Determination of Conductors from Galois Module Structure

Tame multiplicity and conductor for local Galois representations

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $σ$ be an irreducible smooth representation of the absolute Weil group $\Cal W_F$ of $F$ and $\sw(σ)$ the Swan exponent of $σ$. Assume $\sw(σ) \ge1$. Let $\Cal I_F$ be the inertia subgroup of $\Cal W_F$ and $\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\varSigma$, of order prime to $p$, so that $σ(\Cal I_F) = σ(\Cal P_F)\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $σ$ of any character of $\varSigma$ is bounded by $\sw(σ)$.

Galois action on special theta values

Using Shimura's reciprocity law, we calculate the Galois action on the special values of theta functions associated to primitive Dirichlet characters of odd conductor, normalised by the Dedekind eta function, at the point i. As a consequence, we prove some experimental results of Cohen and Zagier and we deduce a partial result on the non-vanishing of these special theta values with prime conductor.

A bound for the conductor of an open subgroup of GL2 associated to an elliptic curve

Given an elliptic curve $E$ without complex multiplication defined over a number field $K$, consider the image of the Galois representation defined by letting Galois act on the torsion of $E$. Serre's open image theorem implies that there is a positive integer $m$ for which the Galois image is completely determined by its reduction modulo $m$. In this note, we prove a bound on the smallest such $m$ in terms of standard invariants associated with $E$. The bound is sharp and improves upon previous results.

On ramification filtrations and $p$-adic differential modules, I: equal characteristic case

Let $k$ be a complete discretely valued field of equal characteristic $p > 0$ with possibly imperfect residue field and let $G_k$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on $G_k$ coincide with the differential Artin conductors and Swan conductors of Galois representations of $G_k$. As a consequence, we give a Hasse-Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse-Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors a…

Strong modularity of reducible Galois representations

In this paper, we call strongly modular those reducible semi-simple odd mod $l$ Galois representations for which the conclusion of the strongest form of Serre's original modularity conjecture holds. Under the assumption that the Serre weight $k$ satisfies $l\textgreater{}k+1$, we give a precise characterization of strongly modular representations, hence generalizing a classical theorem of Ribet pertaining to the case of conductor $1$.When the representation $ρ$ is not strongly modular, we give a necessary and sufficient condition on the primes $p$ not dividing $Nl$ for which it arises in level…

素数导子的模2二面体伽罗瓦表示

Mod-2 dihedral Galois representations of prime conductor

针对不超过500000的奇素数N，计算Hecke算子T₂在S₂(Γ₀(N), Q)上的作用并分析其模2约化的特征值，结合类域论与模2伽罗瓦表示解释结果，还扩展了关于特定模2约化的椭圆曲线和模形式的不存在性结果。

Explicit determination of the images of the Galois representations attached to abelian surfaces with End(A)=\Z

We give an effective version of a result reported by Serre asserting that the images of the Galois representations attached to an abelian surface with $\End(A)= \mathbb{Z}$ are as large as possible for almost every prime. Our algorithm depends on the truth of Serre's conjecture for two dimensional odd irreducible Galois representations. Assuming this conjecture we determine the finite set of primes with exceptional image.

Swan conductors for p-adic differential modules, I: A local construction

We define a numerical invariant, the differential Swan conductor, for certain differential modules on a rigid analytic annulus over a p-adic field. This gives a definition of a conductor for p-adic Galois representations with finite local monodromy over an equal characteristic discretely valued field, which agrees with the usual Swan conductor when the residue field is perfect. We also establish analogues of some key properties of the usual Swan conductor, such as integrality (the Hasse-Arf theorem), and the fact that the graded pieces of the associated ramification filtration on Galois groups…

Conductors and the moduli of residual perfection

Let A be a complete discrete valuation ring with possibly imperfect residue field. The purpose of this paper is to give a notion of conductor for Galois representations over A that generalizes the classical Artin conductor. The definition rests on two general results: there is a moduli space that parametrizes the ways of modifying A so that its residue field is perfect, and any information about a Galois-theoretic object over A can be recovered from its pullback to the (residually perfect) discrete valuation ring corresponding to the generic point of this moduli space.

