Coates–Wiles Homomorphism

• Coates–Wiles homomorphism is a construction that extracts p-adic coefficients from norm-compatible local units using Coleman power series and logarithmic derivatives.
• It serves as a bridge connecting explicit reciprocity laws, Iwasawa cohomology, and p-adic L-values across cyclotomic, Lubin–Tate, and function-field settings.
• Generalizations of the homomorphism facilitate explicit computation of polylogarithmic characters, generalized Gauss sums, and dual exponential maps in arithmetic applications.

to=arxiv_search.search 彩票主管 彩神争霸快_args code='{"query":"Coates-Wiles homomorphism Lubin-Tate Coleman Ihara anticyclotomic function fields", "max_results": 10, "sort_by": "relevance"}' 菲律宾申博json {"result":[{"arxiv_id":"(Schneider et al., 2015)","title":"Coates-Wiles homomorphisms and Iwasawa cohomology for Lubin-Tate\n extensions","authors":["P. Schneider","O. Venjakob"],"abstract":"For the $p$-cyclotomic tower of $\mathbb{Q}_p$ Fontaine established a\ndescription of local Iwasawa cohomology with coefficients in a local Galois\nrepresentation $V$ in terms of the $\psi$-operator acting on the attached\netale $(\varphi,\Gamma)$-module $D(V)$. In this article we generalize\nFontaine's result to the case of arbitratry Lubin-Tate towers $L_\infty$ over\nfinite extensions $L$ of $\mathbb{Q}_p$ by using the Kisin-Ren/Fontaine\nequivalence of categories between Galois representations and\n$(\varphi_L,\Gamma_L)$-module and extending parts of [Herr L.: Sur la\ncohomologie galoisienne des corps $\mathbb{Q}_p$0-adiques. Bull. Soc. Math. France 126,\n563-600 (1998)], [Scholl A. J.: Higher fields of norms and $\mathbb{Q}_p$1-modules. Documenta Math.\ 2006, Extra Vol., 685-709]. Moreover, we prove a kind of explicit reciprocity law which calculates the Kummer map over\n$\mathbb{Q}_p$2 for the multiplicative group twisted with the dual of the Tate\nmodule $\mathbb{Q}_p$3 of the Lubin-Tate formal group in terms of Coleman power series and\nthe attached $\mathbb{Q}_p$4-module. The proof is based on a\ngeneralized Schmid-Witt residue formula. Finally, we extend the explicit\nreciprocity law of Bloch and Kato [Bloch S., Kato K.: $\mathbb{Q}_p$5-functions and\nTamagawa numbers of motives. The Grothendieck Festschrift, Vol. I, 333-400,\nProgress Math., 86, Birkh\\"auser Boston 1990] Thm. 2.1 to our situation\nexpressing the Bloch-Kato exponential map for $\mathbb{Q}_p$6 in terms of\ngeneralized Coates-Wiles homomorphisms, where the Lubin-Tate characater\n$\mathbb{Q}_p$7 describes the Galois action on $\mathbb{Q}_p$8","categories":["math.NT","math.AG"]},{"arxiv_id":"(Bandini et al., 2010)","title":"Aspects of Iwasawa theory over function fields","authors":["I. Longhi","K.-S. Tan","F. Trihan","Z. L. Wan"],"abstract":"We consider $\mathbb{Q}_p$9-extensions $V$0 of a global\nfunction field $V$1 and study various aspects of Iwasawa theory with emphasis on\nthe two main themes already (and still) developed in the number fields case as\nwell. When dealing with the Selmer group of an abelian variety $V$2 defined over\n$V$3, we provide all the ingredients to formulate an Iwasawa Main Conjecture\nrelating the Fitting ideal and the $V$4-adic $V$5-function associated to $V$6 and\n$V$7. We do the same, with characteristic ideals and $V$8-adic\n$V$9-functions, in the case of class groups (using known results on\ncharacteristic ideals and Stickelberger elements for $\psi$0-extensions). The final section provides more details for the cyclotomic\n$\psi$1-extension arising from the torsion of the Carlitz\nmodule: in particular, we relate cyclotomic units with Bernoulli-Carlitz\nnumbers by a Coates-Wiles homomorphism.","categories":["math.NT"]},{"arxiv_id":"(Kumar et al., 2014)","title":"A cohomological interpretation of derivations on graded algebras","authors":["Alexander Schenkel"],"abstract":"We present a cohomological interpretation of derivations on graded algebras in\nterms of sheaves on projective varieties. Among others, this recovers a\ngeneralization of a theorem of Wahl describing graded derivations on a\nquasihomogeneous isolated complete intersection singularity. In the projective\nnormal cone over a normal projective variety $\psi$2, a homogeneous $\psi$3-linear\nderivation of degree $\psi$4 on the section ring may be interpreted by a regular\nsection of a coherent reflexive sheaf with poles of order $\psi$5. As an\napplication, generalized Euler sequences on weighted projective spaces and\nnegative derivations of