Let $E$ be an elliptic curve over an algebraically closed field of characteristic 0. We prove that the pre-Bloch group of the function field of $E$ can be generated by the functions of degree not higher than 3. We apply this result to the elliptic dilogarithm function defined by S. Bloch. He has shown that any element of the pre-Bloch group gives a (so-called elliptic Bloch) relation between the values of the so-called Elliptic dilogarithm. We conclude that any elliptic Bloch relation can be reduced to the antisymmetry relation and the elliptic Bloch relations for the functions of degree 3.
Nontrivial lower bounds for the $p$-adic valuations of some type of rational numbers and an application for establishing the integrality of some rational sequences
A relation between the zeros of different two $L$-functions which have the Euler product and functional equation
As automorphic $L$ -functions or Artin $L$ -functions, several classes of $L$ -functions have Euler products and functional equations . In this paper we study the zeros of $L$ -functions which have the Euler products and functional equations . We show that there exists some relation between the zeros of the Riemann zeta-function and the zeros of such $L$ -functions. As a special case of our results, we find the relations between the zeros of the Riemann zeta-function and the zeros of automorphic $L$ -functions attached to elliptic modular forms or the zeros of Rankin-Selberg $L$ -functions att…
Distortion maps for elliptic curves over finite fields
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；University of Münster（明斯特大学）；National and Kapodistrian University of Athens（雅典国立卡波蒂斯坦大学）；University of Copenhagen（哥本哈根大学）；Rijksuniversiteit Groningen（格罗宁根大学）；Sabancı University（萨班哲大学）；University of Bristol（布里斯托大学）
