De Rham representations and universal norms
Laurent Berger
我们计算模的普遍规范的德拉姆p-adic表示。计算使用的理论（φ，伽玛）-模块（Cherbonnier-Colmez的互易公式）和微分方程附加到德拉姆表示。
We compute the module of universal norms for a de Rham p-adic representation. The computation uses the theory of (phi,Gamma)-modules (Cherbonnier-Colmez's reciprocity formula) and the differential equation attached to a de Rham representation.
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Universal norms and de Rham representations
本文证明Iwasawa上同调中通用范数计算从分圆扩张可推广至任意包含它的p-adic扩张，并给出适用于Lubin-Tate扩张的扭曲变体。
Universal norms of partially de Rham $B$-pairs and perfectness in Lubin--Tate Iwasawa theory
该研究探讨B-对的部分de Rham Selmer群在交换p进李扩张下的行为，对解析Lubin–Tate (φ,Γ)-模证明Ψ=1特征空间的余可容性，建立其李代数余不变量与对应Selmer群射影极限的比较结果。
Sharp de Rham realization
我们介绍了“尖锐”（普遍）的扩展1动机（添加剂的因素和扭转）的领域的特征为零。我们通过传递到李代数来定义“夏普德拉姆实现”。在复数上，我们证明了形式Hodge结构范畴中的一个（sharp de Rham）比较定理.对于一个自由的1-动机及其Cartier对偶，我们得到了一个关于它们的尖锐扩张的正则联系，从而产生了一个关于尖锐实现的完美配对。因此，我们提供了“一维sharp de Rham上同调”的代数簇。
This thesis establishes a geometric approach to the de Rham realization of the polylogarithm. As a central result we construct the logarithm sheaves of rational abelian schemes in terms of the birigidified Poincaré bundle with universal integrable connection on the product of the abelian scheme and the universal vectorial extension of its dual. This is achieved essentially by restricting the mentioned data of the Poincaré bundle along the infinitesimal neighborhoods of the zero section of the universal extension. We also clarify how these constructions naturally express within the language of …
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 …
We study the module of universal norms associated with a de Rham $p$-adic Galois representation in a perfectoid field extension. In particular, we compute precisely this module when the Hodge-Tate weights of a representation are greater than or equal to $0$. This generalises a result by Coates and Greenberg for Abelian varieties, and partially answers a question of theirs. Our method relies on the classification of vector bundles over the Fargues-Fontaine curve.
In [BS07] Breuil and Schneider formulated a conjecture on the equivalence of the existence of invariant norms on certain $p$-adically locally algebraic representations of $GL_n(F)$ and the existence of certain de-Rham representations of $Gal(\bar{F}/F)$, where $F$ is a finite extension of $\mathbb{Q}_p$. In [Bre03b, DI13] Breuil and de Ieso proved that in the case $n = 2$ and under some restrictions, the existence of certain admissible filtrations on the $ϕ$-module associated to the two-dimensional de-Rham representation of $Gal(\bar{F}/F)$ implies the existence of invariant norms on the corre…
In this dissertation, we discuss mainly the corresponding geometric and representation theoretic aspects of relative $p$-adic Hodge theory and $p$-adic motives. To be more precise, we study the corresponding analytic geometry of the corresponding spaces over and attached to period rings in the relative $p$-adic Hodge theory, including derived topological de Rham complexes and derived topological logarithmic de Rham complexes after Bhatt, Gabber, Guo and Illusie which is in some sense equivalent to the derived prismatic cohomology of Bhatt-Scholze as shown in the work of Li-Liu, $\mathcal{O}\ma…
Normes invariantes et existence de filtrations admissibles
设L是Q_p的有限扩张，d是正整数.一个猜想，由于C. Breuil和P. Schneider认为GL_{d+1}（L）的某些局部代数表示上的不变范数的存在性应该等价于Gal（\bar{L} /L）的某些（d+1）维de Rham表示的存在性。我们证明了这个猜想的简单方向：不变范数的存在意味着存在的容许过滤，通过推广的思想M. Emerton。
First steps towards $p$-adic Langlands functoriality
根据Colmez和Fontaine的理论，局部域的伽罗瓦群的de Rham表示大致对应于在基础向量空间上配备容许滤子的Weil-Deligne群的表示。使用经典的局部朗兰兹对应的修改，我们与任何一对组成的Weil-Deligne群表示和过滤（允许或不允许）的一个特定的局部代数表示的一般线性群的类型。我们宣传的猜想，这对来自德拉姆表示当且仅当相应的局部代数表示进行不变的规范。在晶体的情况下，Weil-Deligne群表示是非分歧的，相关的局部代数表示可以使用经典的佐竹同构来研究。通过将后者扩展到Hecke代数的一个特定的范数完备化，我们证明了一个不变范数的存在意味着我们的对确实来自于一个结晶表示。我们还表明，通过使用Tannakian范畴的形式主义，后者的事实是兼容的经典unramified朗兰兹函，因此推广到任意分裂还原群。
On the universal norm distribution
本文引入并研究了泛范数分布，它是泛常分布和泛欧拉系概念的推广。研究了泛范分布的Anderson型分解，并利用此分解研究了泛范分布的群上同调。
We investigate the group of universal norms attached to the cyclotomic Z {\ell}-tower of a totally real number field in connection with Grenberg's conjecture on Iwasawa invariants of such a field.
We begin a systematic investigation of universal norms for $p$-adic representations in higher rank Iwasawa theory. After establishing the basic properties of the module of higher rank universal norms we construct an Iwasawa-theoretic pairing that is relevant to this setting. This allows us, for example, to refine the classical Iwasawa Main Conjecture for cyclotomic fields, and also to give applications to various well-known conjectures in arithmetic concerning Iwasawa invariants and leading terms of $L$-functions.
For any positive integer M we show that there are infinitely many real quadratic fields that do not admit M-ary universal quadratic forms (without any restriction on the parity of their cross coefficients).
For an odd prime p and a number field F containing a primitive p-th root of unity, we describe the Kummer radical A_F of the first layers of all the Z_p-extensions of F in terms of universal norms of p-units along the cyclotomic tower of F . We also study "twisted" radicals related to AF .
On a characterisation of perfectoid fields by Iwasawa theory
本文通过证明与特定 de Rham Galois 表示相关的泛范数模的消失可刻画完备化为 perfectoid 域的 $p$ 进数域代数扩张，推广了 Coates-Greenberg 和 Bondarko 的相关结果。
Universal quadratic forms, small norms and traces in families of number fields
本文研究多类全实数域上通用二次型的秩，通过刻画不可分解整数与余微分中小迹元素，并估计小范数主理想数量，给出秩的良好界。
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