论文标题
mod- $ $ p $ galois表示，不是由阿伯利亚品种引起的
Mod-$p$ Galois representations not arising from abelian varieties
论文作者
论文摘要
众所周知，任何Galois表示$ρ：g _ {\ Mathbb {Q}} \ rightArrow \ Mathrm {gl}（2，2，\ Mathbb {f} _p）$，确定性等于mod- $ $ p $ cyclotomic特征，从$ p $ cyclotomic the $ p $ p $ - ifipt $ -tors $ ifipt $ ellipt curveve，如果$ p \ leq 5 $。在尺寸$ g = 2 $中，当$ p \ le 3 $时，众所周知，在$ \ mathrm {gsp}中价值的任何galois表示形式（4，\ m athbb {f} _p）$带有环体齐友的字符。在本文中，我们研究了所有Primes $ p $和dimensions $ g \ ge 2 $的问题。当$ g \ ge 2 $和$（g，p）\ neq（2,2）$，$（2,3）$，$（3,2）$时，我们证明在$ \ mathbb {q} $中存在$ \ mathbb {q} $ ym $ \ sim v，gsp}（gsp}（2g，\ mathbbbb {f sim sim and and anclial can y an Cy an Cy anclial can ar can ar can ar can）任何$ g $ dimensional Abelian品种的$ p $ torsion表示，超过$ \ mathbb {q} $。
It is known that any Galois representation $ρ: G_{\mathbb{Q}} \rightarrow \mathrm{GL}(2,\mathbb{F}_p)$ with determinant equal to the mod-$p$ cyclotomic character, arises from the $p$-torsion of an elliptic curve over $\mathbb{Q}$, if and only if $p \leq 5$. In dimension $g = 2$, when $p \le 3$, it is again known that any Galois representation valued in $\mathrm{GSp}(4,\mathbb{F}_p)$ with cyclotomic similitude character arises from an abelian surface. In this paper, we study this question for all primes $p$ and dimensions $g \ge 2$. When $g \ge 2$ and $(g,p) \neq (2,2)$, $(2,3)$, $(3,2)$, we prove the existence of a Galois representation over $\mathbb{Q}$ valued in $\mathrm{GSp}(2g,\mathbb{F}_p)$ with cyclotomic similitude character, that cannot arise as the $p$-torsion representation of any $g$-dimensional abelian variety over $\mathbb{Q}$.
