Maria Vlasenko
For an abelian extension of number fields we show that the Stark conjecture for all Artin L-functions with zero of order r is equivalent to existence of a special element in the rational span of the r-th exterior power of the Galois module of units of the bigger field.
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In a previous paper the second author developed a new approach to the abelian p-adic Stark Conjecture at s=1 and stated some related conjectures. This paper develops and applies techniques using p-adic measures and continued fractions to numerically investigate one of these conjectures to high precision in fifteen cases with a real quadratic base field.
We show that for an odd prime p, the p-primary parts of refinements of the (imprimitive) non-abelian Brumer and Brumer-Stark conjectures are implied by the equivariant Iwasawa main conjecture (EIMC) for totally real fields. Crucially, this result does not depend on the vanishing of the relevant Iwasawa mu-invariant. In combination with the authors' previous work on the EIMC, this leads to unconditional proofs of the non-abelian Brumer and Brumer-Stark conjectures in many new cases.
We formulate a conjecture which generalizes Darmon's "refined class number formula". We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the "except 2-part" of Darmon's conjecture, which was first proved by Mazur and Rubin.
Given an abelian, CM extension K of any totally real number field k, we consider two conjectures `of Stark type'. The `Integrality Conjecture' concerns the image of a p-adic map `\mathfrak{s}_{K/k,S}' determined by the minus-part of the S-truncated equivariant L-function for K/k at s=1. It is connected to the Equivariant Tamagawa Number Conjecture of Burns and Flach. The `Congruence Conjecture' says that \mathfrak{s}_{K/k,S} gives an explicit reciprocity law for the element predicted by the corresponding Rubin-Stark Conjecture for K^+/k. We study the general properties of these conjectures and…
We discuss three different formulations of the equivariant Iwasawa main conjecture attached to an extension K/k of totally real fields with Galois group G, where k is a number field and G is a p-adic Lie group of dimension 1 for an odd prime p. All these formulations are equivalent and hold if Iwasawa's μ-invariant vanishes. Under mild hypotheses, we use this to prove non-abelian generalizations of Brumer's conjecture, the Brumer-Stark conjecture and a strong version of the Coates-Sinnott conjecture provided that μ= 0.
We study the behaviour of the Stark conjecture for an abelian extension K/k of totally real number fields as K varies in a cyclotomic Z_p-tower. We consider possible strengthenings of the natural norm-coherence in the tower of putative solution of the complex conjecture and,especially, the consequences for the analogous p-adic conjecture. More precisely, under two principal assumptions - (i) that these solutions are given by exterior powers of norm-coherent sequences of global (or p-semilocal) units and (ii) that p splits in k - we show how the values of p-adic twisted zeta-functions for K/k a…
Let $F$ be a totally real number field, $p$ a rational prime, and $χ$ a finite order totally odd abelian character of Gal$(\bar{F}/F)$ such that $χ(\mathfrak{p})=1$ for some $\mathfrak{p}|p$. Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the $p$-adic $L$-function associated to $χ$ at its exceptional zero and the $\mathfrak{p}$-adic logarithm of a $p$-unit in the $χ$ component of $F_χ^\times$. In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture assuming two conditions: that Leopoldt's conjecture holds for $F$ and $p$, and that…
Let $K/k$ be a finite Galois CM-extension of number fields whose Galois group $G$ is monomial and $S$ a finite set of places of $k$.\ Then the "Stickelberger element" $θ_{K/k,S}$ is defined.\ Concerning this element,\ Andreas Nickel formulated the non-abelian Brumer and Brumer-Stark conjectures and their "weak" versions.\ In this paper,\ when $G$ is a monomial group,\ we prove that the weak non-abelian conjectures are reduced to the weak conjectures for abelian subextensions.\ We write $D_{4p},\ Q_{2^{n+2}}$ and $A_4$ for the dihedral group of order $4p$ for any odd prime $p$,\ the generalized…
In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form whose spectral coefficients are $p$-adic logarithms of Gross-Stark units and Stark-Heegner points. Here we describe how this construction gives rise to a practical algorithm for explicitly computing these logarithms to specified precision, and how to recover the exact values of the Gross-Stark units and Stark-Heegner points from them. Key tools are overconvergent modular forms, reduction theory of quadratic forms and Newton polygons. As an application, we tabulate Brumer-Stark units in narrow Hilbert class fields of rea…
Let $K/k$ be an abelian extension of number fields with a distinguished place of $k$ that splits totally in $K$. In that situation, the abelian rank one Stark conjecture predicts the existence of a unit in $K$, called the Stark unit, constructed from the values of the $L$-functions attached to the extension. In this paper, assuming the Stark unit exists, we prove index formulae for it. In a second part, we study the solutions of the index formulae and prove that they admit solutions unconditionally for quadratic, quartic and sextic (with some additional conditions) cyclic extensions. As a resu…
The aim of this paper is to compare the orders of the class groups and the quotients of the $r$-th exterior power of units modulo Rubin-Stark units.
