Laurent Berger
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。
We introduce the "sharp" (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the "sharp de Rham realization" by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. We thus provide "one-dimensional sharp de Rham cohomology" of algebraic varieties.
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…
Let L be a finite extension of Q_p and d a positive integer. A conjecture, due to C. Breuil and P. Schneider, says that the existence of invariant norms on certain locally algebraic representations of GL_{d+1}(L) should be equivalent to the existence of certain (d+1)-dimensional de Rham representations of Gal(\bar{L}/L). We prove the easy direction of this conjecture: the existence of invariant norms implies the existence of admissible filtrations, by generalizing an idea of M.Emerton.
By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the …
We introduce and study the universal norm distribution in this paper, which generalizes the concepts of universal ordinary distribution and the universal Euler system. We study the Anderson type resolution of the universal norm distribution and then use this resolution to study the group cohomology of the universal norm distribution.
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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