On factorizations of certain Kummer characters associated to once-punctured elliptic curves with complex multiplication

In this paper, we study certain Kummer characters, which we call the elliptic Soulé characters, arising from Galois actions on the pro-$p$ fundamental groups of once-punctured elliptic curves with complex multiplication. In particular, we prove that elliptic Soulé characters having values in Tate twists can be written in terms of the Soulé characters and generalized Bernoulli numbers. We apply this result to give a criterion for surjectivity of the elliptic Soulé characters and an analogue of the Coleman-Ihara formula.

💡 Research Summary

The paper investigates a family of Kummer characters that arise from the Galois action on the pro‑p geometric fundamental group of a once‑punctured elliptic curve with complex multiplication (CM). The author calls these characters “elliptic Soulé characters”. The setting is an imaginary quadratic field K of class number one, a CM elliptic curve E/K, and a prime p ≥ 5 that splits in K as (p)=𝔭·\bar{𝔭}. The p‑adic Tate modules TₚE and T_{\bar{𝔭}}E give rise to two one‑dimensional continuous Galois characters ε_{𝔭} and ε_{\bar{𝔭}}; for a pair of non‑negative integers m=(m₁,m₂) the combined character is ε_m=ε_{𝔭}^{m₁}ε_{\bar{𝔭}}^{m₂}. The special case (1,1) coincides with the usual p‑adic cyclotomic character ε_cyc.

A key ingredient is the fundamental theta function θ(z,L) attached to the lattice L=Ω·O_K that uniformises E(ℂ). For each n≥1 the pⁿ‑division points ω_{𝔭,n}=Ω/pⁿ and ω_{\bar{𝔭},n}=Ω/pⁿ give an O_K‑basis of E