What question did this study set out to answer?

The study aims to provide a sufficient condition for extensions of Qp that yield non-empty principal homogeneous spaces.

June 19, 2026

On the trace map of Lubin-Tate formal groups and a result of Lang-Tate

Key Points

The study aims to provide a sufficient condition for extensions of Qp that yield non-empty principal homogeneous spaces.
Analyzed the trace map of Lubin-Tate formal groups in relation to p-adic fields.
Established conditions under which a totally ramified abelian extension splits a principal homogeneous space.
Investigation focused on degree p extensions when n=1, linking valuation and degree.
Showed that a sufficiently large totally ramified abelian extension F/Qp splits the principal homogeneous space C.
For n=1, identified that F splits C if and only if vF(DF/Qp)=2p-1.

Abstract

Let p be a prime number, E be an elliptic curve over Qₚ with good supersingular reduction, and C be a principal homogeneous space of E/Qₚ with period pⁿ. In this paper we give a sufficient condition for extensions F/Qₚ so that C (F). In particular, we show that a totally ramified abelian extension F/Qₚ splits C if F: Qₚ is sufficiently large. Moreover, in case n=1, we show that a degree p extension F/Qₚ splits C if and only if vF (D₅/ₐ䂹) =2p-1. This result is an analogue and also a complement to a result of Lang-Tate on splitting fields of principal homogeneous spaces of abelian varieties.

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