On 2-dimensional 2-adic Galois representations of local and global fields

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.

Let K be a field, Br(K) its Brauer group. If L/K is a field extension, then the relative Brauer group Br(L/K) is the kernel of the restriction map resL/K : Br(K)→ Br(L). Relative Brauer groups have been studied by Fein and Schacher. Every subgroup of Br(K) is a relative Brauer group Br(L/K) for some extension L/K, and the question arises as to which subgroups of Br(K) are algebraic relative Brauer groups, i.e. of the form Br(L/K) with L/K an algebraic extension. For example if L/K is a finite extension of number fields, then Br(L/K) is infinite, so no finite subgroup of Br(K) is an algebraic relative Brauer group. In [1] the question was raised as to whether or not the n-torsion subgroup Brn(K) of the Brauer group Br(K) of a field K is an algebraic relative Brauer group. For example, if K is a (p-adic) local field, then Br(K) ∼= Q/Z, so Brn(K) is an algebraic relative Brauer group for all n. A counterexample was given in [1] for n = 2 and K a formal power series field over a local field. For global fields K, the problem is a purely arithmetic one, because of the fundamental local-global description of the Brauer group of a global field. In particular, for a Galois extension L/K of global fields, if the local degree of L/K at every finite prime is equal to n, and is equal to 2 at the real primes for n even, then Br(L/K) = Brn(K). In [1], it was proved that Brn(Q) is an algebraic relative Brauer group for all squarefree n. In [2], the arithmetic criterion above was verified for any number field K Galois over Q and any n prime to the class number of K, so in particular, Brn(Q) is an algebraic relative Brauer group for all n. In [3], Popescu proved that for a global function field K of characteristic p, the arithmetic criterion holds for n prime to the order of the non-p part of the Picard group of K. In this paper we settle the question completely, by verifying the arithmetic criterion for all n and all global fields K. In particular, the n-torsion subgroup of the Brauer group of K is an algebraic relative Brauer group for all n and all global fields K. The proof, an extension of the ideas in [2], reduces to the case n a prime power `. We first carry out the proof for number fields K. The proof for the function field case when ` 6= char(K) is essentially the same as the proof in the number field case. The proof for ` = char(K) appears in [3].

On the Tits indices of absolutely almost simple algebraic groups over local and global fields

Статья посвящена памяти Олега Николаевича Введенского (1937 – 1981 гг.). О. Н. Введенский был учеником академика И. Р. Шафаревича. Исследования О. Н. и полученные им результаты связаны с двойственностью в эллиптических кривых и с соответствующими когомологиями Галуа над локальными полями, со спариванием Шафаревича-Тэйта и с другими спариваниями, с локальной и квази-локальной теорией полей классов эллиптических кривых, с теорией абелевых многообразий размерности больше 1, с теорией коммутативных формальных групп над локальными полями. Представлены как результаты, полученные О. Н. Введенским, так и новые избранные результаты, развивающие исследования в направлениях фундаментальных групп схем, главных однородных пространств (торсеров) и двойственности. Первая часть статьи, представлення здесь, является введением как в результаты, полученные О. Н. Введенским в направлении двойственности абелевых многообразий и формальных групп, так и в новые избранные результаты, развивающие исследования в направлениях фундаментальных групп схем, главных однородных пространств (торсеров) и двойственности. Во Введении приведены предварительные сведения и представлено содержание статьи. В первом разделе дан краткий обзор избранных результатов по теории алгебраических, квазиалгебраические и проалгебраические группы и групповых схем. Далее, в разделе 2 преставлены избранные результаты по фундаментальным группам алгебраических многообразий, по фундаментальным группам схем, а в разделе 3 - избранные результаты о главных однородных пространствах (торсерах), развивающие исследования О. Н. и других авторов. Термин торсер мы используем как перевод на русский язык в редакции И.Р. Шафаревича английского термина torsor. В разделе 4 даны сведения о двойственности, а в разделе 5 представлены результаты О. Н. по арифметической теории формальных групп и их развитие. Результаты, этого раздела, представленные над локальными и квази-локальными полями K, над их кольцами целых, и над их полями вычетов k, связанны (1) с формальной структурой абелевых многообразий, (2) с коммутативными формальными группами, (3) с соответствующими гомоморфизмами и изогениями. В статье алгебраические многообразия, абелевы схемы и коммутативные формальные групповые схемы определены, как правило, над локальными и квази-локальными полями, над их кольцами целых, и над их полями вычетов. Но кратко рассматриваются эти объектыи и над глобальными полями, так как О. Н. интересовала тематика алгебраических многообразий над глобальными полями и он проводил соответствующие исследования. Предполагается, что характеристика полей вычетов больше 3, если не оговаривается иное.Я признателен В.Н. Чубарикову за предложение опубликовать статью в сборнике.Особая признательность Н. М. Добровольскому за помощь и поддержку в процессе подготовки статьи к печати.

For arithmetical schemes $X$, K. Kato \[J. Reine Angew. Math. 366, 142--183 (1986; Zbl 0576.12012)] introduced certain complexes $C^{r,s}(X)$ of Gersten-Bloch-Ogus type whose components involve Galois cohomology groups of all the residue fields of $X$. For specific $(r,s)$, he stated some conjectures on their homology generalizing the fundamental isomorphisms and exact sequences for Brauer groups of local and global fields. We prove some of these conjectures in small degrees and give applications to the class field theory of smooth projective varieties over local fields, and finiteness questions for some motivic cohomology groups over local and global fields.

