In this report, I will talk about two topics. One is about the cube sum problem; the other is a generalization of Kolyvagin's work on bounding Selmer groups.

I). A nonzero rational number is called a cube sum if it is of form $a^3+b^3$, $a,b/in /BQ^/times$. We prove that for any integer $k/geq 1$, there exists infinitely many odd integer $n$ with exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a cube sum). The cube sum problems are naturally related to certain elliptic curves, and we obtain some results on Birch and Swinnerton-Dyer conjectures of these elliptic curves. This is a joint work with L. Cai and Y. Tian.

II). Given a modular form $f$ and an algebraic Hecke character $/chi$, let $V_f$ and $V_/chi$ be the $2$-dimensional Galois representations of $G_/BQ$ associated to $f$ and $/chi$. Then $V=V_f/otimes V_/chi$ is a $4$-dimensional Galois representation. Our goal is to use the Euler system of generalized Heegner cycles to bound the Bloch-Kato Selmer gorup $H^1_f(/BQ,V)$. We prove, under suitable conditions, that if the generalized Heegner cycle associated to $f$ and $/chi$ is not torsion, then $H_f^1(/BQ,V)$ is one-dimensional and is generated by the generalized Heegner cycle. This should be consistent with the Beilinson-Bloch-Kato conjecture. This is a generalization of Kolyvagin's work on bounding the Selmer groups of elliptic curves using Euler system of Heegner points. This is joint with Y. Tian.

海报