We investigate the arithmetic of special values of a new class of $L$-functions recently introduced by the second author. We prove that these special values are encoded in some particular polynomials which we call Anderson-Stark units. We then use these Anderson-Stark units to prove that $L$-functions can be expressed as sums of polylogarithms.
We introduce an integral version of the Eisenstein cocycle. As applications we prove a conjecture of Gross regarding the "order of vanishing" of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross--Stark units.
The purpose of this paper is to formulate and study a common refinement of a version of Stark's conjecture and its $p$-adic analogue, in terms of Fontaine's $p$-adic period ring and $p$-adic Hodge theory. We construct period-ring-valued functions under a generalization of Yoshida's conjecture on the transcendental parts of CM-periods. Then we conjecture a reciprocity law on their special values concerning the absolute Frobenius action. We show that our conjecture implies a part of Stark's conjecture when the base field is an arbitrary real field and the splitting place is its real place. It al…
We prove that a refinement of Stark's Conjecture formulated by Rubin is true up to primes dividing the order of the Galois group, for finite, abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions a statement stronger than Rubin's holds true.
On the Root of Unity Ambiguity in a Formula for the Brumer--Stark Units
本文解决了布默-斯塔克单位公式中单位根歧义的问题，并去除了Dasgupta-Kakde结果中的假设。
We show that the module of Stark units associated to a sign-normalized rank one Drinfeld module can be obtained from Anderson's equivariant $A$-harmonic series. We apply this to obtain a class formula à la Taelman and to prove a several variable log-algebraicity theorem, generalizing Anderson's log-algebraicity theorem. We also give another proof of Anderson's log-algebraicity theorem using shtukas and obtain various results concerning the module of Stark units for Drinfeld modules of arbitrary rank.
On the Equality of Three Formulas for Brumer--Stark Units
本文证明了三个Brumer-Stark单位公式的等价性，验证了其中两个公式的有效性。
In this article, we study the p-ordinary Iwasawa theory of the (conjectural) Rubin-Stark elements defined over abelian extensions of a CM field F and develop a rank-g Euler/Kolyvagin system machinery (where 2g is the degree of F), refining and generalizing Perrin-Riou's theory and the author's prior work. This has several important arithmetic consequences: Using the recent results of Hida and Hsieh on the CM main conjectures, we prove a natural extension of a theorem of Yager for the CM field F, where we relate the Rubin-Stark elements to the several-variable Katz p-adic L-function. Furthermor…
The Shintani--Faddeev modular cocycle: Stark units from $q$-Pochhammer ratios
本文将与实二次域相关的Stark单位解释为模上闭链的实乘法值，并证明精化的Shintani的Kronecker极限公式将Stark类不变量平方根与该上同调不变量联系起来。
Let $K$ be a totally real number field of degree $r$. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{2}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over $K$. The goal of this paper is to compare the characteristic ideal of the $χ$-quotient of the projective limit of the narrow class groups to the $χ$-quotient of the projective limit of the $r$-th exterior power of totally positive units modulo a subgroup of Rubin-Stark units, for some $\overline{\mathbb{Q}_{2}}$-irreducible characters $χ$ of $\mathrm{Gal}(L_{\infty}/K_{\infty})$.
In this paper we construct, using Stark elements of Rubin [Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 33-62], Kolyvagin systems for certain modified Selmer structures (that are adjusted to have core rank one in the sense of [Mem. Amer. Math. Soc. 168 (2004), no. 799] and prove a Gras-type conjecture, relating these Kolyvagin systems to appropriate ideal class groups, refining the results of Rubin [J. Reine Angew. Math. 425 (1992), 141-154].
