中的各种群轨道族变得等分布。攻击这些问题的方法一般分为分析方法（杜克，Iwaniec，.）和遍历理论方法（Lindenstrauss，Einsiedler，.）。最简单的未解决的情况之一是所谓的“混合猜想”的Venkatesh和米歇尔关于对Heegner点的增长判别。在最近的工作与Vivek Shende，我们表明，功能领域的模拟，这些acquitures有一个美丽的几何描述，涉及模空间的向量丛曲线的低gonality。在混合猜想的情况下，我们展示了如何从超椭圆曲线的Brill-Noether轨迹的上同调稳定化结果中得出该问题。通过建立这一点，我们证明了混合猜想在函数域设置（结果目前是有条件的指数界的总和，这些空间，我们目前只能建立在特征0;这似乎是一个指数界的这些空间的Betti数，