Abstract Let K be a totally real field and G K : = Gal ( K ‾ / K ) its absolute Galois group, where K ‾ is a fixed algebraic closure of K. Let l be a prime and E a finite extension of Q l . Let S be a finite set of finite places of K not dividing l. Assume that K, S, Hodge-Tate type h and a positive integer n are fixed. In this paper, we prove that if l is sufficiently large, then, for any fixed E, there are only finitely many isomorphism classes of crystalline representations r : G K → GL n ( E ) unramified outside S ∪ { v : v | l } , with fixed Hodge-Tate type h, such that r | G K ′ ≃ ⊕ r i ′ for some finite totally real field extension K ′ of K unramified at all places of K over l, where each representation r i ′ over E is an 1-dimensional representation of G K ′ or a totally odd irreducible 2-dimensional representation of G K ′ with distinct Hodge-Tate numbers.

中文翻译：

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全实域的绝对伽罗瓦群的晶体表示的有限性

摘要 令 K 为全实域，GK : = Gal ( K ‾ / K ) 为绝对伽罗瓦群，其中 K ‾ 是 K 的固定代数闭包。令 l 为素数，E 为 Q l 的有限外延。令 S 是 K 的有限位的有限集，不除 l。假设 K、S、Hodge-Tate 类型 h 和一个正整数 n 是固定的。在本文中，我们证明如果 l 足够大，那么对于任何固定的 E，只有有限多个晶体表示的同构类 r : GK → GL n ( E ) 在 S ∪ { v : v | 之外未分枝 l } ，具有固定的 Hodge-Tate 类型 h，使得 r | GK ′ ≃ ⊕ ri ′ 对于一些有限的全实域扩展 K ′ ， K ′ 在 K 上的所有地方都没有分枝，