论文标题
与非CM椭圆曲线产物相关的Frobenius痕迹分布的边界
Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves
论文作者
论文摘要
令$ g \ geq 1 $为整数，让$ a/\ mathbb {q} $是一个亚伯族品种，超过$ \ mathbb {q} $ to％to％product $ e_1 \ times \ times \ ldots \ ldots \ times e_g e_g e_g e_g e_g的椭圆形curves $ e_1/\ e_1/\ mathbb $ e_1/\ mathbb { $ e_g/\ mathbb {q} $，没有复杂的乘法，并且在$ \ overline {\ mathbb {q}} $上均不复杂。 $ g $椭圆曲线的产物定义在$ \ mathbb {q} $上，对$ \ overline {\ mathbb {q}} $以及每个都没有复杂的乘法而对$ \ overline {\ mathbb {q}} $上定义。 ％成对的非遗传性$ \ +叠加{\ mathbb {q}} $。对于整数$ t $和一个正实数$ x $，用$π_a（x，t）$表示Primes $ p \ leq x $的数量，以％的abelian valivert $ a $减少，frobenius trace $ a_ {1，p}，p}（a，p}（a）$与$ a $ a $ modulo $ equals $ equals $ equals $ equals $ equals $ equals相关。假设Dedekind Zeta功能的一般性Riemann假设，我们证明$π_A（x，0）\ ll_a x^{1 - \ frac {1} {1} {3 g+1}}}}}}/（\ log x） X^{1- \ frac {1} {3 G + 2}}}/（\ log x）^{1- \ frac {2} {3 G + 2}} $如果$ t \ neq 0 $。这些界限在很大程度上改善了H. Chen，N。Jones和V. Serban的$ G = 2 $的最新界限，并可能被视为对M.R. Murty，V.K. $ G = 1 $获得的任意$ G $的概括。 Murty和N. Saradha，结合了D. Zywina的$ \ log X $的优化。在相同的假设下，我们还证明了一组Primes $ p $满足$ | a_ {1，p}（a）|> p^{\ frac {\ frac {1} {3 g + 1} - \ varepsilon} $的密度。
Let $g \geq 1$ be an integer and let $A/\mathbb{Q}$ be an abelian variety that is isogenous over $\mathbb{Q}$ to %the product $E_1 \times \ldots \times E_g$ of elliptic curves $E_1/\mathbb{Q}$, $\ldots$, $E_g/\mathbb{Q}$, without complex multiplication and pairwise non-isogenous over $\overline{\mathbb{Q}}$. a product of $g$ elliptic curves defined over $\mathbb{Q}$, pairwise non-isogenous over $\overline{\mathbb{Q}}$ and each without complex multiplication. %pairwise non-isogenous over $\overline{\mathbb{Q}}$. For an integer $t$ and a positive real number $x$, denote by $π_A(x, t)$ the number of primes $p \leq x$, of good reduction for %the abelian variety $A$, for which the Frobenius trace $a_{1, p}(A)$ associated to the reduction of $A$ modulo $p$ equals $t$. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that $π_A(x, 0) \ll_A x^{1 - \frac{1}{3 g+1 }}/(\log x)^{1 - \frac{2}{3 g+1}}$ and $π_A(x, t) \ll_A x^{1 - \frac{1}{3 g + 2}}/(\log x)^{1 - \frac{2}{3 g + 2}}$ if $t \neq 0$. These bounds largely improve upon recent ones obtained for $g = 2$ by H. Chen, N. Jones, and V. Serban, and may be viewed as generalizations to arbitrary $g$ of the bounds obtained for $g=1$ by M.R. Murty, V.K. Murty, and N. Saradha, combined with a refinement in the power of $\log x$ by D. Zywina. Under the same assumptions, we also prove the existence of a density one set of primes $p$ satisfying $|a_{1, p}(A)|>p^{\frac{1}{3 g + 1} - \varepsilon}$ for any fixed $\varepsilon>0$.
