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Feb 9 at 16:32 comment added Wojowu Thank you David, Hida's work does indeed make sense as a sort of bridge between the two ideas. Would you happen to know what Wiles meant when he talks about "Hecke rings for the varying fields in a cyclotomic tower"? Perhaps this just refers to the towers of varying level $Np^r$ but I'm not sure in what way they're related to cyclotomic towers.
Feb 9 at 8:35 history answered David Loeffler CC BY-SA 4.0