We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark's conjecture over $\mathbb{Q}$, Rubin's conjecture, Popescu's conjecture, and a conjecture due to Burns that constitutes a generalization of Brumer's classical conjecture on annihilation of class groups. Our approach is general and could be used for any abelian extension of number fields, independent of the signature and type of places (finite or infinite) that split completely in the extension. We then employ our techniques in the situation where $K$ is a …
We study a relation between two refinements of the rank one abelian Gross-Stark conjecture: For a suitable abelian extension $H/F$ of number fields, a Gross-Stark unit is defined as a $p$-unit of $H$ satisfying some proporties. Let $τ\in \mathrm{Gal}(H/F)$. Yoshida and the author constructed the symbol $Y_p(τ)$ by using $p$-adic $\log$ multiple gamma functions, and conjectured that the $\log_p$ of a Gross-Stark unit can be expressed by $Y_p(τ)$. Dasgupta constructed the symbol $u_T(τ)$ by using the $p$-adic multiplicative integration, and conjectured that a Gross-Stark unit can be expressed by…
Comparing Two Formulas for the Gross-Stark Units
本文证明了当F为三次域时，两个关于格罗斯-斯塔克单位的公式在格罗斯调节矩阵的对角线元素上是相同的。
Let $K$ be a totally real number field of degree $r=[K:\mathbb{Q}]$ and let $p$ be an odd rational prime. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over $K$. In this article, we extend results of \cite{Kazim109} relating characteristic ideal of the $χ$-quotient of the projective limit of the ideal class groups to the $χ$-quotient of the projective limit of the $r$-th exterior power of units modulo Rubin-Stark units, in the non semi-simple case, for some $\overline{\mathbb{Q}_{p}}$-irreductible ch…
Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F. We prove this conjecture unconditionally when E/Q is abelian. We also show that for certain non-abelian extensions E/F the p-adic Stark conjecture at s=1 is implied by Leopoldt's conjecture for E at p. Moreover, we prove that for a fixed prime p, the p-adic Stark conjecture at s=1 for E/F implies Stark's conjecture at s=1 for E/F. This leads to a `prim…
Main theorem of [Buyukboduk, arXiv:0706.0377v1] suggests that it should be possible to lift the Kolyvagin systems of Stark units constructed in [Buyukboduk, arXiv:math/0703426v1] to a Kolyvagin system over the cyclotomic Iwasawa algebra. This is what we prove in this paper. This construction gives the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement of local Iwasawa theory. This s…
On Weil-Stark elements, I: general properties
本文在全局域的单位群的约化外幂格中构造典范元素族，证明其恢复实阿贝尔域中的分圆元素理论，并建立一般情形下的详细算术性质。
In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of `zeta element', of `Weil-étale cohomology complexes' and of `integral Selmer groups' for the multiplicative group $\mathbb{G}_m$ over finite abelian extensions of number fields. We then conjecture a precise connection between zeta elements and Weil-étale cohomology complexes, we show this conjecture is equivalent to a special case of the equivariant Tamagawa number conjecture and we give an unconditional proof of the analogous statement for global function fie…
We describe a conjectural construction (in the spirit of Hilbert's 12th problem) of units in abelian extensions of certain base fields which are neither totally real nor CM. These base fields are quadratic extensions with exactly one complex place of a totally real number field F, and are referred to as Almost Totally real (ATR) extensions. Our construction involves certain null-homologous topological cycles on the Hilbert modular variety attached to F. The special units are the images of these cycles under a map defined by integration of weight two Eisenstein series on GL_2(F). This map is fo…
Brumer-Stark Units and Explicit Class Field Theory
本文证明了全实域CM阿贝尔扩张中Brumer-Stark p-单位的整Gross-Stark猜想的p部分，结合n-1个F中元素的平方根生成全实域极大阿贝尔扩张，给出其无条件构造，并推广Ribet方法结合新伽罗瓦模完成证明。
The `Congruence Conjecture' was developed by the second author in a previous paper. It provides a conjectural explicit reciprocity law for a certain element associated to an abelian extension of a totally real number field whose existence is predicted by earlier conjectures of Rubin and Stark. The first aim of the present paper is to design and apply techniques to investigate the Congruence Conjecture numerically. We then present complete verifications of the conjecture in 48 varied cases with real quadratic base fields.
In 1980, Gross conjectured a formula for the expected leading term at $s=0$ of the Deligne--Ribet $p$-adic $L$-function associated to a totally even character $ψ$ of a totally real field $F$. The conjecture states that after scaling by $L(ψω^{-1}, 0)$, this value is equal to a $p$-adic regulator of units in the abelian extension of $F$ cut out by $ψω^{-1}$. In this paper, we prove Gross's conjecture.
