저작근 근전도에 관한 정상교합자와 ii급 부정교합자의 비교 연구

13 EADERS of Mr. Bertrand Russell's philosophical £v works know that one of his pupils before the outbreak of the war, an Austrian, Mr. Ludwig Wittgenstein, caused him to change his views in some important particulars. Curiosity can now be satisfied. The “Tractatus Logico-Philosophicus “which Mr. Ogden has included in his new library of philosophy is a remarkable and strikingly original work. It is published in German and English in parallel pages. It is difficult to appreciate the reason for this, seeing that the author is evidently familiar with our language and has himself carefully revised the proofs of the translation. Also we should have liked to have the Tractatus without Mr. Russell's Introduction, not, we hasten to add, on account of any fault or shortcoming in that introduction, which is highly appreciative and in part a defence of himself, in part explanatory of the author, but for the reason that good wine needs no bush and that Mr. Russell's bush has the unfortunate effect of dulling the palate instead of whetting the appetite. In his penultimate sentence Mr. Russell says; “To have constructed a theory of logic which is not at any point obviously wrong is to have achieved a work of extraordinary difficulty and importance.” We agree, but how uninspiring when compared with Mr. Wittgenstein's own statement of aim: “What can be said at all can be said clearly, and whereof one cannot speak, thereof one must be silent.” Tractatus Logico-Philosophicus. By Ludwig Wittgenstein. (International Library of Psychology, Philosophy and Scientific Method.) Pp. 189. (London: Kegan Paul and Co., Ltd.; New York: Harcourt, Brace and Co., Inc., 1922.) 10s. 6d. net.

Tractatus Logico-Philosophicus

We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.

/pdf/automorphy-for-some-l-adic-lifts-of-automorphic-mod-l-galois-1lron6w183.pdf

Automorphy for some l-adic lifts of automorphic mod l Galois representations

We prove potential modularity theorems for l-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral j -invariant defined over a totally real field.

https://www.math.ias.edu/~rtaylor/cy2fin.pdf

A Family of Calabi–Yau Varieties and Potential Automorphy II

Modular elliptic curves and fermat's last theorem

We prove an automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call \potentential diagonalizability." This result allows for \change of weight" and seems to be substantially more exible than previous theorems along the same lines. We derive several applications. For instance we show that any irreducible, totally odd, essentially self-dual, regular, weakly compatible system of l-adic representations of the absolute Galois group of a totally real eld is potentially automorphic and hence is pure and its L-function has meromorphic continuation to the whole complex plane and satises the expected functional equation.

/pdf/potential-automorphy-and-change-of-weight-2y91dhutev.pdf

Potential automorphy and change of weight

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of GL2(AF ), F a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et. al., but makes use of automorphy lifting theorems over ramified fields, together with a “topological” argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of potential automorphy theorems for non-ordinary n-dimensional Galois representations.

http://wwwf.imperial.ac.uk/~tsg/Index_files/satotatetotallyreal.pdf

The Sato-Tate conjecture for Hilbert modular forms

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor–Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions—one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on the automorphic side, applies to automorphic forms on the group $$\mathrm {GL}(n)$$
 over a general number field; it is contingent on a conjecture which, in particular, predicts the existence of Galois representations associated to torsion classes in the cohomology of the associated locally symmetric space. We show that if this conjecture holds, then our main theorem implies the following: if E is an elliptic curve over an arbitrary number field, then E is potentially automorphic and satisfies the Sato–Tate conjecture. In addition, we also prove some unconditional results. For example, in the setting of $$\mathrm {GL}(2)$$
 over $$\mathbf {Q}$$
 , we identify certain minimal global deformation rings with the Hecke algebras acting on spaces of p-adic Katz modular forms of weight 1. Such algebras may well contain p-torsion. Moreover, we also completely solve the problem (for p odd) of determining the multiplicity of an irreducible modular representation $$\overline{\rho }$$
 in the Jacobian $$J_1(N)$$
 , where N is the minimal level such that $$\overline{\rho }$$
 arises in weight two.

/pdf/modularity-lifting-beyond-the-taylor-wiles-method-1az4m3dfxu.pdf

Modularity lifting beyond the Taylor–Wiles method

We use the patching method of Taylor–Wiles and Kisin to construct a candidate for the p-adic local Langlands correspondence for GLn(F ), F a finite extension of Qp. We use our construction to prove many new cases of the Breuil–Schneider conjecture.

http://math.uchicago.edu/~emerton/pdffiles/Gpatching.pdf

David Geraghty

Papers

A Family of Calabi–Yau Varieties and Potential Automorphy II

Potential automorphy and change of weight

The Sato-Tate conjecture for Hilbert modular forms

Modularity lifting beyond the Taylor–Wiles method

Patching and the p-adic local Langlands correspondence