Explicit determination of images of Galois representations attached to Hilbert modular forms

In a previous article, the second author proved that the image of the Galois representation mod $λ$ attached to a Hilbert modular newform is large or all but finitely many primes $λ$, if the form is not a theta series. In this brief note, we give an explicit bound for this exceptional finite set of primes and determine the images in three different examples. Our examples are of Hilbert newforms on real quadratic fields, of parallel or non-parallel weight and of different levels.

Reducible Galois representations and the homology of GL(3,Z)

We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. Let $ρ$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the conductor of $ρ$ is squarefree, we prove that $ρ$ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup $Γ$ of $\GL(3,\Z)$. In addition, we prove that the coefficient module needed is, in fact, predicted by a conjecture of Ash, Doud, P…

Conductors of wild extensions of local fields, especially in mixed characteristic (0,2)

If K_0 is the fraction field of the Witt vectors over an algebraically closed field k of characteristic p, we calculate upper bounds on the conductor of higher ramification for (the Galois closure of) extensions K_0(zeta_{p^r}, sqrt[p^r]{a})/K_0, where a is in K_0(zeta_{p^r}). Here zeta_{p^r} is a primitive p^r-th root of unity. In certain cases, including when a is in K_0 and p=2, we calculate the conductor exactly. These calculations can be used to determine the discriminants of various extensions of Q obtained by adjoining roots of unity and radicals.

Malle's conjecture with multiple invariants

We define invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$ of Galois extensions of number fields with a fixed Galois group. Then, we propose a heuristic in the spirit of Malle's conjecture which asymptotically predicts the number of extensions that satisfy $\operatorname{inv}_i\leq X_i$ for all $X_i$. The resulting conjecture is proved for abelian Galois groups. We also describe refined Artin conductors that carry essentially the same information as the invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$.

Kato's conductor and generic residual perfection

Let A be a complete discrete valuation ring with possibly imperfect residue field, and let $χ$ be a one-dimensional Galois representation over A. I show that the non-logarithmic variant of Kato's Swan conductor is the same for $χ$ and the pullback of $χ$ to the generic residual perfection of A. This implies the conductor from "Conductors and the moduli of residual perfection" (math.NT/0112305) extends the non-logarithmic variant of Kato's.

偶Galois变形环的精细结构

Fine structure of even Galois deformation rings

该论文研究素导体为ℓ的偶Galois表示族，证明其对应水平S={ℓ,3}的偶模3变形环的商环不存在特定形式，进而推导出全局偶变形环在Z_3模范畴中有限且平坦的结论。

On the class numbers of real cyclotomic fields of conductor $pq$

The class numbers $h^{+}$ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime conductor, and we make it applicable to real cyclotomic fields of conductor equal to the produ…

On the images of modular and geometric three-dimensional Galois represetations

We address the problem of the determination of the images of three-dimensional geometric and modular Galois representations. In the modular case the existence of these representations is only conjectural. We give conditions to ensure that the images are as large as possible for almost every prime. We compute some examples and stress the applications to inverse Galois theory.

On the images of the Galois representations attached to genus 2 Siegel modular forms

We address the problem of the determination of the images of the Galois representations attached to genus 2 Siegel cusp forms of level 1 having multiplicity one. These representations are symplectic. We prove that the images are as large as possible for almost every prime, if the Siegel cusp form is not a Maass spezialform and verifies two easy to check conditions.