singularities are studied.","categories":["math.AG","13N15","14M05","14J17","17B66"]},{"arxiv_id":"(Hussain et al., 2022)","title":"A note on the $\psi$6-equivariant Chow ring of $\psi$7","authors":["Hyeonho Cho","Taejin Kim"],"abstract":"We study the $\psi$8-equivariant Chow ring of $\psi$9, where the\nsymmetric group acts by permuting the factors. As an application, we prove a\ncriterion on equivariant rational equivalence of codimension 1 cycles on\n$(\varphi,\Gamma)$0.","categories":["math.AG"]},{"arxiv_id":"(Nakamura et al., 2014)","title":"Polylogarithmic analogue of the Coleman-Ihara formula, I","authors":["Hiroaki Nakamura","Zhaofeng Wojtkowiak"],"abstract":"The Coleman-Ihara formula expresses Soule's $(\varphi,\Gamma)$1-adic characters restricted to\n$(\varphi,\Gamma)$2-local Galois group as the Coates-Wiles homomorphism multiplied by $(\varphi,\Gamma)$3-adic\n$(\varphi,\Gamma)$4-values at positive integers. In this paper, we show an analogous formula\nthat $(\varphi,\Gamma)$5-adic polylogarithmic characters for $(\varphi,\Gamma)$6 restrict to the\nCoates-Wiles homomorphism multiplied by Coleman's $(\varphi,\Gamma)$7-adic polylogarithms at\nany roots of unity of order prime to $(\varphi,\Gamma)$8.","categories":["math.NT","11R23"]},{"arxiv_id":"(Burungale et al., 10 Jul 2025)","title":"The $(\varphi,\Gamma)$9-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $D(V)$0-values of imaginary quadratic fields at inert primes","authors":["Shingo Kobayashi","Adebisi Agboola"],"abstract":"We prove an asymptotic formula for the $D(V)$1-adic valuation of Hecke $D(V)$2-values\nof an imaginary quadratic field at an inert prime $D(V)$3 along the anticyclotomic\n$D(V)$4-tower. The key is determination of the $D(V)$5-adic valuation of\ngeneralized Gauss sums defined using Coates-Wiles homomorphism, and of local\nresolvents in $D(V)$6-extensions. This answers a question of Rubin.","categories":["math.NT"]},{"arxiv_id":"(Hong et al., 2017)","title":"On $D(V)$7-adic Stark systems","authors":["Ryotaro Sakamoto"],"abstract":"The theory of Kolyvagin systems was generalized to the theory of Stark systems\nby Mazur and Rubin. We define the notion of $D(V)$8-adic Stark systems over\ncomplete Gorenstein local rings and prove theorems analogous to theorems in the\nclassical Stark systems theory. We also prove a control theorem and discuss an\napplication to weak Leopoldt conjectures.","categories":["math.NT"]},{"arxiv_id":"(Vovchenko et al., 2015)","title":"The automorphism group of certain higher dimensional categories","authors":["Toshiro Kuwabara"],"abstract":"We study the automorphism group of the category $D(V)$9, which was introduced\nby Segal. We first determine the group of automorphisms of the category\n$L_\infty$0. Then we show that every automorphism of the category\n$L_\infty$1 is induced by an automorphism of a group, either the trivial\ngroup or the cyclic group of order 2. Based on that result, we determine the\nautomorphism group of the category $L_\infty$2.","categories":["math.CT","20L05","18D50"]},{"arxiv_id":"(Schmidbauer et al., 2024)","title":"Unobstructedness of deformations of weak Fano manifolds","authors":["Giulio Codogni","Niels Lindner"],"abstract":"We prove that weak Fano manifolds with logarithmic tangent bundle having no\nhigher cohomology have unobstructed deformations. In particular, if $L_\infty$3 is a\nweak Fano manifold and $L_\infty$4 a smooth divisor in $L_\infty$5, then the deformations\nof the pair $L_\infty$6 are unobstructed for $L_\infty$7 sufficiently large. The proof\nuses that deformations of line bundles are unobstructed.","categories":["math.AG"]}]} The Coates–Wiles homomorphism is a family of maps that extracts $L_\infty$8-adic or local-arithmetic coefficients from norm-compatible systems of local units by means of Coleman power series and their logarithmic derivatives. In the cyclotomic setting, it is defined by the coefficients of $L_\infty$9 after the change of variable $L$0; in Lubin–Tate settings it is reformulated using the invariant derivation attached to a formal $L$1-module; in function-field settings it admits a Carlitz-module analogue via Hasse–Teichmüller derivatives. Across these settings, the homomorphism serves as a bridge between local units, explicit reciprocity laws, Iwasawa cohomology, $L$2-adic $L$3-values, polylogarithmic characters, and generalized Gauss sums (Nakamura et al., 2014, Schneider et al., 2015, Bandini et al., 2010, Burungale et al., 10 Jul 2025).

