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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635526
• 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

我的建议是在函数域设置中制定并证明数论中几个著名猜想的类似物。这些类似物既美丽又自然，但在文献中却被忽视了。为攻击此类类似物而创建的技术在数论中具有丰富的意想不到的应用。非常突出的是，德利涅在韦伊猜想上的工作以及随后在指数和上的结果导致了整个数论的重大突破，甚至在组合学和遍历理论中被证明是有用的。下面我讨论我正在进行的两个研究项目，它们例证了上述哲学：它们还说明了这样一个原则：研究函数域类比通常对于在原始问题上取得进展很有用，无论是直接作为解决方案的一步，还是在 1) Frey-Mazur 猜想指出，对于任何素数 p > 17，有理数上的椭圆曲线可以通过简单地将其 p 挠率视为伽罗瓦表示来分类为同源性。这是一个非常深刻的猜想，它表明对 Mazur 和其他人之前关于椭圆曲线扭转的工作的广泛概括。人们可以将 Frey-Mazur 猜想重新表述为某一族模空间 M_p 不具有有理点的陈述。我们与本杰明·巴克 (Benjamin Bakker) 一起研究了在函数域（任何特征）上定义的椭圆曲线的猜想。该类比相当于 M_p 不包含任何低亏格曲线的陈述。以 Bombieri-Lang 猜想为条件，这意味着簇 M_p 有理点的有限性，为原始猜想迈出了第一步。正如所预料的，该猜想的函数域版本本身涉及一些非常有趣的数学：特别是，通过结合代数几何、双曲几何和丢番图近似的方法，巴克和我有 成功证明了“假椭圆曲线”的类似猜想，即允许四元数乘法的阿贝尔曲面。由于空间 M_p 是非紧的，最初的猜想目前还难以捉摸，但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，并且正在进一步研究这一点。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制定阿贝尔簇的 Frey-Mazur 猜想，其中情况因泰特模辛群的群论而变得更加复杂。 2) 数论中有许多猜想指出，齐质空间中的各种群轨道族是均匀分布的。解决这些问题的方法通常分为分析方法（Duke、Iwaniec，...）和遍历理论方法（Lindenstrauss、Einsiedler，...）。最简单的未解决案例之一是 Venkatesh 和 Michel 关于增长判别式 Heegner 点对的所谓“混合猜想”。在最近与 Vivek Shende 的合作中，我们证明了这些猜想的函数场类比具有美丽的几何描述，涉及低线性曲线上向量丛的模空间。在混合猜想的情况下，我们展示了如何从超椭圆曲线的布里尔-诺特轨迹的上同调稳定性结果中得出该问题。通过建立这一点，我们证明了函数域设置中的混合猜想（结果目前以这些空间的 Betti 数之和的指数界为条件，目前我们只能在特征 0 中建立；这似乎是 t