Galois representations attached to elliptic curves with complex multiplication

The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$. Let $p\geq 2$ be a prime, let $G_{\mathbb{Q}(j(E))}$ be the absolute Galois group of $\mathbb{Q}(j(E))$, and let $ρ_{E,p^\infty}\colon G_{\mathbb{Q}(j(E))}\t…

利用局部常值性确定某些模p伽罗瓦表示

Determination of certain mod $p$ Galois representations using local constancy

该研究针对素数p≥5及特定区间内的权值k，证明了二维晶态伽罗瓦表示的模p约化在权空间中的局部常值性，借助GL₂(ℚₚ)的模p局部朗兰兹对应显式计算约化并给出常值半径下界，还得到多组新参数下的显式模p约化结果。

Mod p representations on elliptic curves

Modular Galois representations into GL_2(F_p) with cyclotomic determinant arise from elliptic curves for p = 2,3,5. We show (by constructing explicit examples) that such elliptic curves cannot be chosen to have conductor as small as possible at all primes other than p. Our proof involves finding all elliptic curves of conductor 85779, a custom computation carried out for us by Cremona. This leads to a counterexample to a conjecture of Lario and Rio. For p > 5, we construct irreducible representations with cyclotomic determinant that do not arise from any elliptic curve over Q.

On the ramification of non-abelian Galois coverings of degree $p^3$

The refined Swan conductor is defined by K.\ Kato \cite{KK2}, and generalized by T.\ Saito \cite{wild}. In this part, we consider some smooth $l$-adic étale sheaves of rank $p$ such that we can be define the $rsw$ following T.\ Saito, on some smooth dense open subscheme $U$ of a smooth separated scheme X of finite type over a perfect fields $κ$ of characteristic $p>0$. We give an explicit expression of $rsw(\mathcal{F})$ in some situation. As a consequence, we show that it is integral.

One-dimensional elementary abelian extensions have Galois scaffolding

We define a variant of normal basis, called a {\em Galois scaffolding}, that allows for an easy determination of valuation, and has implications for Galois module structure. We identify fully ramified, elementary abelian extensions of local function fields of characteristic $p$, called {\em one-dimensional}, that, in a particular sense, are as simple as cyclic degree $p$ extensions, and prove the statement in the title above.

整体函数域上模 $p$ Galois 表示的有限性定理

A finiteness theorem for mod $p$ Galois representations over global function fields

对于特征不同于 $p$ 的整体函数域 $K$，证明了 Artin 导子有界的连续几何半单表示 $\rho: G_K \to \mathrm{GL}_n(\overline{\mathbb{F}}_p)$ 只有有限多个同构类，且无需假设 $p$ 不整除 $n$。

Equivariant Birch-Swinnerton-Dyer conjecture for the base change of elliptic curves: An example

Let E be an elliptic curved defined over $\Q$ and let $K/\Q$ be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for $h^1(E\times_{\Q} K)(1)$ viewed as a motive over $\Q$ with coefficients in $\Q[G]$ relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper we prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S_3-extension of $\Q$.

具有伽罗瓦群 $GL_2(\mathbb{F}_\ell)$ 的数域数量的下界

Lower bounds for the number of number fields with Galois group $GL_2(\mathbb{F}_\ell)$

本文证明了伽罗瓦群为 $GL_2(\mathbb{F}_\ell)$ 且判别式有界的数域数量具有明确的渐近下界，并对按 Artin 导子排序的相应满射伽罗瓦表示给出了类似下界。

等特征情形下的有界导子有限性猜想

On the bounded-conductor finiteness conjecture in equal characteristic

本文研究等特征整体函数域上模p Galois表示的有界导子有限性猜想，建立了带提升假设的有限性定理，并构造反例证明无提升时猜想一般不成立，从而完全解决二维情形。

On refined ramification filtrations in the equal characteristic case

Let k be a complete discrete valuation field of equal characteristic p>0. Using the tools of p-adic differential modules, we define refined Artin and Swan conductors for a representation of the absolute Galois group $G_k$ with finite local monodromy; this leads to a description of the subquotients of the ramification filtration on $G_k$. We prove that our definition of the refined Swan conductors coincide with that is given by Saito, which uses etale cohomology. We also study its relation with the toroidal variation of the Swan conductors.