1. Basic construction and coefficient-extraction principle

In the cyclotomic local setting of Nakamura–Wojtkowiak, one takes an odd prime $L$4, a finite unramified extension $L$5, and the inverse limit $L$6 of principal units in $L$7. Coleman’s map sends a norm-compatible unit $L$8 to a power series

$L$9

whose value $\mathbb{Q}_p$0 is characterized by

$\mathbb{Q}_p$1

The $\mathbb{Q}_p$2-th Coates–Wiles homomorphism is then defined by the expansion

$\mathbb{Q}_p$3

so that $\mathbb{Q}_p$4 is literally the $\mathbb{Q}_p$5-th Taylor coefficient of $\mathbb{Q}_p$6 in the cyclotomic parameter $\mathbb{Q}_p$7 (Nakamura et al., 2014).

This coefficient-extraction principle persists in the other settings represented in the literature cited here. Schneider–Venjakob formulate generalized Coates–Wiles homomorphisms for Lubin–Tate towers by replacing the cyclotomic derivation with the invariant derivation of a Lubin–Tate formal group. Longhi–Tan–Trihan–Wan formulate a function-field analogue for the Carlitz tower by applying $\mathbb{Q}_p$8 to a Coleman power series, composing with the Carlitz exponential, and extracting coefficients via Hasse–Teichmüller derivatives. Kobayashi–Agboola use logarithmic derivatives of Coleman series at $\mathbb{Q}_p$9 and at Lubin–Tate torsion points to build the generalized Gauss sums $(\varphi_L,\Gamma_L)$0 that control valuations of anticyclotomic Hecke $(\varphi_L,\Gamma_L)$1-values (Schneider et al., 2015, Bandini et al., 2010, Burungale et al., 10 Jul 2025).

A common restriction of the notion to the cyclotomic tower is therefore too narrow. The papers considered here show that the Coates–Wiles construction is stable under substantial changes of formal group, local tower, and arithmetic context.

2. Cyclotomic normalization and the Coleman–Ihara framework

In the cyclotomic normalization used by Nakamura–Wojtkowiak, the Coates–Wiles homomorphism is attached to the logarithm of the Coleman power series, not directly to the unit system itself. The normalization follows Bloch–Kato rather than Coleman’s original conventions: the constant term of $(\varphi_L,\Gamma_L)$2 is congruent to $(\varphi_L,\Gamma_L)$3, but need not equal $(\varphi_L,\Gamma_L)$4, and consequently the defining series starts at $(\varphi_L,\Gamma_L)$5, not $(\varphi_L,\Gamma_L)$6 (Nakamura et al., 2014).

This normalization places the homomorphism inside the classical Coleman–Ihara formula. For $(\varphi_L,\Gamma_L)$7 odd, the paper recalls the formula

$(\varphi_L,\Gamma_L)$8

and explains that the original Coleman–Ihara formula is recovered from its more general theorem by specializing to $(\varphi_L,\Gamma_L)$9 and $\mathbb{Q}_p$00. In the same framework, the restricted and unrestricted polylogarithmic characters satisfy

$\mathbb{Q}_p$01

and

$\mathbb{Q}_p$02

for $\mathbb{Q}_p$03 in the image of local reciprocity and $\mathbb{Q}_p$04 of order prime to $\mathbb{Q}_p$05 (Nakamura et al., 2014).