$p$-adic Higher Green's Functions for Stark-Heegner Cycles
本文将 Darmon-Vonk 的刚性亚纯上闭链理论推广至高权，构造了实二次除子的高阶 Green 配对，为代数 Stark-Heegner 闭链的存在性提供了间接证据。
In an earlier article we proved the existence of a canonical Kolyvagin derivative homomorphism between the modules of Euler and Kolyvagin systems (in any given rank) that are associated to $p$-adic representations over number fields. We now explain how the existence of such a homomorphism leads to new results on the structure of the Selmer modules of Galois representations over Gorenstein orders and to a strategy for verifying (refinements of) the Tamagawa number conjecture of Bloch and Kato. We describe concrete applications relating to the multiplicative group over arbitrary number fields an…
Let A be a CM abelian variety defined over a number field K. We compute congruence relations on units in fields generated by adjoining torsion points of A to K. For elliptic curves, congruence relations of the type we compute were used in the proof of the Coates-Wiles theorem. In general, we compute congruence relations on exterior products of units in division fields. This result seems to fit more naturally into the framework of the Stark/Rubin conjectures.
The strong Stark conjecture for totally odd characters
本文在有限Galois CM-扩张且Galois群具有交换Sylow $p$-子群的条件下，证明了完全奇特征和奇素数$p$的强Stark猜想的$p$-部分，并给出了等变高木数猜想减$p$-部分的无条件证明。
Let $k$ be a fixed finite geometric extension of the rational function field $\mathbb{F}_q(t)$. Let $F/k$ be a finite abelian extension such that there is an $\Fq$-rational place $\infty$ in $k$ which splits in $F/k$ and let $\mathcal{O}_F$ denote the integral closure in $F$ of the ring of functions in $k$ that are regular outside $\infty$. We describe algorithms for computing the divisor class number and in certain cases for computing the structure of the divisor class group and discrete logarithms between Galois conjugate divisors in the divisor class group of $F$. The algorithms are efficie…
We give a new proof of a conjecture of Darmon, Lauder and Rotger regarding the computation of the $\mathcal L$-invariant of the adjoint of a weight one modular form in terms of units and $p$-units. While in our previous work with Rotger the essential ingredient was the use of Galois deformations techniques following the computations of Bellaïche and Dimitrov, we propose a new approach exclusively using the properties of Beilinson--Flach classes. One of the key ingredients is the computation of a cyclotomic derivative of a cohomology class in the framework of Perrin-Riou theory, which can be se…
Our first goal in this note is to explain that a weak form of Perrin-Riou's conjecture on the non-triviality of Beilinson-Kato classes follows as an easy consequence of the Iwasawa main conjectures, and deduce its refined versions in the supersingular case from this fact and a variety of Gross-Zagier formulae. Our second goal is to set up a conceptual framework in the context of $Λ$-adic Kolyvagin systems to treat analogues of Perrin-Riou's conjectures for higher motives of higher rank. We apply this general discussion in order to establish a link between Heegner points on a general class of C…
The Harris-Venkatesh conjecture for derived Hecke operators II: a unified Stark conjecture
本文证明了权为1的虚二面体模形式的Stark猜想与Harris-Venkatesh猜想间的兼容性，利用二元Siegel-Weil公式将导出Hecke算子作用重新解释为模Stark单位数据，并提出统一的猜想。
How Stark units enter SIC overlaps
本文通过精确和数值证据表明，SIC-POVM 向量间的重叠单位总是基域最大整数环上射线类域中 Stark 单位平方根的整数幂乘积，并解释了非最小情形下的格结构及重叠单位等于±1的条件。
We propose a recipe for constructing a SIC fiducial vector in complex Hilbert space of dimension of the form $d=n^2+3$, focussing on prime dimensions $d=p$. Such structures are shown to exist in thirteen prime dimensions of this kind, the highest being $p=19603$. The real quadratic base field $K$ (in the standard SIC terminology) attached to such dimensions has fundamental units $u_K$ of norm $-1$. Let $\mathbb{Z}_K$ denote the ring of integers of $K$, then $p\mathbb{Z}_K$ splits into two ideals $\mathfrak{p}$ and $\mathfrak{p}'$. The initial entry of the fiducial is the square $ξ^2$ of a geom…
On generalized Iwasawa main conjectures and $p$-adic Stark conjectures for Artin motives
本文提出关于Artin动机的广义Iwasawa主猜想和$p$-adic Stark猜想，并证明其与Rubin-Stark元素的等价性及虚二次域中相关猜想的无条件成立。
