Explicit Higher Arithmetic Geometry

PI的研究主要关注丢番图方程：丢番图方程是一个我们寻求整数（整数）或有理数（分数）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-由费马在350年前提出，直到1995年才被怀尔斯证明-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将费马方程的假设解通过弗雷椭圆曲线与椭圆模形式相关联来进行的。在贾维斯（谢菲尔德）和达蒙（麦吉尔）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弗雷椭圆曲线或通过弗雷超几何阿贝尔簇与希尔伯特模形式相关。我们建议研究这种方法，并使其显式的几个家庭的丢番图方程，这可能是解决的帮助下，最近的计算突破，由于Dembele.Another方向的建议研究涉及显式算法的阿贝尔簇的子簇。这些品种的主题是最近的理论进展的Faltings，Vojta，Buium等在许多方面，这些品种是最自然的推广曲线的高等属谁明确的算术已深入研究了卡塞尔，弗林，谢弗，Poonen，斯托尔，Bruin等在过去的15年。拟议的研究将寻求转移到阿贝尔品种的子品种领域的许多适用于曲线的技术。特别是，我们将寻求类似的科尔曼界，Chabauty，莫德尔-韦伊和明确的方法来确定合理的点。

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.