代数K-理论为研究算术代数几何,特别是椭圆曲线提供了新思路新方法.本项目利用椭圆曲线的K-理论和Mahler测度之间的关系来研究椭圆曲线的算术性质，椭圆双对数与L-函数之间的关系（Zagier猜想）以及Beilinson猜想.利用带复乘的椭圆曲线的理论来研究二次多项式表素数问题. 发展我们已有的独创方法研究数域的代数整数环的K群与高阶Regulator，L函数，Zeta函数，Iwasawa不变量方面的关系.

Algebraic K-theory provide new ideas and methods to study arithmetic algebraic geometry, in particular elliptic curves.In this poject, we shall study the arithmetic properties of elliptic curves，the relation between elliptic dilogarithm and L-functions of elliptic curves (Zagier's conjecture）and Beilinson's conjecture by the relationship between Mahler measure and K-theory of elliptic curve. By the theory of CM elliptic curve ,we will study the problem of primes captured by quadratic polynomials. Developing of our original methods, we further study the relationship among K-groups of the ring of algebraic integers, higher Regulators, L functions, Zeta functions and Iwasawa invariants of number fields.