论文标题
关于算术除数的不可还原性和分布
On the Irreducibility and Distribution of Arithmetic Divisors
论文作者
论文摘要
我们介绍了算术循环的$ε$ - $ - $ - $ - $ - $可记录，这意味着其分析部分的程度与其不可还原的经典部分相比很小。我们将证明，每$ε> 0 $，算术上充足的Hermitian线束的任何足够高的张量功率都可以由$ε$ - $ rirredubible-mirreducible Arithmetic Divisor表示。我们的证明方法还使我们能够研究算术上足够的Hermitian Line Bundle $ \ overline {\ Mathcal {l}} $的小部分的分布分布。我们将证明，为了增加张量功率，$ \叠加{\ Mathcal {l}}^{\ otimes n} $在弱的意义上，这些除数的归一化dirac测量值几乎总是收敛到$ c_1（\ overline {\ mathcal {l}}}）$。使用数字的几何形状，我们将从复杂分析中的正线束随机部分的分布分布中推断出这一结果。作为一个应用程序，我们将为零整数多项式集提供新的等分结果。最后，我们将表达算术相交的算术相交数量，算术上充足的遗传线束束作为有限纤维上经典几何相交数的限制。
We introduce the notion of $ε$-irreducibility for arithmetic cycles meaning that the degree of its analytic part is small compared to the degree of its irreducible classical part. We will show that for every $ε>0$ any sufficiently high tensor power of an arithmetically ample hermitian line bundle can be represented by an $ε$-irreducible arithmetic divisor. Our methods of proof also allow us to study the distribution of divisors of small sections of an arithmetically ample hermitian line bundle $\overline{\mathcal{L}}$. We will prove that for increasing tensor powers $\overline{\mathcal{L}}^{\otimes n}$ the normalized Dirac measures of these divisors almost always converge to $c_1(\overline{\mathcal{L}})$ in the weak sense. Using geometry of numbers we will deduce this result from a distribution result on divisors of random sections of positive line bundles in complex analysis. As an application, we will give a new equidistribution result for the zero sets of integer polynomials. Finally, we will express the arithmetic intersection number of arithmetically ample hermitian line bundles as a limit of classical geometric intersection numbers over the finite fibers.