局部伽罗瓦表示与商奇点的质量公式 I：计数函数的比较

Mass formulas for local Galois representations and quotient singularities I: a comparison of counting functions

本文研究局部伽罗瓦表示中 Artin 导子与野 Deligne-Mumford 栈 motivic 积分权的关系，并借此证明 McKay 对应的一种形式，将 Bhargava 质量公式与点的 Hilbert 概型联系起来。

A short basis of the Stickelberger ideal of a cyclotomic field

We exhibit an explicit short basis of the Stickelberger ideal of cyclotomic fields of any conductor $m$, i.e., a basis containing only short elements. By definition, an element of $\mathbb{Z}[G_m]$, where $G_m$ denotes the Galois group of the field, is called short whenever it writes as $\sum_{σ\in G_m} \varepsilon_σσ$ with all $\varepsilon_σ\in\{0,1\}$. One ingredient for building such a basis consists in picking wisely generators $α_m(b)$ in a large family of short elements. As a direct practical consequence, we deduce from this short basis an explicit upper bound on the relative class numbe…

On the embedding problem for $2^+S_4$ representations

Let $2^+S_4$ denote the double cover of $S_4$ corresponding to the element in $H^2(S_4,\Z/2\Z)$ where transpositions lift to elements of order 2 and the product of two disjoint transpositions to elements of order 4 (denoted $\tilde S_4$ in \cite{Serre}). Given an elliptic curve $E$, let $E[2]$ denote its 2-torsion points. Under some conditions on $E$ (as in \cite{Bayer}) elements in $H^1(\Gal_\Q,E[2])\backslash \{0 \}$ correspond to Galois extensions $N$ of $\Q$ with Galois group (isomorphic to) $S_4$. On this work we give an interpretation of the addition law on such fields, and prove that th…

The Hilbert-Grunwald specialization property over number fields

Given a finite group $G$ and a number field $K$, we investigate the following question: Does there exist a Galois extension $E/K(t)$ with group $G$ whose set of specializations yields solutions to all Grunwald problems for the group $G$, outside a finite set of primes? Following previous work, such a Galois extension would be said to have the "Hilbert-Grunwald property". In this paper we reach a complete classification of groups $G$ which admit an extension with the Hilbert-Grunwald property over fields such as $K=\mathbb{Q}$. We thereby also complete the determination of the ``local dimension…

Drinfeld模的局部单值性

Local monodromy of Drinfeld modules

针对Drinfeld模局部单值性复杂的问题，提出局部导子不变量，用局部周期格体积给出其上界，发展函数域赋范格理论并联系向量丛正则性，还描述大剩余秩Drinfeld模的惯性像性质。

On differential modules associated to de Rham representations in the imperfect residue field case

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$, whose residue field may not be perfect, and $G_K$ the absolute Galois group of $K$. In the first part of this paper, we prove that Scholl's generalization of fields of norms over $K$ is compatible with Abbes-Saito's ramification theory. In the second part, we construct a functor $\mathbb{N}_{\mathrm{dR}}(V)$ associating a de Rham representation $V$ with a $(φ,\nabla)$-module in the sense of Kedlaya. Finally, we prove a compatibility between Kedlaya's differential Swan conductor of $\mathbb{N}_{\mathrm{dR}}(V)$ and …

CM点的p-adic等分布

p-adic equidistribution of CM points

研究模曲线和Shimura曲线上CM点在p-adic Berkovich解析空间中的等分布问题，通过划分显式“盆地”并证明局部Galois轨道序列存在极限的条件，结合拟典范提升理论与相交数新公式得出结论。

Large 2-adic Galois image and non-existence of certain abelian surfaces over Q

Motivated by our arithmetic applications, we required some tools that might be of independent interest. Let $\mathcal E$ be an absolutely irreducible group scheme of rank $p^4$ over $\mathbb Z_p$. We provide a complete description of the Honda systems of $p$-divisible groups $\mathcal G$ such that $\mathcal G[p^{n+1}]/\mathcal G[p^n] \simeq \mathcal E$ for all $n$. Then we find a bound for the abelian conductor of the second layer $\mathbb Q_p(\mathcal G[p^2])/\mathbb Q_p(\mathcal G[p])$, stronger in our case than can be deduced from Fontaine's bound. Let $π\!: \, {\rm Sp}_{2g}(\mathbb Z_p) \t…

