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Title:听The Wiles-Lenstra-Diamond numerical criterion in higher codimensions.

础产蝉迟谤补肠迟:听I will report on recent joint work with Srikanth Iyengar and Jeff Manning. We give a development of numerical criterion that was used by Wiles as an essential ingredient in his approach to modularity of elliptic curves over $Q$. The patching method introduced by Wiles and Taylor has been developed considerably while the numerical criterion has lagged behind. We prove new commutative algebra results that lead to a generalisation of the Wiles-Lenstra-Diamond numerical criterion in situations of positive defect (as arise when proving modularity of elliptic curves over number fields with a complex place). A key step in our work is the definition of congruence modules in higher codimensions which should be relevant to studying eigenvarieties at classical points.

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