We consider partial zeta functions $ζ(s,c)$ associated with ray classes $c$'s of a totally real field. Stark's conjecture implies that an appropriate product of $\exp(ζ'(0,c))$'s is an algebraic number which is called a Stark unit. Shintani gave an explicit formula for $\exp(ζ'(0,c))$ in terms of Barnes' multiple gamma function. Yoshida ``decomposed'' Shintani's formula: he defined the symbol $X(c,ι)$ satisfying that $\exp(ζ'(0,c))=\prod_ι \exp(X(c,ι))$ where $ι$ runs over all real embeddings of $F$. Hence we can decompose a Stark unit into a product of $[F:\mathbb Q]$ terms. The main result i…
Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element $Θ^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $\text{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with $\mathbf{Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $\text{Cl}^T(H) \…
We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at $s=1$ (equivalently, $s=0$). Brumer's perspective arose by generalizing Stickelberger's work regarding the factorization of Gauss sums and the annihilation of class groups of cyclotomic fields. These viewpoints were synthesized by Tate, who stated the Brumer-Stark conjecture in its current form. The conjecture considers a totally real field $F$ and a finite abelian C…
Balanced triple product $p$-adic $L$-functions and Stark points
本研究推广Hsieh的三变量平衡三重积$p$-进$L$-函数构造，针对秩一情形提出联系其插值域外值与Stark点、Stark单位的椭圆Stark猜想，并对同一虚二次域的二面体表示证明该猜想。
A Conjecture on Almost Flat SIC-POVMs
本文探讨具有反单位元对称性的SIC-POVMs的恒等式是否能确定Stark单位，发现其不足以完全确定但可能影响较小。
In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements, utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. We show how to apply this construction to prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and deduce the main conjecture of Iwasawa theory for totally real fields (for totally odd characters). Although the main results of this paper have already been established by Wiles, our approach provides another example (which slightly differs from the case of Sta…
We give a formula for the class number of an arbitrary CM algebraic torus over $\mathbb{Q}$. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.
For an abelian number field of odd degree, we study the structure of its 2-Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed.
For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension.
The existence of a set of d^2 pairwise equiangular complex lines (equivalently, a SIC-POVM) in d-dimensional Hilbert space is currently known only for a finite set of dimensions d. We prove that, if there exists a set of real units in a certain ray class field (depending on d) satisfying certain congruence conditions and algebraic properties, a SIC-POVM may be constructed when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s=0 and is closely connected t…
Let $F$ be a totally real field. For each ideal class $c$ of $F$ and each real embedding $ι$ of $F$, Hiroyuki Yoshida defined an invariant $X(c,ι)$ as a finite sum of log of Barnes' multiple gamma functions with some correction terms. Then the derivative value of the partial zeta function $ζ(s,c)$ has a canonical decomposition $ζ'(0,c)=\sum_ιX(c,ι)$, where $ι$ runs over all real embeddings of $F$. Yoshida studied the relation between $\exp(X(c,ι))$'s, Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the $p$-adic analogue $X_p(c,ι)$: In particular, we di…
Let $K$ be a real quadratic field and let $p$ be a prime number which is inert in $K$. Let $K_p$ be the completion of $K$ at $p$. In a previous paper, we constructed a $p$-adic invariant $u_C\in K_p$, and we proved a $p$-adic Kronecker limit formula relating $u_C$ to the first derivative at $s=0$ of a certain $p$-adic zeta function. By analogy with the $p$- adic Gross-Stark conjectures, we conjectured that $u_C$ is a $p$-unit in a suitable narrow ray class field of $K$. Recently, Dasgupta has proposed an exact $p$-adic formula for the Gross-Stark units of an arbitrary totally real number field…
In this work we prove the so-called "torsion congruences" between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results in the special cases of $n=1$ and $n=2$. In both of these cases we also explain their implications for some particular…
Let F/k be a finite abelian extension of global function fields, totally split at a distinguished place \infty. We prove that a complex Gras conjecture holds for a suitable group of Stark units, and we derive a refined analytic class number formula.
Local Minkowski units in non-abelian extensions with cyclic Sylow $p$-subgroups
本文针对伽罗瓦群为非p群与循环p群直积的伽罗瓦扩张，建立了p处局部Minkowski单位的存在准则，应用该准则构造了满足条件的非阿贝尔扩张，分析了相关Iwasawa模结构并推广了研究范围。
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