论文标题
从等级2 GALOIS代表构建Abelian品种
Constructing abelian varieties from rank 2 Galois representations
论文作者
论文摘要
Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \ bar {\ mathbb q} _ \ ell $，并且在某个关闭点的$ x $ $ x \ setminus u $上有糟糕的无限减少。我们表明，$ \ Mathbb l $作为$ U $ $ U $的Abelian品种家族的共同体的总结。该论点遵循了雪登-tsimerman最近定理的证明的结构，后者表明当$ e = \ mathbb q $时，$ \ mathbb l $是对椭圆曲线$ e_u \ rightarrow u $的共同体的同构。
Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \bar{\mathbb Q}_\ell$, and has bad, infinite reduction at some closed point $x$ of $X\setminus U$. We show that $\mathbb L$ occurs as a summand of the cohomology of a family of abelian varieties over $U$. The argument follows the structure of the proof of a recent theorem of Snowden-Tsimerman, who show that when $E=\mathbb Q$, then $\mathbb L$ is isomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$.
