Potential Automorphy and the Leopoldt conjecture

By Chandrashekhar B. Khare, J. Thorne. 2014

53 citations recorded by the source corpus.

We study in this paper Hida's $p$-adic Hecke algebra for ${\rm GL}_n$ over a CM field $F$. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his conjecture in the case $F={\Bbb Q}$ implies the classical Leopoldt conjecture for a number field $K$ of degree $n$ over ${\Bbb Q}$, if one assumes further the existence of automorphic induction of characters for the extension $K/{\Bbb Q}$. We study Hida's conjecture using the automorphy lifting techniques adapted to the ${\rm GL}_n$ setting by Calegari-Geraghty. We prove an automorphy lifting result in this setting, conditional on existence and local-global compatibility of Galois representations arising from torsion classes in the cohomology of the corresponding symmetric manifolds. Under the same conditions we show that one can deduce the classical (abelian) Leopoldt conjectures for a totally real number field $K$ and a prime $p$ using Hida's non-abelian Leopoldt conjecture for $p$-adic Hecke algebra for ${\rm GL}_n$ over CM fields without needing to assume automorphic induction of characters for the extension $K/{\Bbb Q}$. For this methods of potential automorphy results are used.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e4904dacd1f7bdbc49235672812accb56c4b573

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In Algebra and Number Theory

1. T. Gee
2. Frank Calegari
3. Akshay Venkatesh
4. J. Thorne
5. P. Allen
6. James Newton
7. Søren Galatius
8. Ana Caraiani

In Geometry and Topology

1. T. Gee
2. J. Thorne
3. Peter Scholze
4. Frank Calegari
5. Ana Caraiani
6. James Newton
7. Akshay Venkatesh
8. P. Allen

In Mathematical Physics

1. T. Gee
2. J. Thorne
3. Peter Scholze
4. Frank Calegari
5. Ana Caraiani
6. James Newton
7. Akshay Venkatesh
8. P. Allen

Citations in the corpus

• Potential automorphy over CM fields (2018, methodology): “Rather, we sidestep this difficulty by a new technical innovation; a derived version of “Ihara avoidance” which simultaneously generalizes the main idea of [Tay08] as well as a localization in characteristic zero idea first used in [KT14].”
• Derived Galois deformation rings (2016, methodology): “sor product, see [28, Theorem 6]). 13. BACKGROUND ON THE OBSTRUCTED TAYLOR–WILES METHOD, AFTER CALEGARI–GERAGHTY We present the “obstructed” Taylor–Wiles method, in the form given by Khare and Thorne [20]; it is originally discovered by Calegari–Geraghty [6], and another version has been developed b”