Nonvanishing of generalised Kato classes and Iwasawa main conjectures

2023/12/03 by Castella, Francesc
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2312.01481

Abstract

A construction due to Darmon--Rotger gives rise to generalised Kato classes κ_p(E) in the p-adic Selmer group \rm Sel(Q,V_pE) of elliptic curves E/Q of positive even analytic rank, where p>3 is any prime of good ordinary reduction for E. In some cases, their conjectured that κ_p(E)≠ 0 precisely when \rm Sel(Q,V_pE) is two-dimensional. The first cases of this conjecture were obtained in a joint work of the author with M.-L. Hsieh. In this note we give a new proof of the implication κ_p(E)≠ 0 \Longrightarrow \rm dim_Qp\rm Sel(Q,V_pE)=2 established in op. cit., and show that the converse implication holds if and only if the restriction map \rm loc_p:\rm Sel(Q,V_pE)→ E(Q_p)\otimesQ_p is nonzero. The present approach is an adaptation to the non-CM case of the method introduced by the author in the case of CM elliptic curves.

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