论文标题
Abelian品种中的Hypersurfaces的Shafarevich猜想
The Shafarevich conjecture for hypersurfaces in abelian varieties
论文作者
论文摘要
Faltings证明，在数字$ k $上有有限的Abelian $ g $属，在有限的Primes $ s $之外进行了很好的减少。修复了其中一种Abelian品种$ a $，我们证明，$ a $中有有限的平滑性突出表面，在$ s $之外有很好的减少，代表了néron-Severi $ a $ a $ a $的特定级别，只要$ 4 $ $ 4 $ 4 $ $ 4 $。我们的方法建立在Arxiv：1807.02721的方法基础上，研究$ P $ -ADIC结构的变化，以将$ P $ -ADIC GALOIS代表的有限结果转变为几何有限符。一种关键的新成分是一种通过使用坦纳基人的阿贝里亚品种卷积理论来证明由这些超浮标的中间共同体产生的霍奇结构变化的大型单构方法的方法。
Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many smooth hypersurfaces in $A$, with good reduction outside $S$, representing a given ample class in the Néron-Severi group of $A$, up to translation, as long as the dimension of $A$ is at least $4$. Our approach builds on the approach of arXiv:1807.02721 which studies $p$-adic variations of Hodge structure to turn finiteness results for $p$-adic Galois representations into geometric finiteness statements. A key new ingredient is an approach to proving big monodromy for the variations of Hodge structure arising from the middle cohomology of these hypersurfaces using the Tannakian theory of sheaf convolution on abelian varieties.