The conceptual significance is explicit in the paper: the restriction of $\mathbb{Q}_p$06-adic or $\mathbb{Q}_p$07-adic polylogarithmic characters to the local cyclotomic Galois group is controlled by a product of a $\mathbb{Q}_p$08-adic polylogarithm value and a Coates–Wiles homomorphism. In the special case $\mathbb{Q}_p$09, the polylogarithmic side collapses to Kubota–Leopoldt values, so the same formalism recovers the Soulé-character/$\mathbb{Q}_p$10-adic-$\mathbb{Q}_p$11-value relation.

3. Lubin–Tate generalization and Iwasawa cohomology

Schneider–Venjakob replace the cyclotomic tower of $\mathbb{Q}_p$12 with an arbitrary Lubin–Tate tower $\mathbb{Q}_p$13, where $\mathbb{Q}_p$14 is finite, $\mathbb{Q}_p$15 is a Lubin–Tate formal $\mathbb{Q}_p$16-module for a uniformizer $\mathbb{Q}_p$17, and $\mathbb{Q}_p$18 is its Tate module. If $\mathbb{Q}_p$19, Coleman’s theorem gives a unique Laurent series

$\mathbb{Q}_p$20

such that $\mathbb{Q}_p$21 for all $\mathbb{Q}_p$22. The relevant derivative is the invariant derivation

$\mathbb{Q}_p$23

and the Lubin–Tate logarithmic derivative is

$\mathbb{Q}_p$24

The generalized Coates–Wiles homomorphism is then defined by

$\mathbb{Q}_p$25

and more generally, for $\mathbb{Q}_p$26,

$\mathbb{Q}_p$27

The associated $\mathbb{Q}_p$28-valued map is denoted $\mathbb{Q}_p$29 (Schneider et al., 2015).

This is a genuine extension of the classical picture. When $\mathbb{Q}_p$30 and $\mathbb{Q}_p$31, one has $\mathbb{Q}_p$32, $\mathbb{Q}_p$33, $\mathbb{Q}_p$34, and $\mathbb{Q}_p$35, so the Lubin–Tate definition reduces to the familiar cyclotomic construction. The paper emphasizes that the extra twist by $\mathbb{Q}_p$36 in the Iwasawa-cohomological exact sequence is a genuinely new Lubin–Tate phenomenon and disappears in the cyclotomic case (Schneider et al., 2015).

The same paper also identifies the place of these homomorphisms in Lubin–Tate Iwasawa cohomology. Its $\mathbb{Q}_p$37-description of $\mathbb{Q}_p$38, together with the Kisin–Ren/Fontaine equivalence, shows that the generalized Coates–Wiles homomorphisms are not merely formal coefficients: they are the local quantities through which the Kummer map and the Bloch–Kato exponential become explicit.

4. Explicit reciprocity, Kummer maps, and Galois characters

A central theme across the cited works is that the Coates–Wiles homomorphism is the local coefficient that makes explicit reciprocity laws computable. In the cyclotomic setting, Nakamura–Wojtkowiak use Coleman’s explicit reciprocity law to evaluate Hilbert symbols in terms of the logarithm of an auxiliary series $\mathbb{Q}_p$39 and the logarithmic derivative of the Coleman series $\mathbb{Q}_p$40. The operational appearance of the Coates–Wiles homomorphism is through the identity

$\mathbb{Q}_p$41

which is the mechanism by which the local reciprocity pairing extracts the Coates–Wiles coefficient (Nakamura et al., 2014).

In the Lubin–Tate setting of Schneider–Venjakob, the Kummer map over the tower $\mathbb{Q}_p$42 is identified with an explicit logarithmic derivative map

$\mathbb{Q}_p$43

and the resulting diagram

$\mathbb{Q}_p$44

is commutative. The Bloch–Kato exponential is then expressed directly in terms of generalized Coates–Wiles homomorphisms: $\mathbb{Q}_p$45 and

$\mathbb{Q}_p$46

The paper states that this extends the explicit reciprocity law of Bloch and Kato to the Lubin–Tate situation (Schneider et al., 2015).

The upshot is that the Coates–Wiles homomorphism is not only a device for encoding units analytically. It is also the exact term through which local class field theory, Kummer theory, and $\mathbb{Q}_p$47-adic Hodge-theoretic exponentials are related.