In this article we explain how the results in our previous article on ‘algebraic Hecke characters and compatible systems of mod$p$Galois representations over global fields’ allow one to attach a Hecke character to every cuspidal Drinfeld modular eigenform from its associated crystal that was constructed in earlier work of the author. On the technical side, we prove along the way a number of results on endomorphism rings of${\it\tau}$-sheaves and crystals. These are needed to exhibit the close relation between Hecke operators as endomorphisms of crystals on the one side and Frobenius automorphisms acting on étale sheaves associated to crystals on the other. We also present some partial results on the ramification of Hecke characters associated to Drinfeld modular eigenforms. An important phenomenon absent from the case of classical modular forms is that ramification can also result from places of modular curves of good but non-ordinary reduction. In an appendix, jointly with Centeleghe we prove some basic results on$p$-adic Galois representations attached to$\text{GL}_{2}$-type cuspidal automorphic forms over global fields of characteristic $p$.

An iterative construction of irreducible polynomials reducible modulo every prime

It is widely accepted that magnetic reconnection releases a large amount of energy during solar flares. Studies of reconnection usually assume that the length scale over which the global (macroscopic) magnetic field reverses is identical to the thickness of the reconnection site. However, in spatially extended high-Lundquist number plasmas such as the solar corona, this scenario is untenable; the reconnection site is microscopic and embedded inside the macroscopic current set up by global fields. We use numerical simulations and scaling arguments to show that embedded effects on reconnection could have a profound influence on energy storage before a flare. From large-scale high-Lundquist number resistive magnetohydrodynamics simulations of reconnection with a diffusion region on a much smaller scale than the macroscopic current sheet, we find that the generation of secondary islands is governed by the local magnetic field immediately upstream of the diffusion region rather than the (potentially much larger) global field. This diminishes the production of secondary islands and leads to a thicker diffusion region than those predicted using the global field strength. Such considerations are crucial for understanding the onset of solar eruptions and how energy accumulates before such eruptions. We argue that if reconnection with secondary islands is fast, the energy storage times before an eruption are too small to explain observations. If reconnection with secondary islands remains slow, embedded effects cause the diffusion region to begin far wider than kinetic scales, so energy storage before a flare can occur while collisional (Sweet-Parker) reconnection with secondary islands proceeds.

For a connected reductive group $G$ over a local or global field $K$, we define a diamond (or power) operation $$ \begin{align*} &(\xi,n)\mapsto \xi^{\Diamond n}\,\colon\, \mathrm{H}^1\kern -0.8pt(K,G)\times{\mathbb Z}\to \mathrm{H}^1\kern -0.8pt(K,G)\end{align*} $$ of raising to power $n$ in the Galois cohomology pointed set. This operation is new when $K$ is a number field. We show that this power operation has many good properties. When $G$ is a torus, the set $\mathrm{H}^{1}\kern -0.8pt(K,G)$ has a natural group structure, and $\xi ^{\Diamond n}$ then coincides with the $n$-th power of $\xi $ in this group. On the other hand, we show that a power operation on $\mathrm{H}^{1}\kern -0.8pt(K,G)$, functorial in $G$, which we define over local and global fields, cannot be defined for an arbitrary field $K$. Our proof of this assertion relies on the results of Appendix B written by Philippe Gille. Using the power operation, for a cohomology class $\xi $ in $\mathrm{H}^{1}\kern -0.8pt(K,G)$ over local or global field, we define the period $\operatorname{per}(\xi )$ to be the least integer $n\geqslant 1$ such that $\xi ^{\Diamond n}=1$. We define the index $\operatorname{ind}(\xi )$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $\xi $. The period and index of a cohomology class generalize the period and index a central simple algebra over $K$. For any connected reductive group $G$ over a local or global field $K$, we show that $\operatorname{per}(\xi )$ divides $\operatorname{ind}(\xi )$ and that $\operatorname{ind}(\xi )$ may be strictly greater than $\operatorname{per}(\xi )$, but they always have the same prime factors.

Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

The coefficient space is a kind of resolution of singularities of the universal flat deformation space for a given Galois representation of some local field. It parametrizes (in some sense) the finite flat models for the Galois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.

Lifting [formula omitted] representations to characteristics p2

ABSTRACTTwo fields are Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic. Witt equivalent fields can be understood to be fields having the same quadratic form theory. The behavior of finite fields, local fields, global fields, as well as function fields of curves defined over Archimedean local fields under Witt equivalence is well understood. Numbers of classes of Witt equivalent fields with finite numbers of square classes are also known in some cases. Witt equivalence of general function fields over global fields was studied in the earlier work [13] by the authors and applied to study Witt equivalence of function fields of curves over global fields. In this paper, we extend these results to local case, i.e. we discuss Witt equivalence of function fields of curves over local fields. As an application, we show that, modulo some additional assumptions, Witt equivalence of two such function fields implies Witt equivalence of underlying local fields.

On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.