REVIEW
CM Elliptic Analogue of the Coleman--Ihara Formula
Number Theory · Submitted 15 Oct 2026 · 43 references
Pith's verdict: Solid two-variable generalization of the Coleman–Ihara formula to CM elliptic curves; technically careful and correct within its assumptions.
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Referee report
The paper
The paper claims
Proof of the two-variable CM elliptic Coleman–Ihara formula: each elliptic Soulé character restricts at p to a two-variable Coates–Wiles map times a two-variable p-adic L-value divided by a power of the p-adic period.
The reading
What carries the argument
The argument runs on three coordinated pieces. First, the objects: the elliptic Soulé characters of Definition 3.6 are the Kummer characters of the elliptic units $\epsilon_{(i,j),n}$ built from the fundamental $\theta$ function $\theta(z;L)$ evaluated at $p$-power torsion points, and by Nakamura's construction the same characters describe the Galois action on the maximal metabelian quotient of the pro-$p$ geometric fundamental group of the once-punctured curve. Second, the bridge between the cyclotomic and elliptic worlds: the formal-group isomorphism $\eta:\widehat{\mathbb{G}}_m\to\widehat{E}$, normalized by $\omega_n=\eta^{\varphi^{-n}}(\zeta_{p^n}-1)$, together with Proposition 5.2, which shows that the cyclotomic Coleman power series $f_\epsilon$ and the elliptic Coleman power series $g_\beta\circ\eta$ agree modulo $(1+S)^{p^m}-1$; this is what allows Coleman's explicit reciprocity law (Theorem 4.6, in the semilocal form of [NSW17a]) to be applied to elliptic units. Third, the translation into $L$-values: Lemma 4.10 and Definition 4.12 extract the two-variable Coates–Wiles homomorphism $\varphi^{\mathrm{CW}}_{(i,j)}$ (after Yager) from logarithmic derivatives of elliptic Coleman power series, while Proposition 4.17 identifies the measure attached to the elliptic-unit system $\theta(a)$ with de Shalit's measure $\mu_{dS}$ that defines the two-variable $p$-adic $L$-function. The whole computation is packaged in Theorem 5.4 as the evaluation of a single Hilbert norm residue symbol in two ways.
Extended reading notes
Core claim
The central claim is Theorem 1.1 (Corollary 5.5). For $K$ an imaginary quadratic field of class number one, $E$ an elliptic curve with complex multiplication by $\mathcal{O}_K$, and $p\ge 5$ a prime that splits as $(p)=\mathfrak{p}\bar{\mathfrak{p}}$ with $E$ of good reduction at $p$, the elliptic Soulé character $\chi^{\mathrm{ell}}_{(i,j)}$ is the Kummer character attached to the products $\epsilon_{(i,j),n}=\prod_{0\le a,b<p^n,\ p\nmid\gcd(a,b)}\theta(a\omega_{p,n}+b\omega_{\bar{p},n};L)^{a^{i-1}b^{j-1}}$ of fundamental $\theta$ values at $p$-power torsion points. The theorem states that for every pair of positive integers $(i,j)\neq(1,1)$ with $i\equiv j\pmod{w_K}$ and every norm-compatible system $\beta\in U^{\mathrm{ell}}_\infty$, the identity $$\frac{\$chi^{{\mathrm{ell}}$}_{(i,j)}(\mathrm{rec}_\infty(\$\beta$))}{1-\$pi^{{j-1}}$\bar{\pi}^{i-1}} = \frac{1}{2w_K}\left(1-\frac{1}{1-\$pi^{{i-1}}$\bar{\pi}^{j-1}}-\frac{1}{1-\$pi^{{j-1}}$\bar{\pi}^{i-1}}\right)\cdot\frac{L_p(1-i,1-j,\omega_{(i,j)}^{-1})}{\$Omega_p^{{j-i}}$}\,\$varphi^{{\mathrm{CW}}$}_{(i,j)}(\$\beta$)$$ holds, where $\varphi^{\mathrm{CW}}_{(i,j)}$ is the two-variable Coates–Wiles homomorphism introduced by Yager, $L_p(s_1,s_2,\chi)$ is de Shalit's two-variable $p$-adic $L$-function, and $\Omega_p$ is the $p$-adic period fixed by the normalization (9). In words: the local restriction of each elliptic Soulé character is exactly the product of the $L$-value, the Coates–Wiles map, and two Euler-factor-like corrections built from $\pi$ and $\bar{\pi}$. The proof computes one Hilbert norm residue symbol in two ways — once through the Kummer property of elliptic units, once through Coleman's explicit reciprocity law — and identifies the Coleman-power-series expression with the $L$-function through a measure comparison.
Load-bearing premise
The load-bearing premise is that $K$ has class number one, so that complex multiplication supplies a single p-adic period $\Omega_p$ and a formal-group isomorphism $\eta$ with the normalization $\omega_n=\eta^{\varphi^{-n}}(\zeta_{p^n}-1)$; if the class number were larger than one, the comparison of Coleman power series and the normalization of the two-variable $L$-function would have to be redone class by class, and the paper also assumes $p\ge 5$ splits in $K$ and that $E$ has good reduction at $p$.
Editorial extensions
If this is right
- The local restriction of every elliptic Soulé character $\chi^{\mathrm{ell}}_{(i,j)}$ with $i\equiv j\pmod{w_K}$ is completely determined by the two-variable Coates–Wiles homomorphism and the two-variable $p$-adic $L$-function, with no further hidden invariants.
