Number Theory and Algebraic Geometry

研究人员Noam Elkies研究了数论中的问题，特别是椭圆曲线和曲面等变种的算法，以及它们与纠错码和欧几里德格等其他数学领域的关系和应用。具体项目包括：证明和计算利用重量为3/2的p-进制模形式和奇解析秩二次扭曲的算术之间的新关系；改进将Mordell-Weil格与Weyl群联系起来的Shioda-Usui理论，并将其推广到某些复反射群；继续研究最优递归塔和研究人员对这类塔的模性猜想；扩展双线性(Somos)递归、theta序列和显式模空间之间的联系的研究；以及格约化在研究和有效计算丢番图方程和不等式的解的新应用。调查者继续研究与方程的代数解的数学结构有关的数论问题。这将使我们更好地理解这些方程及其解(椭圆曲线等)，并与其他主题进一步联系，如效率计算、纠错码和球体填充。

Abstract Noam Elkies The investigator studies problems in number theory, notablythe arithmetic of varieties such as elliptic curves and surfacesand their relations and applications to other areas of mathematicssuch as error-correcting codes and Euclidean lattices. Specificprojects include: proof and computational exploitation of newrelations between p-adic modular forms of weight 3/2 and the arithmeticof quadratic twists of odd analytic rank; refinement of the Shioda-Usuitheory relating Mordell-Weil lattices with Weyl groups, andgeneralization to certain complex reflection groups; continued studyof optimal recursive towers and the investigator's modularity conjecturefor such towers; extended study of the connection between bilinear(Somos) recurrences, theta sequences, and explicit moduli spaces;and novel applications of lattice reduction to study and efficientlycompute solutions of Diophantine equations and inequalities.The investigator continues his study of problems in number theoryconcerning the mathematical structure of algebraic solutionsof equations. This should lead both to better understandingof those equations and their solutions (elliptic curves, etc.),and to further connections with other topics such as efficientcomputation, error-correcting codes, and sphere packing.