Eva Bayer-Fluckiger, Raman Parimala
We define degree two cohomological invariants for G-Galois algebras over fields of characteristic not 2, and use them to give necessary conditions for the existence of a self--dual normal basis. In some cases, we show that these conditions are also sufficient.
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We prove that several properties of absolute Galois groups are preserved under a profinite completion.
Let p be a prime and F a field containing a primitive pth root of unity. Then for n in N, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is <=n if and only if the corestriction maps H^n(H,Fp) -> H^n(G,Fp) are surjective for all open subgroups H of index p. Using this result we derive a surprising generalization to dim_Fp H^n(H,Fp) of Schreier's formula for dim_Fp H^1(H,Fp).
Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. We determine a new canonical quotient $\mathrm{Gal}(F_{(3)}/F)$ of $G_F$ which encodes the full mod-$q$ cohomology ring $H^*(G_F,\mathbb{Z}/q)$ and is minimal with respect to this property. We prove some fundamental structure theorems related to these quotients. In particular, it is shown that when $q=p$ is an odd prime, $F_{(3)}$ is the compositum of all Galois extensions $E$ of $F$ such that $\mathrm{Gal}(E/F)$ is isomorphic to $\{1\}$, $\mathbb{Z}/p$ or to the nonabelian g…
Let p be a prime and F a field containing a primitive pth root of unity. Let E/F be a cyclic extension of degree p and G_E < G_F the associated absolute Galois groups. We determine precise conditions for the cohomology group H^n(E)=H^n(G_E,Fp) to be free or trivial as an Fp[Gal(E/F)]-module. We examine when these properties for H^n(E) are inherited by H^k(E), k>n, and, by analogy with cohomological dimension, we introduce notions of cohomological freeness and cohomological triviality. We give examples of H^n(E) free or trivial for each n in N with prescribed cohomological dimension.
For a list $\cal{L}$ of finite groups and for a profinite group $G$, we consider the intersection $T(G)$ of all open normal subgroups $N$ of $G$ with $G/N$ in $\cal{L}$. We give a cohomological characterization of the epimorphisms $π\colon S\to G$ of profinite groups (satisfying some additional requirements) such that $π[T(S)]=T(G)$. For $p$ prime, this is used to describe cohomologically the profinite groups $G$ whose $n$th term $G_{(n,p)}$ (resp., $G^{(n,p)}$) in the $p$-Zassenhaus filtration (resp., lower $p$-central filtration) is an intersection of this form. When $G=G_F$ is the absolute …
Let l be an odd prime and K/k a Galois extension of totally real number fields with Galois group G such that K/k_\infty and k/Q are finite. We give a full description of the algebraic structure of the semisimple algebra QG=Quot(ΛG) for pro-l Galois groups G with ΛG = Z_l[[G]] the Iwasawa algebra. We moreover compute the cohomological dimension of the centres of the Wedderburn components of QG and state some results on the completion of QG.
Transfer principles for Galois cohomology and Serre's conjecture II
本文证明域的上同调维数的转移原理，构造低维扩张，并应用于Serre猜想II，证明特征0情形蕴含正特征情形。
Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity
本文通过Galois上同调方法研究偶Galois表示族中Selmer秩的分布，利用Wiles公式和全球互反律证明了Selmer秩增加的素数密度为1/192。
Let $π$ be a cuspidal, cohomological automorphic representation of an inner form $G$ of $\mathrm{PGL}_2$ over a number field $F$ of arbitrary signature. Further, let $\mathfrak{p}$ be a prime of $F$ such that $G$ is split at $\mathfrak{p}$ and the local component $π_\mathfrak{p}$ of $π$ at $\mathfrak{p}$ is the Steinberg representation. Assuming that the representation is non-critical at $\mathfrak{p}$ we construct automorphic $\mathcal{L}$-invariants for the representation $π$. If the number field $F$ is totally real, we show that these automorphic $\mathcal{L}$-invariants agree with the Font…
In this paper we investigate the cohomological obstruction for the field of moduli of a $G$-cover to be a field of definition, in the case of local fields and covers with tame admissible reduction. This applies in particular to $p$-adic fields where $p$ does not divide the order of the group $G$. We give examples of $G$-covers with field of moduli $\QQ_p$ that cannot be defined over $\QQ_p$, for all primes $p>5$.
A cohomological translation of the Kaplansky radical for profinite groups
引入任意profinite群和素数p的Kaplansky根的上同调类比，证明对于域，群论版本的H-猜想等价于原猜想，并建立其对于多种域类的成立性。
Let $E$ be an elliptic curve defined over a number field $K$ with supersingular reduction at all primes of $K$ above $p$. If $K_{\infty}/K$ is a $\mathbb{Z}_p$-extension such that $E(K_{\infty})[p^{\infty}]$ is finite and $H^2(G_S(K_{\infty}), E[p^{\infty}])=0$, then we prove that the $Λ$-torsion subgroup of the Pontryagin dual of $\text{Sel}_{p^{\infty}}(E/K_{\infty})$ is pseudo-isomorphic to the Pontryagin dual of the fine Selmer group of $E$ over $K_{\infty}$. This is the Galois-cohomological analog of a flat-cohomological result of Wingberg.
Tame Galois Groups, Linking Numbers and Mildness
通过结合加权Zassenhaus滤过、图论方法和Koch型表示，证明对于任意有限驯服素数集S0（|S0|≥2），存在两个驯服素数组成的集合S1使得G_{S0∪S1}的上同调维数为2，从而改进了Labute的结果并回答了其关于温和群的问题。
Let $K$ be the fraction field of a two-dimensional henselian, excellent, equi-characteristic local domain. We prove a local-global principle for Galois cohomology with finite coefficients over $K$. We use classical machinery from étale cohomology theory, drawing upon an idea in Saito's work on two-dimensional local class field theory. This approach works equally well over the function field of a curve over an equi-characteristic henselian discrete valuation field, thereby giving a different proof of (a slightly generalized version of) a recent result of Harbater, Hartmann and Krashen. We also …
Euler systems for GSp(4)
本文利用 GL(2) x GL(2) 的 Eisenstein 类的推前，为与 GSp(4) 上同调尖点自守表示相关的 Galois 表示构造了 Euler 系统。
Massey products and the Iwasawa theory of fine Selmer groups
本文用Massey积重新表述Coates-Sujatha猜想，证明μ-不变量的消失可传递至邻域扩张，并建立形变环无阻碍性与精细Selmer群μ-不变量消失的等价关系。
We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2,3,5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological obstruction that measures if a given twist is birational to a moduli space of abelian varieties. This obstruction has implications for the rational points on these varieties. As a result, we see that all possible 3-level structures can be realized by abelian surfaces, whereas Kummer 3-level structures that group-theoretically may be admissible, may not be realizable…
2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves I
本文提出了一种通过 Galois 协同论方法确定超椭圆曲线 Jacobian 的 Bloch-Kato Selmer 组秩的方法，并利用该方法回答了关于有理点的问题。
Cohomological flatness over discrete valuation rings: numerical and logarithmic criteria
本文提出在离散赋值环上实现上同调平坦性的数值和对数准则，应用于曲线和具有良好减少的阿贝尔簇torsors，给出正则模型的对数光滑性条件。
We determine conditions for the invariance of the gonality under base extension, depending on the numeric invariants of the curve. More generally, we study the Galois descent of morphisms of curves to Brauer-Severi varieties, and also of rational normal scrolls.
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