49 research outputs found

    CM elliptic curves of rank 2 and nonvanishing of generalised Kato classes

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    Let E/QE_{/\mathbf{Q}} be a CM elliptic curve and p>3p>3 a prime of good ordinary reduction for EE. Assume L(E,1)=0L(E,1)=0 with sign +1+1 (so ords=1L(E,s)≥2{\rm ord}_{s=1}L(E,s)\geq 2) and EE has a rational point of infinite order. We prove that if Sel(Q,VpE){\rm Sel}(\mathbf{Q},V_pE) is 22-dimensional, then a certain generalised Kato class κp∈Sel(Q,VpE)\kappa_p\in{\rm Sel}(\mathbf{Q},V_pE) is nonzero; and conversely, the nonvanishing of κp\kappa_p implies that Sel(Q,VpE){\rm Sel}(\mathbf{Q},V_pE) is 22-dimensional. For non-CM elliptic curves, a similar result was proved in a joint work with M.-L. Hsieh. The proof in this paper is completely different, and should extend to other settings. Moreover, combined with work of Rubin, we can exhibit explicit bases for 22-dimensional Sel(Q,VpE){\rm Sel}(\mathbf{Q},V_pE).Comment: 20 pages, comments welcom

    Variation of anticyclotomic Iwasawa invariants in Hida families

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    Building on the construction of big Heegner points in the quaternionic setting, and their relation to special values of Rankin-Selberg LL-functions, we obtain anticyclotomic analogues of the results of Emerton-Pollack-Weston on the variation of Iwasawa invariants in Hida families. In particular, combined with the known cases of the anticyclotomic Iwasawa main conjecture in weight 22, our results yield a proof of the main conjecture for pp-ordinary newforms of higher weights and trivial nebentypus.Comment: Essentially final version, to appear in Algebra & Number Theor
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