Arithmetic of Elliptic Curves

9970382 DR.西尔弗曼将对椭圆曲线的算法进行研究。他将对一种求解椭圆曲线上离散对数问题的新算法进行理论分析，并对算法所需的各种参数进行优化。他将研究椭圆曲线上正则高度的下界，并证明非互易点的Lehmer型下界。他将研究椭圆曲线的Frobenius迹值与曲线的阶数之间的关系，从而研究曲线族中阶数的变化，并给出Mestre产生高阶椭圆曲线的方法的概率定量证明。所有这些项目都将极大地提高我们对椭圆曲线算术理论的了解，这一数学领域本身就具有广泛的兴趣，因为它适用于数论和其他数学领域。这一理论已经被广泛研究了很长时间，既是因为它的内在美，也是因为它适用于数论和数学的其他领域。近年来，现代算术和代数几何的力量得到了充分的发挥，取得了优异的效果。所有这三个主要项目都将回答有关椭圆曲线算法的重要理论和实践问题，并将推动该领域的进一步研究。

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.