5. Generalized Gauss sums and the anticyclotomic inert-prime setting

Kobayashi–Agboola place the Coates–Wiles homomorphism at the center of a local theory over the unramified quadratic extension $\mathbb{Q}_p$48 at an inert prime $\mathbb{Q}_p$49. Let $\mathbb{Q}_p$50 be a Lubin–Tate formal group over $\mathbb{Q}_p$51 for $\mathbb{Q}_p$52, with formal logarithm $\mathbb{Q}_p$53, and let $\mathbb{Q}_p$54 be the principal units of $\mathbb{Q}_p$55. For

$\mathbb{Q}_p$56

with Coleman power series $\mathbb{Q}_p$57, the paper defines

$\mathbb{Q}_p$58

and states that these maps are well-defined and Galois equivariant. For a finite character $\mathbb{Q}_p$59 factoring through $\mathbb{Q}_p$60, it then defines

$\mathbb{Q}_p$61

with the explicit remark that the definition does not depend on the choice of $\mathbb{Q}_p$62 (Burungale et al., 10 Jul 2025).

The paper describes $\mathbb{Q}_p$63 as “analogous to the Gauss sum in the cyclotomic case, defined via Coates-Wiles homomorphism (or the dual exponential map).” This generalized Gauss sum is then built into the local Iwasawa module

$\mathbb{Q}_p$64

and used to define the sign submodules

$\mathbb{Q}_p$65

Rubin had shown these are free rank-one $\mathbb{Q}_p$66-modules, and Rubin’s conjecture is stated as

$\mathbb{Q}_p$67

An important structural formula identifies $\mathbb{Q}_p$68 with a twisted sum of dual exponentials: $\mathbb{Q}_p$69 Accordingly, the Coates–Wiles construction is simultaneously a Coleman-series logarithmic derivative and a cohomological dual-exponential object.

6. Valuations, special values, and the function-field analogue

In the inert anticyclotomic setting, Kobayashi–Agboola use the identity

$\mathbb{Q}_p$70

to tie the generalized Coates–Wiles Gauss sums to local resolvents. Their local resolvent theorem gives

$\mathbb{Q}_p$71

for $\mathbb{Q}_p$72 of order $\mathbb{Q}_p$73, with equality if $\mathbb{Q}_p$74 is a uniformizer, while Proposition $\mathbb{Q}_p$75 implies

$\mathbb{Q}_p$76

when $\mathbb{Q}_p$77 is a uniformizer. From this, they derive the explicit valuation formula

$\mathbb{Q}_p$78

for $\mathbb{Q}_p$79 of order $\mathbb{Q}_p$80 and $\mathbb{Q}_p$81. The abstract states that the determination of these valuations is the key local input in the asymptotic formula for the $\mathbb{Q}_p$82-adic valuation of Hecke $\mathbb{Q}_p$83-values, and that this answers a question of Rubin (Burungale et al., 10 Jul 2025).

A parallel but characteristic-$\mathbb{Q}_p$84 analogue appears in the Carlitz setting. Longhi–Tan–Trihan–Wan consider $\mathbb{Q}_p$85, a prime $\mathbb{Q}_p$86, the tower $\mathbb{Q}_p$87, its local completions $\mathbb{Q}_p$88, and the inverse limit of units $\mathbb{Q}_p$89. For $\mathbb{Q}_p$90, they define

$\mathbb{Q}_p$91

equivalently through

$\mathbb{Q}_p$92

These maps satisfy the weight-$\mathbb{Q}_p$93 equivariance

$\mathbb{Q}_p$94

For the cyclotomic units

$\mathbb{Q}_p$95

the paper proves

$\mathbb{Q}_p$96

and equivalently

$\mathbb{Q}_p$97

The paper presents this as the function-field analogue of the classical statement that Coates–Wiles homomorphisms applied to cyclotomic units recover special zeta values and Bernoulli numbers (Bandini et al., 2010).

Taken together, these results show a coherent pattern. The Coates–Wiles homomorphism is the local coefficient extractor attached to Coleman theory; its realizations vary with the formal group and the tower, but its arithmetic role is stable. It encodes local units in a form compatible with explicit reciprocity, and it supplies the local factors that govern polylogarithmic characters, Bloch–Kato exponentials, generalized Gauss sums, and special-value formulas.