Explicit reciprocity laws for formal Drinfeld modules without the congruence hypothesis, and their extension to higher local fields

For a formal Drinfeld module of stable reduction of height one over a local field of characteristic $p$, the Kummer pairing constructed from torsion points admits an explicit reciprocity law: a computable formula for the pairing in terms of the module logarithm, a Coleman-type power series or derivation, and the trace. The published formula of Ala Eddine (2022/2023) requires a congruence hypothesis $\rho_{\eta}\equiv\tau^{m_0}\pmod{\mathfrak{p}_H}$ on the twisted multiplication power series, and a follow-up by Ala Eddine and Omidi Koma removes this hypothesis at the level of a general derivation formula but still evaluates the key derivation at torsion uniformizers only under the same congruence. Two scientifically linked directions remain open: (i) an explicit reciprocity law with fully computable ingredients, in particular an explicit value of the derivation at a torsion uniformizer (equivalently an explicit $\delta_n$-type homomorphism), for modules not satisfying the congruence hypothesis; and (ii) the extension of these explicit reciprocity laws to higher-dimensional local fields in the sense of Flórez, as explicitly posed by the author in the July 2026 revision of the follow-up paper.

Background

Local class field theory describes the abelianized absolute Galois group of a local field through the norm residue (reciprocity) map $\Phi_L:L^{\times}\to\mathrm{Gal}(L^{ab}|L)$. An explicit reciprocity law is a concrete, computable formula for this map — or for a Kummer pairing built from it — in terms of analytic ingredients such as logarithms, power series, torsion points, and the field trace, in the spirit of Hilbert's ninth problem. In characteristic zero, Wiles (1978) proved such laws for general Lubin–Tate formal groups, i.e. one-dimensional commutative formal group laws whose endomorphism ring realizes a local ring of integers.

In characteristic $p$, the analogous objects are Drinfeld modules: a Drinfeld $A$-module over a field $L$ of characteristic $p$ is a ring homomorphism from a coefficient ring $A$ (such as $\mathbb{F}_q[T]$) into the skew ring $L\{\tau\}$ of twisted polynomials, where $\tau$ is the Frobenius $x\mapsto x^q$ and $\tau a = a^q \tau$, with the constant term of the image of $a$ equal to $a$. A formal Drinfeld $\mathcal{O}$-module, as studied here, is the completion-theoretic analogue: a continuous representation $\mathcal{O}\to\mathcal{O}_H\{\{\tau\}\}$ of the ring of integers $\mathcal{O}=\mathcal{O}_K$ of a local field $K$ of characteristic $p$ (with maximal ideal $\mathfrak{p}$ and residue field of order $q$) by twisted formal power series, where $H$ is a finite unramified extension of $K$ with valuation ring $\mathcal{O}_H$ and maximal ideal $\mathfrak{p}_H$, over which the module is defined; unlike the Carlitz module or rank-one Drinfeld modules, the acting series $\rho_a$ are genuine power series, not polynomials. The module has stable reduction of height one when the coefficients of all $\rho_a$ are integral and the reduction is again a Drinfeld module of height one; such modules exist in abundance by work of Rosen. The Carlitz module (the rank-one analogue of $\mathbb{G}_m$) and sign-normalized rank-one Drinfeld modules play the role of Lubin–Tate groups in this setting.

Anglès (1999) proved an explicit reciprocity law for the local Carlitz module, and Bars and Longhi (2009) extended it to formal Drinfeld modules arising from sign-normalized rank-one Drinfeld modules. Ala Eddine (arXiv:2202.02348, published in J. Théor. Nombres Bordeaux 2023) generalized both: for a formal Drinfeld module with stable reduction of height one satisfying the congruence hypothesis $\rho_{\eta}\equiv\tau^{m_0}\pmod{\mathfrak{p}_H}$ (all coefficients of the twisted series $\rho_{\eta}=\sum b_i\tau^i$ lie in the maximal ideal $\mathfrak{p}_H$ except $b_{m_0}$, where $\eta$ is a fixed uniformizer-type element of valuation $m_0$), her Theorem 5.6 expresses the Kummer pairing $(\alpha,\beta)_{\rho,L,n}=\Phi_L(\beta)(\xi)-\xi$, with $\xi$ a root of $\rho_{\eta^n}(X)=\alpha$, as $(\alpha,\beta)_{\rho,L,n}=\frac{1}{\eta^n}T_{L|K}(\lambda_\rho(\alpha)\delta_n(\beta))\cdot_\rho v_n$, where $\lambda_\rho$ is the module logarithm, $T_{L|K}$ the trace, $v_n$ a generator of the $\mathfrak{p}^{nm_0}$-torsion submodule $W_\rho^n=\{x\in\mathfrak{p}_{\bar{\Omega}}:\rho_{\eta^n}(x)=0\}$ (with $\bar{\Omega}$ the completion of a fixed algebraic closure of $K$, and $E_\rho^n=H(W_\rho^n)$ the torsion field), $\cdot_\rho$ the action of the module on its torsion points, and $\delta_n(\beta)=f'(v_n)/\beta\bmod\mathcal{D}_n$ (modulo a suitable congruence subgroup $\mathcal{D}_n$) a Coleman-type derivative homomorphism built from a power series $f$ with $f(v_n)=\beta$ (Coleman power series, whose existence in this setting is due to Oukhaba). The paper closes by asking whether such explicit reciprocity laws can be proved for all formal Drinfeld modules having stable reduction of height one — i.e. without the congruence hypothesis — and by proposing the extension to local fields of higher dimension in the sense of Flórez.

