Lifting and automorphy of reducible mod p Galois representations over global fields
Lifting and automorphy of reducible mod p Galois representations over global fields
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DOI：
10.1007/s00222-021-01085-7
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发表时间：
2020-08
影响因子：
3.1
通讯作者：
N. Fakhruddin;Chandrashekhar B. Khare;Stefan Patrikis
中科院分区：
文献类型：
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作者：
N. Fakhruddin;Chandrashekhar B. Khare;Stefan Patrikis
我们证明了具有特征为奇素数的有限域的最可约奇表示的模性。这是一个类似的塞尔著名的模块性猜想（其中涉及不可约，奇数表示）的可约，奇数表示。我们的证明提升到一个不可约的几何p-adic表示，这是已知的，从一个新的形式由Skinner-Wiles和潘的结果。当F是一个特征不同于m的整体函数域时，通过建立近似进提升定理并引用L. Lafforgue的工作，我们同样证明了许多可约表示的自同构性.至关重要的是，在这两种情况下，我们表明，实际的表示，而不仅仅是它的半简化，产生于减少的几何表示附加到尖形自守表示。我们的主要定理建立了整体域F的Galois群的模表示的几何提升结果，F的值在约化群G（k）中，并且当F是数域时，假设F是奇数。因此，我们发现，提升定理，结合自同构提升结果开创的怀尔斯在数域的情况下，并在全球Langlands对应证明的Drinfeld和L. Lafforgue在函数域的情况下，给出了唯一已知的方法来访问modpGalois表示的模在可约和不可约的情况下。
We prove the modularity of most reducible, odd representationswithka finite field of characteristic an odd primep. This is an analogue of Serre’s celebrated modularity conjecture (which concerned irreducible, odd representations) for reducible, odd representations. Our proof liftsto an irreducible geometricp-adic representationwhich is known to arise from a newform by results of Skinner–Wiles and Pan. We likewise prove automorphy of many reducible representationswhenFis a global function field of characteristic different fromp, by establishing ap-adic lifting theorem and invoking the work of L. Lafforgue. Crucially, in both cases we show that the actual representation, rather than just its semisimplification, arises from reduction of the geometric representation attached to a cuspidal automorphic representation. Our main theorem establishes a geometric lifting result for modprepresentationsof Galois groups of global fieldsF, valued in reductive groupsG(k), and assumed to be odd whenFis a number field. Thus we find that lifting theorems, combined with automorphy lifting results pioneered by Wiles in the number field case and the results in the global Langlands correspondence proved by Drinfeld and L. Lafforgue in the function field case, give the only known method to access modularity of modpGalois representations both in reducible and irreducible cases.