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  1. ; ; (, Journal of the European Mathematical Society)
    In the late 1990s, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those p-ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at p. It is expected that such p-ordinary eigenforms are precisely those with complex multiplication. In this paper, we study Coleman-Greenberg's question using Galois deformation theory. In particular, for p-ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the p-indivisibility of a certain class group. 
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  2. (, Journal of the Institute of Mathematics of Jussieu)
    In this paper, we prove an ‘explicit reciprocity law’ relating Howard’s system of big Heegner points to a two-variable p-adic L-function (constructed here) interpolating the p-adic Rankin L-series of Bertolini–Darmon–Prasanna in Hida families. As applications, we obtain a direct relation between classical Heegner cycles and the higher weight specializations of big Heegner points, refining earlier work of the author, and prove the vanishing of Selmer groups of CM elliptic curves twisted by 2-dimensional Artin representations in cases predicted by the equivariant Birch and Swinnerton-Dyer conjecture. 
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