We extend the classical duality results of Poitou and Tate for finite discrete Galois modules over local and global fields (local duality, nine-term exact sequence, etc.) to all affine commutative group schemes of finite type, building on the recent work of Česnavičius extending these results to all finite commutative group schemes. We concentrate mainly on the more difficult function field setting, giving some remarks about the number field case along the way.
In this paper, we establish a Poitou-Tate's global duality for totally positive Galois cohomology. We illustrate this result in the case of the twisted module "à la Tate" $\mathbb{Z}_{2}(i)$, $i$ integer.
我们给出了局部类域论的一个完备的证明，通过Lubin-Tate理论和Hasse-Arf定理，完善了Iwasawa的书中的论点。在修正后的版本中，（i）包括了正特征情形，（ii）简化了基变换的证明，（iii）简化并加强了Artin映射的构造，使得我们可以从完全极大非分歧扩张上的任意Lubin-Tate群开始. (to出现在Annales de la Faculte des Sciences de Toulouse，17-2（2008））
Pellarin introduced the deformation of multiple zeta values of Thakur as elements over Tate algebras. In this paper, we relate these values to a certain coordinate of a higher dimensional Drinfeld module over Tate algebras which we will introduce. Moreover, we define multiple polylogarithms in our setting and represent deformation of multiple zeta values as a linear combination of multiple polylogarithms.
We give an algorithm that takes a smooth hypersurface over a number field and computes a $p$-adic approximation of the obstruction map on the Tate classes of a finite reduction. This gives an upper bound on the "middle Picard number" of the hypersurface. The improvement over existing methods is that our method relies only on a single prime reduction and gives the possibility of cutting down on the dimension of Tate classes by two or more. The obstruction map comes from $p$-adic variational Hodge conjecture and we rely on the recent advancement by Bloch-Esnault-Kerz to interpret our bounds.
Variétés abéliennes sur les corps de fonctions de courbes sur des corps locaux supérieurs
设$k$是高维局部域，$X$是$k$上的光滑射影几何整的曲线.令$K$为$X$的函数字段。我们通过上同调类定义Tate-Shafarevich群的阿贝尔簇局部平凡的$K$的每个完成来自一个封闭点的$X$。证明了$k$上阿贝尔簇的局部对偶定理，以及$K$上交换簇的Tate-Shafarevich群的全局对偶定理. Soient $k$ un corps local supérieur et $X$ une courbe projective lisse géométriquement intègre de corps de functions $K$ .在考虑$K$的完备性由$X$的一个端点给出的上同调类的情况下，对Tate-Shafarevich的一个Abélienne变量群进行了定义。关于$k$上阿贝林种二元性区域理论的建立，也是关于$K$上阿贝林种Tate-Shafarevich群体的二元性全球理论。
We show that the $\ell$-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$-adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher co-dimension are given.
For each prime $p$, we show that there exist geometrically simple abelian varieties $A/\mathbb Q$ with non-trivial $p$-torsion in their Tate-Shafarevich groups. Specifically, for any prime $N\equiv 1 \pmod{p}$, let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point $P$ of order $p$, and let $B = A_f/\langle P \rangle$. Then the number of positive integers $d \leq X$, such that the Tate-Shafarevich group of $\hat B_d$ has non-trivial $p$-torsion, is $\gg X/\log X$, where $\hat B_d$ is the dual of the $d$-th quadratic twist of $B$. We prove this more generally for abelian varieties o…
On Spectral Sequences for Iwasawa Adjoints à la Jannsen for Families
在\citenospec{MR1097615}中，建立了$\Z_p$上（不一定是交换的）Iwasawa代数（主要是$p$ -adic李群）上的（整体和局部）Iwasawa模的几个谱序列，这些谱序列对于在算术应用中确定这类模的某些性质是非常有用的.上述结果的稍微推广可以在\citenospec{MR2333680}（阿贝尔群和更一般的系数环）、\citenospec{MR1924402}（不一定阿贝尔群的乘积，但具有$\Z_p$系数）和\citenospec{MR3084561}中找到。不幸的是，一些詹森的谱序列的家庭表示的系数（本地）岩泽上同调仍然失踪。我们解释和遵循的哲学，所有这些谱序列的后果或类似物的地方上同调和对偶à la Grothendieck（和泰特的对偶群）。
We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of $2$-descent and $3$-descent in a certain family of Mordell curves $E_k \colon y^2 = x^3 + k$. As a by-product of our methods, we show that, for every $r \geq 0$, a positive proportion of curves $E_k$ have Tate--Shafarevich group with $3$-rank at least $r$.
