论文标题
有效的Abelian品种和应用的Sato-Tate猜想
Effective Sato-Tate conjecture for abelian varieties and applications
论文作者
论文摘要
从对动机功能的广义假设假设，我们得出了Abelian品种A的有效版本，该版本是通过连接的SATO-TATE组定义的Abelian品种A定义的。通过有效，我们的意思是，我们在SATO-TATE量度预测的计数中给出了上限，该计数仅取决于A的某些不变性。我们讨论了此条件结果的三个应用。首先，对于在K上定义的Abelian品种，我们考虑了Linnik的Abelian品种问题的变体，该变体要求在给定间隔的归一化Frobenius Trace的素数的最小规范上建立上限。其次，对于具有复杂乘法的k上定义的椭圆曲线，我们确定（直至通过非零常数）的渐近数量的frobenius trace的渐近数达到了hasse-weil结合的整体部分。第三，对于在K上定义的一对Abelian品种，没有常见的因素，直到k衰减，我们发现了在最小规范的上限上，相应的frobenius痕迹具有相反的符号。
From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato-Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik's problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius trace attain the integral part of the Hasse-Weil bound. Third, for a pair of abelian varieties defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces have opposite sign.
