Explicit Higher Arithmetic Geometry

PI的研究主要涉及丢番图方程：丢番图方程是我们以整数(整数)或有理数(小数)来求解的方程。丢番图方程的一个例子是x^n+y^n=z^n。费马大定理-由Fermat在350年前提出，直到1995年才由Wiles证明-指出没有n至少为3且x，y，z都是非零整数的解。费马大定理的证明是通过一条Frey椭圆曲线将费马方程的假想解与椭圆模形式联系起来。在Jarvis(Shefffield)和Darmon(McGill)的工作中，设想了这种设置的推广，其中丢番图方程的解通过数域上的Frey椭圆曲线或通过Frey超几何Abelian簇与Hilbert模形式相关。我们建议对这种方法进行研究，并将其显式用于几族丢番图方程族，然后利用Dembel最近的计算突破来求解这些方程。另一个建议的研究方向涉及Abelian簇的子簇的显式算法。这些变元是Faltings，Vojta，Buium等人最近理论进展的主题。在许多方面，这些变元是更高亏格曲线的最自然的推广，在过去的15年里，Cassel，Flynn，Schaefer，Poonen，Stoll，Bruin等人对显式算术进行了深入的研究。拟议的研究将寻求将许多适用于曲线的技术转移到阿贝尔变种的亚变种领域。特别是，我们将寻求类似于Coleman界、Chabauty、Mordell-Weil和确定有理点的显式方法。

The PI's research is mainly concerned with Diophantine equations: a Diophantine equation is an equation for which we seek solutions in integers (whole numbers) or rationals (fractional numbers). An example of a Diophantine equation is x^n+y^n=z^n. Fermat's Last Theorem---posed by Fermat 350 years ago and only proved by Wiles in 1995---states that there are no solutions with n at least 3 and x,y,z all non-zero integers. The proof of Fermat's Last Theorem works by relating hypothetical solutions of the Fermat equation to elliptic modular forms via a Frey elliptic curve. In the work of Jarvis (Sheffield) and of Darmon (McGill) a generalization of this setting is envisaged where solutions of Diophantine equations are related to Hilbert modular forms via Frey elliptic curves over number fields or via Frey hypergeometric Abelian varieties. It is proposed to investigate this approach and make it explicit for several families of Diophantine equations, which may then be solved with the help of recent computational breakthroughs due to Dembele.Another direction of the proposed study involves the explicit arithmetic of subvarieties of Abelian varieties. Such varieties are the subject of recent theoretical advances by Faltings, Vojta, Buium, etc. In many ways, these varieties are the most natural generalization of curves of higher genus who explicit arithmetic has been intensively studied by Cassels, Flynn, Schaefer, Poonen, Stoll, Bruin, etc. over the last 15 years. The proposed research will seek to transfer many of the techniques applicable to curves to the realm of subvarieties of Abelian varieties. In particular, we will seek analogues of Coleman bounds, Chabauty, Mordell-Weil and explicit methods for determining rational points.