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的背景下，对数论中几个著名猜想的类比进行公式化和证明。这些类比既美丽又自然，但在文学中却被忽视了。为攻击这种类比而创造的技术在数论中有着意想不到的应用。值得注意的是，Deligne关于Weil猜想的工作和随后关于指数和的结果在整个数论中取得了重大突破，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理，即研究函数域模拟通常有助于在原始问题上取得进展，或者直接作为解决方案的一步，或者通过提供如何进行的直觉以更微妙的方式。1)Frey-Mazur猜想表明，对于任何素数p&gt，17，有理数上的椭圆曲线可以简单地通过将它们的p-挠率作为伽罗瓦表示来分类到同源。这是一个非常深刻的猜想，它暗示了Mazur和其他人以前关于椭圆曲线扭转的工作的广泛推广。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。我们和Benjamin Bakker一起研究了定义在函数域上的椭圆曲线的这个猜想(任何特征)。这个类比相当于说M_p不包含任何低亏格曲线。在Bombieri-Lang猜想的条件下，这意味着变元M_p的有理点是有限的，这是通向原始猜想的第一步。不出所料，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，Bakker和我成功地证明了类似的猜想“伪椭圆曲线”，即允许四元数乘法的阿贝尔曲面。由于空间M_p是非紧的，最初的猜想至今仍难以捉摸，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交换簇有关的高维模空间，我们希望这项工作将有助于建立一个关于交换簇的Frey-Mazur猜想，在这种情况下，情况被Tate模的辛群的群论进一步复杂化。2)数论中有许多猜想表明齐次空间中的各族群轨道是均匀分布的。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等)。最简单的悬而未决的案例之一是Venkatesh和Michel关于Heegner点对日益增长的判别式的所谓“混合猜想”。在最近与Vivek Shende的工作中，我们证明了这些猜想的函数场模拟具有涉及低阶次曲线上的向量丛的模空间的美丽的几何描述。在混合猜想的情况下，我们证明了关于超椭圆曲线的Brill-Noether轨迹上同调的稳定化的结果是如何产生的。通过建立这一点，我们证明了函数域中的混合猜想(目前的结果是以这些空间的Betti数之和的指数界为条件的，目前我们只能在特征0中建立；这似乎是