Function Field Analogues of Questions in Number Theory

我的建议是在函数场设置中形成和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。为攻击这种类似物而创造的技术在数论中有着丰富的意想不到的应用。Deligne关于Weil猜想的工作以及随后关于指数和的结果在数论方面取得了重大突破，甚至在组合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，它们是上述哲学的例证：它们还说明了研究函数场模拟通常有助于在原始问题上取得进展的原则，要么直接作为解决方案的一个步骤，要么以一种更微妙的方式提供如何进行的直觉。**1) Frey-Mazur猜想指出，对于任何素数p bbbb17，在有理数上的椭圆曲线可以简单地通过将其p-扭转视为伽罗瓦表示来分类到等根。这是一个非常深刻的猜想，它对Mazur等人先前关于椭圆曲线扭转的工作进行了广泛的推广。我们可以将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。与Benjamin Bakker一起，我们一直在研究在函数场（任何特征）上定义的椭圆曲线的这个猜想。这种类比相当于M_p不包含任何低属曲线。在Bombieri-Lang猜想的条件下，这将意味着M_p的有理点的有限性，为原始猜想提供了第一步。**正如所料，该猜想的函数域版本本身涉及到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“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是非紧化的，原始猜想仍然难以捉摸，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，并将进一步研究这一问题。**由于我们的方法也适用于与阿贝尔变体相关的高维模空间，我们希望这项工作将有助于建立阿贝尔变体的Frey-Mazur猜想，其中情况因Tate模的辛群的群理论而进一步复杂化。**2)在数论中有许多猜想表明齐次空间中各种族的群轨道是等分布的。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, einsedler，…）。其中一个最简单的未解决的案例是Venkatesh和Michel关于成长判别的Heegner点对的所谓“混合猜想”。在最近与Vivek Shende合作的工作中，我们证明了这些猜想的函数场模拟具有涉及低向性曲线上向量束模空间的漂亮几何描述。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轨迹的上同调的稳定化结果推导出这个问题。通过建立这一点，我们证明了函数场设置下的混合猜想（结果目前以这些空间的Betti数和的指数界为条件，我们目前只能在特征0中建立，这是t）

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