论文标题
Abelian表面多峰曲折的局部全球原理
A local-global principle for polyquadratic twists of abelian surfaces
论文作者
论文摘要
我们说，如果在field $ f $上定义的两个Abelian品种$ a $ a $ a'$是多quadratic twist，如果它们在$ f $的galois延长$ f $上是同一性的，他们的galois集团的指数为$ 2 $。令$ a $和$ a'$为ABELIAN品种，定义在数字$ k $ dimension $ g \ geq 1 $上。在本文中，我们证明，如果$ g \ \ \ leq 2 $，则$ a $和$ a'$是polyquadratic twists，并且仅当几乎所有的primes $ \ p $ $ k $ $ k $ of $ k $的降低模式$ \ p $ p $ as polyquadratic Twists。我们以$ g = 3 $的形式展示了该本地全球原则的反例。这项工作以Khare和Larsen的几何类似物为基础，并以Fité建立的二次曲折标准为基础，依靠Rajan和Ramakrishnan的作品。
We say that two abelian varieties $A$ and $A'$ defined over a field $F$ are polyquadratic twists if they are isogenous over a Galois extension of $F$ whose Galois group has exponent dividing $2$. Let $A$ and $A'$ be abelian varieties defined over a number field $K$ of dimension $g\geq 1$. In this article we prove that, if $g\leq 2$, then $A$ and $A'$ are polyquadratic twists if and only if for almost all primes $\p$ of $K$ their reductions modulo $\p$ are polyquadratic twists. We exhibit a counterexample to this local-global principle for $g=3$. This work builds on a geometric analogue by Khare and Larsen, and on a similar criterion for quadratic twists established by Fité, relying itself on the works by Rajan and Ramakrishnan.