On the structure of the Galois group of the Abelian closure of a number field

Following a paper by Athanasios Angelakis and Peter Stevenhagen on the determination of imaginary quadratic fields having the same absolute Abelian Galois group A, we study this property for arbitrary number fields. We show that such a property is probably not easily generalizable, apart from imaginary quadratic fields, because of some p-adic obstructions coming from the global units. By restriction to the p-Sylow subgroups of A, we show that the corresponding study is related to a generalization of the classical notion of p-rational fields. However, we obtain some non-trivial information abou…

Galois module structure of Milnor K-theory in characteristic p

Let E be a cyclic extension of degree p^n of a field F of characteristic p. Using arithmetic invariants of E/F we determine k_mE, the Milnor K-groups K_mE modulo p, as Fp[Gal(E/F)]-modules for all m in N. In particular, we show that each indecomposable summand of k_mE has Fp-dimension a power of p. That all powers p^i, i=0,1,...,n, occur for suitable examples is shown in a subsequent paper [MSS2], where additionally the main result of this paper becomes an essential induction step in the determination of K_mE/p^sK_mE as (Z/p^sZ)[Gal(E/F)]-modules for all m, s in N.

The reciprocity conjecture of Khare and Wintenberger

We prove a strengthening of the "reciprocity conjecture" of Khare and Wintenberger. The input to the original conjecture is an odd prime p, a CM number field F containing the pth roots of unity, and a pair of primes of the maximal totally real subfield E of F that are inert in the cyclotomic Z_p-extension of E. In analogy to a statement about generalized Jacobians of curves, the conjecture asserts the equality of two procyclic subgroups of the Galois group G of the maximal p-ramified abelian pro-p extension of E. The first is the intersection of the subgroup of G fixing the cyclotomic Z_p-exte…

On the vanishing of Selmer groups for elliptic curves over ring class fields

Let E be a rational elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let H be the ring class field of K of conductor c prime to ND with Galois group G over K. Fix a complex character χof G. Our main result is that if the special value of the χ-twisted L-function of E/K is non-zero then the tensor product (with respect to χ) of the p-Selmer group of E/H with W over Z[G] is 0 for all but finitely many primes p, where W is a suitable finite ext…

$3$-rank of ambiguous class groups of cubic Kummer extensions

Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(ζ_3)$ with $d>1$ a cube-free integer and $ζ_3$ a primitive third root of unity. Denote by $C_{k,3}^{(σ)}$ the $3$-group of ambiguous classes of the extension $k/k_0$ with relative group $G=\operatorname{Gal}(k/k_0)=\langleσ\rangle$. The aims of this paper are to characterize all extensions $k/k_0$ with cyclic $3$-group of ambiguous classes $C_{k,3}^{(σ)}$ of order $3$, to investigate the multiplicity $m(f)$ of the conductors $f$ of these abelian extensions $k/k_0$, and to classify the fields $k$ according to the cohomology…

Deligne-Henniart 扭转公式的推广及其应用

An extension of Deligne-Henniart's twisting formula and its applications

本文推广了 Deligne 及 Deligne-Henniart 的扭转公式，将其扩展到素数维 U-迷向 Heisenberg 表示，并应用于局部根数不变公式与伽罗瓦侧逆定理。

循环三次域的群论方法

Group theoretic approach to cyclic cubic fields

本文用群论方法研究共享共同导子的循环三次数域四元组，通过三次剩余符号和歧主理想条件确定极大亚交换非分歧3-扩张的自同构群，并借助关系秩判定其是否等同于极大非分歧pro-3-扩张的Galois群。

p³次海森伯格扩张的哈塞范数原理

Hasse norm principle for Heisenberg extensions of degree $p^3$

本文针对伽罗瓦闭包群为阶p³海森伯格群的p³或p²次可分域扩张，给出哈塞范数原理成立的充要条件，进而利用小野公式推导得出对应范一环面的塔马加瓦数。