论文标题
与通用的阿贝尔品种相关的Frobenius痕迹的分布的界限
Bounds for the distribution of the Frobenius traces associated to a generic abelian variety
论文作者
论文摘要
令$ g \ geq 1 $为整数，让$ a $为$ \ mathbb {q} $和dimension $ g $定义的Abelian品种。假设，对于每个足够大的prime $ \ ell $，$ a $ a $的剩余模型$ \ ell $ galois表示形式是同构对$ \ text {gsp} _ {2G} _ {2G}（\ m athbb {z}/\ ell \ ell \ ell \ mathbb {z}）$。对于整数$ t $和一个正实数$ x $，用$π_A（x，t）$表示Primes $ p \ leq x $的数量，$ a $的质量降低了，为此，Frobenius trace $ a_ {1，p}，p}（a）与减少$ a $ a $ modulo $ p $ p $ p $ e equals $ e equals $ equals $ e e e e e e e e e e e e e e equals $ a $ e e a $ a $。假设Dedekind Zeta功能的一般性Riemann假设，我们证明$π_A（x，0）\ ll_a x^{1- \ frac {1} {2G^2+g+1}}}}}}}}}}}}}/（\ log x） t）\ ll_a x^{1- \ frac {1} {2g^2+g+2}}}}/（\ log x）^{1- \ frac {2} {2G^2+g+2}} $如果$ t \ neq 0 $。根据上述假设，我们还证明了一组普特$ p $满足$ | a_ {1，p}（a）|> p^{\ frac {\ frac {1} {2g^2+g+1}}}/（\ log p）/（\ log p）假设除了对Dedekind Zeta功能的普遍假设，Artin的Holomorphy猜想以及对Artin L功能的跨相关性猜想之外，我们还证明了$π_A（x，0）\ ll_a x^{1- \ frac {1- \ frac {1} {1} {1} {g+1} {g+1}}}} x）^{1- \ frac {4} {g+1}} $和那个$π_A（x，x，t）\ ll_a x^{1- \ frac {1} {g+2}}}}}/（\ log x）密度一组Primes $ p $满足$ | a_ {1，p}（a）|> p^{\ frac {1} {g+2} - \ \ varepsilon} $，用于任何固定的$ \ varepsilon> 0 $。这些目前是$π_A（x，t）$的最著名的有条件上限，也是$ | a_ {1，p}（a）| $，对于大多数Primes $ p $的最著名有条件下限。
Let $g \geq 1$ be an integer and let $A$ be an abelian variety defined over $\mathbb{Q}$ and of dimension $g$. Assume that, for each sufficiently large prime $\ell$, the image of the residual modulo $\ell$ Galois representation of $A$ is isomorphic to $\text{GSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$. 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 $A$, 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 $π_A(x, 0) \ll_A x^{1-\frac{1}{2g^2+g+1}}/(\log x)^{1-\frac{2}{2g^2+g+1}}$ and that $π_A(x, t) \ll_A x^{1-\frac{1}{2g^2+g+2}}/(\log x)^{1-\frac{2}{2g^2+g+2}}$ if $t \neq 0$. Under the assumptions stated above, we also prove the existence of a density one set of primes $p$ satisfying $|a_{1, p}(A)|> p^{\frac{1}{2g^2+g+1}}/(\log p)^{\varepsilon}$ for any fixed $\varepsilon>0$. Assuming, in addition to the Generalized Riemann Hypothesis for Dedekind zeta functions, Artin's Holomorphy Conjecture and a Pair Correlation Conjecture for Artin L-functions, we prove that $π_A(x, 0) \ll_A x^{1-\frac{1}{g+1}}/(\log x)^{1-\frac{4}{g+1}}$ and that $π_A(x, t) \ll_A x^{1-\frac{1}{g+2}}/(\log x)^{1-\frac{4}{g+2}}$ if $t \neq 0$, and we deduce the existence of a density one set of primes $p$ satisfying $|a_{1, p}(A)|> p^{\frac{1}{g+2}-\varepsilon}$ for any fixed $\varepsilon>0$. These are currently the best known conditional upper bounds for $π_A(x, t)$ and the best known conditional lower bounds for $|a_{1, p}(A)|$, for most primes $p$.
