论文标题
Mod 2 Galois表示的自动形式与Quintic Dwork家族相关的和一些五重的三项式的互惠
Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials
论文作者
论文摘要
In this paper, we determine mod $2$ Galois representations $\barρ_{ψ,2}:G_K:={\rm Gal}(\bar{K}/K)\longrightarrow {\rm GSp}_4(\mathbb{F}_2)$ associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic family $$ x^5_0+x^5_1+x^5_2+x^5_3+x^5_4-5ψx_0x_1x_1x_2x_3x_3x_4= 0，\ψ\在k $ k $ k $下定义在quintic trinomial $f_ψ$下方的数字字段$ k $下定义。应用此结果，当$ k = f $是一个完全真实的领域时，对于某些qaudratic完全真实的扩展$ m/f $，我们证明$ \barρ_{ψ，2} | _ {g_m} $与Hilbert-Siegel模块化的Hecke hecke eigen cusp形式相关平行重量三。 在证据的过程中，我们观察到这种mod $ 2 $表示的图像受Quintic Trinorial $$f_ψ（x）= 4x^5-5ψx^4+1，\ψ\ in k $ 4的互惠控制，其分解字段通常是5- type 5- the immetric Group $ s_5 $。这使我们能够使用$ {\ rm gal}的二维，完全奇怪的Artin表示的模块化（\ bar {f}/f）$，这是由于Shu Sasaki和Hilbert cusp表格的几个Langlands函数升降机。然后，它可以保证存在所需的Hilbert-Siegel模块化模块化牙曲质形式的平行重量三匹配与所讨论的兼容系统的Hodge类型匹配。还讨论了扭曲的版本，并且与一般的五次三项官员有关。
In this paper, we determine mod $2$ Galois representations $\barρ_{ψ,2}:G_K:={\rm Gal}(\bar{K}/K)\longrightarrow {\rm GSp}_4(\mathbb{F}_2)$ associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic family $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4-5ψX_0X_1X_2X_3X_4=0,\ ψ\in K$$ defined over a number field $K$ under the irreducibility condition of the quintic trinomial $f_ψ$ below. Applying this result, when $K=F$ is a totally real field, for some at most qaudratic totally real extension $M/F$, we prove that $\barρ_{ψ,2}|_{G_M}$ is associated to a Hilbert-Siegel modular Hecke eigen cusp form for ${\rm GSp}_4(\mathbb{A}_M)$ of parallel weight three. In the course of the proof, we observe that the image of such a mod $2$ representation is governed by reciprocity of the quintic trinomial $$f_ψ(x)=4x^5-5ψx^4+1,\ ψ\in K$$ whose decomposition field is generically of type 5-th symmetric group $S_5$. This enable us to use results on the modularity of 2-dimensional, totally odd Artin representations of ${\rm Gal}(\bar{F}/F)$ due to Shu Sasaki and several Langlands functorial lifts for Hilbert cusp forms. Then, it guarantees the existence of a desired Hilbert-Siegel modular cusp form of parallel weight three matching with the Hodge type of the compatible system in question.A twisted version is also discussed and it is related to general quintic trinomials.
