Arithmetic of Elliptic Curves

9970382博士西尔弗曼将研究椭圆曲线的算法。他将做一个新的算法，解决椭圆曲线上的离散对数问题的理论分析，并给出优化算法所需的各种参数。 他将研究下限的典型高度椭圆曲线和证明一个莱默型界的非互易点。 他将调查之间的关系Frobenius迹值的椭圆曲线和排名的曲线，以便研究的变化排名家庭的曲线，并给予概率定量理由梅斯特的方法生产椭圆曲线的高排名。 所有这些项目将大大推进我们的知识的算术理论的椭圆曲线，一个领域的数学，这是广泛的兴趣，在其本身的权利，因为其适用于数论和其他领域的mathematics.The拟议的研究项目符合一般框架的理解数论性质的椭圆曲线。 这个理论已经被广泛研究了很长一段时间，既因为它的内在美，也因为它对数论和数学其他领域的适用性。近年来，充分发挥了现代算术和代数几何的威力，取得了很好的效果。 这三个主要项目将回答有关椭圆曲线算法的重要理论和实践问题，并将推动这一领域的进一步研究。

9970382Dr. Silverman will conduct research on the arithmetic of elliptic curves. He will do a theoretical analysis of a new algorithm for solving the discrete logarithm problem on elliptic curves and give optimizations for the various parameters required by the algorithm. He will study lower bounds for the canonical height on elliptic curves and prove a Lehmer-type bound for non-reciprocal points. He will investigate the relationship between the Frobenius trace values of an elliptic curve and the rank of the curve so as the study the variation of the rank in families of curves and give a probabilistic quantitative justification for Mestre's method of producing elliptic curves of high rank. All of these projects will significantly advance our knowledge of the arithmetic theory of elliptic curves, an area of mathematics which is of widespread interest in its own right and because of its applicability to number theory and other areas of mathematics.The proposed research project fits into the general framework of understanding the number theoretic properties of elliptic curves. This theory has been extensively studied for a long time, both for its intrinsic beauty and its applicability to other areas of number theory and mathematics. In recent years, the full power of modern arithmetic and algebraic geometry have been brought into play, with excellent results. All three main projects will answer important theoretical and practical questions concerning the arithmetic of elliptic curves and will spur further research in this field.