Higher-dimensional local fields (higher local fields) are complete discrete valuation fields whose residue field is itself a local field, iterated; class field theory for them involves the reciprocity map on higher K-groups in the sense of Kato and Parshin, and Flórez (2020, 2022) established explicit reciprocity laws for Lubin–Tate and general formal groups over such fields in mixed characteristic. Whether the characteristic-$p$ Drinfeld-module theory admits analogous multilevel explicit formulas is the second open direction: a higher local field of characteristic $p$ with last residue field finite carries no Drinfeld-module structure directly, so the shape such laws should take (choice of level structure, role of the trace along the tower, and the higher reciprocity map) is itself part of the problem.

Problem Statement

For a formal Drinfeld $\mathcal{O}$-module $\rho$ having stable reduction of height one over a local field of characteristic $p$, two explicitly posed directions of the program of Anglès, Bars–Longhi, and Ala Eddine remain open:

(i) Without the congruence hypothesis $\rho_{\eta}\equiv\tau^{m_0}\pmod{\mathfrak{p}_H}$ on the twisted power series $\rho_{\eta}$, establish an explicit reciprocity law for the Kummer pairing $(\alpha,\beta)_{\rho,L,n}$ whose ingredients are computable from the module's data. Concretely: for modules not satisfying the congruence, determine the value at a uniformizer — in particular at the torsion generators $v_m$ of the torsion fields $E_\rho^m=H(W_\rho^m)$ — of the $\mathcal{O}_K$-derivation $\bar{D}_{L,v_n}$ constructed in the follow-up paper (which is determined by its value at a uniformizer). Equivalently, construct and compute a $\delta_n$-type Coleman-derivative homomorphism so that a formula of the shape $(\alpha,\beta)_{\rho,L,n}=\frac{1}{\eta^n}T_{L|K}(\lambda_\rho(\alpha)\,\delta_n(\beta))\cdot_\rho v_n$ (or its derivation form $(\alpha,\beta)_{L,n}=T_{L|K}(\lambda_\rho(\alpha)\,\operatorname{dlog}\bar{D}_{L,v_n}(\beta))\cdot_\rho v_n$, the notation of the follow-up) can be evaluated, not merely asserted to exist, for the full class of formal Drinfeld modules with stable reduction of height one.

(ii) Extend these explicit reciprocity laws — the trace–logarithm–Coleman/derivation formulas for the Kummer pairing and the compatible Iwasawa maps on norm-compatible systems — to higher-dimensional local fields (higher local fields) in the sense of Flórez, i.e. produce analogous multilevel explicit formulas for the higher norm residue symbol in the characteristic-$p$ Drinfeld-module setting, specifying the level structures and reciprocity maps involved.

These two directions jointly serve one objective: make the explicit reciprocity law for formal Drinfeld modules fully computable across its natural domain — all stable-reduction height-one modules, and base fields beyond the one-dimensional local case.

References

1. Marwa Ala Eddine, 'Explicit Reciprocity Laws for Formal Drinfeld Modules', arXiv:2202.02348 (2022); published version: Journal de théorie des nombres de Bordeaux 35, No. 3, 675–695 (2023), doi:10.5802/jtnb.1260. https://arxiv.org/abs/2202.02348
2. Marwa Ala Eddine and Behzad Omidi Koma, 'The norm residue symbol for formal Drinfeld modules', arXiv:2211.10368 (v1 2022, v2 13 July 2026, preprint). https://arxiv.org/abs/2211.10368
3. Otmar Venjakob, 'Explicit Reciprocity Laws in Iwasawa Theory — A survey with some focus on the Lubin-Tate setting', arXiv:2311.08237 (2023). https://arxiv.org/abs/2311.08237
4. Jorge Flórez, 'Explicit reciprocity laws for higher local fields', Journal of Number Theory 213 (2020), 400–444, doi:10.1016/j.jnt.2019.12.016. https://doi.org/10.1016/j.jnt.2019.12.016
5. Jorge Flórez, 'The norm residue symbol for higher local fields', Journal of Number Theory 235 (2022), 157–218, doi:10.1016/j.jnt.2021.06.031. https://doi.org/10.1016/j.jnt.2021.06.031
6. Bruno Anglès, 'On Explicit Reciprocity Laws for the Local Carlitz–Kummer Symbols', Journal of Number Theory 78, No. 2 (1999), 228–252, doi:10.1006/jnth.1999.2403. https://doi.org/10.1006/jnth.1999.2403
7. Francesc Bars and Ignazio Longhi, 'Coleman's power series and Wiles' reciprocity for rank 1 Drinfeld modules', Journal of Number Theory 129, No. 4 (2009), 789–805, doi:10.1016/j.jnt.2008.12.005. https://doi.org/10.1016/j.jnt.2008.12.005
8. A. Wiles, 'Higher Explicit Reciprocity Laws', Annals of Mathematics (2) 107, No. 2 (1978), 235–254, doi:10.2307/1971143. https://doi.org/10.2307/1971143