- On the cyclotomic line $i=j$, the two-variable formula specializes to the earlier one-variable formula of [IG26], after the paper's correction to Gross's factorization formula (the extra factor $\chi(\sqrt{-d_K})$); Remark 5.6 records the corrected identities and verifies the agreement.
- Because the elliptic Soulé characters also arise from the Galois action on the pro-$p$ geometric fundamental group of the once-punctured elliptic curve, the formula describes that geometric Galois action locally in $L$-function terms.
- The corrections stated in Remarks 5.6–5.7 — to Gross's factorization formula and to the proof of the functional equation in de Shalit's treatise — imply that the corresponding formulas in those references acquire the additional factors recorded here.
Reading between the lines
- My inference: the two-variable family $\chi^{\mathrm{ell}}_{(i,j)}$ with $i\equiv j\pmod{w_K}$ should interpolate as $(i,j)$ varies into an Iwasawa-theoretic object governed by the full Katz–de Shalit two-variable $L$-function; a natural test is whether the family satisfies a two-variable functional equation under $(i,j)\mapsto(2-i,2-j)$ mirroring Katz's one-variable identity (17).
- My inference: the class-number-one hypothesis looks removable at the cost of attaching a period and a formal-group isomorphism to each ideal class, and the shape of the formula then predicts how $\Omega_p$ and the $\pi,\bar{\pi}$ factors change as the class varies.
- My inference: because Nakamura's Kummer characters are defined for arbitrary elliptic curves without complex multiplication, a natural question is how much of the formula survives without CM, where the $L$-function factor would have to be replaced by a conjectural analogue.
- My inference: the paper's note that Sano and Venjakob independently obtained the same result suggests the identity is now the canonical bridge between Yager's Coates–Wiles formalism and de Shalit's measure formalism; comparing the two proofs could pin down which normalizations are forced by the $L$-function interpolation.
Original abstract
The Soul\'{e} characters are Kummer characters constructed from cyclotomic units and also arising from the Galois action on the pro-$p$ geometric fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty\}$. The Coleman--Ihara formula describes the restrictions of these characters to the local Galois group at $p$ as products of the Coates--Wiles homomorphisms and Kubota--Leopoldt $p$-adic $L$-values at positive odd integers. In this paper, we prove an elliptic analogue of this formula for elliptic curves with complex multiplication by the ring of integers of an imaginary quadratic field $K$. We consider certain Kummer characters associated with elliptic units, which we call elliptic Soul\'{e} characters. These characters are elliptic counterparts of the classical Soul\'{e} characters and also arise from the Galois action on the pro-$p$ geometric fundamental groups of once-punctured elliptic curves. For suitable pairs of positive integers $(i,j)$, we give an explicit description of their restrictions to the relevant local Galois groups as products of the two-variable Coates--Wiles homomorphisms introduced by Yager and the two-variable $p$-adic $L$-values of $K$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The Soul\'{e} characters are Kummer characters constructed from cyclotomic units and also arising from the Galois action on the pro-$p$ geometric fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty\}$. The Coleman--Ihara formula describes the restrictions of these characters to the local Galois group at $p$ as products of the Coates--Wiles homomorphisms and Kubota--Leopoldt $p$-adic $L$-values at positive odd integers. In this paper, we prove an elliptic analogue of this formula for elliptic curves with complex multiplication by the ring of integers of an imaginary quadratic field $K$. We consider certain Kummer characters associated with elliptic units, which we call elliptic Soul\'{e} characters. These characters are elliptic counterparts of the classical Soul\'{e} characters and also arise from the Galois action on the pro-$p$ geometric fundamental groups of once-punctured elliptic curves. For suitable pairs of positive integers $(i,j)$, we give an explicit description of their restrictions to the relevant local Galois groups as products of the two-variable Coates--Wiles homomorphisms introduced by Yager and the two-variable $p$-adic $L$-values of $K$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Soul\'{e} characters are Kummer characters constructed from cyclotomic units and also arising from the Galois action on the pro-$p$ geometric fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty\}$. | background 1.00 | 0.06 | 0.03 | asserted 1.00 |
| s1 | The Coleman--Ihara formula describes the restrictions of these characters to the local Galois group at $p$ as products of the Coates--Wiles homomorphisms and Kubota--Leopoldt $p$-adic $L$-values at positive odd integers. | background 1.00 | 0.04 | 0.04 | asserted 1.00 |
| s2 | In this paper, we prove an elliptic analogue of this formula for elliptic curves with complex multiplication by the ring of integers of an imaginary quadratic field $K$. | contribution 0.89 | 0.95 | 0.80 | asserted 1.00 |
| s3 | We consider certain Kummer characters associated with elliptic units, which we call elliptic Soul\'{e} characters. | contribution 0.98 | 0.78 | 0.04 | asserted 0.99 |
| s4 | These characters are elliptic counterparts of the classical Soul\'{e} characters and also arise from the Galois action on the pro-$p$ geometric fundamental groups of once-punctured elliptic curves. | contribution 0.89 | 0.41 | 0.04 | asserted 1.00 |
| s5 | For suitable pairs of positive integers $(i,j)$, we give an explicit description of their restrictions to the relevant local Galois groups as products of the two-variable Coates--Wiles homomorphisms introduced by Yager and the two-variable $p$-adic $L$-values of $K$. | result 0.88 | 0.94 | 0.78 | asserted 1.00 |
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References
Discussion
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