Burns' equivariant Tamagawa invariant T Omega^{loc}(N/\Q,1) for some quaternion fields

The local equivariant Tamagawa number conjecture for almost abelian extensions

We prove the local equivariant Tamagawa number conjecture for the motive of an abelian extension of an imaginary quadratic field with the action of the Galois group ring for all split primes p not equal to 2 or 3 at all negative integer values s.

The local Tamagawa number conjecture for Hecke characters, II

In this paper we prove the weak local Tamagawa number conjecture for the remaining non-critical cases for the motives associated to Hecke characters $ψ_θ:\mathbb{A}_K\to K^*$ of the author's previous paper, where $K$ is an imaginary quadratic field with $cl(K)=1$, under certain restrictions which originate mainly from the Iwasawa theory of imaginary quadratic fields.

A cohomological Tamagawa number formula

For smooth linear group schemes over $\bbZ$ we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori the cohomological and the motivic Tamagawa measures coincide, which reproves the Bloch-Kato conjecture for motives associated to tori.

On the relative K-group in the ETNC

We consider the Burns-Flach formulation of the equivariant Tamagawa number conjecture (ETNC). In their setup, a Tamagawa number is an element of a relative K-group. We show that this relative group agrees with an ordinary K-group, namely of the category of locally compact topological modules over the order. Its virtual objects are an equivariant Haar measure in a precise sense. We expect that all relative K-groups in the ETNC will have analoguous interpretations. At present, we need to restrict to regular orders, e.g. hereditary.

特征 $p$ 下可构造层的 Weil-étale 上同调与等变玉川数猜想

Weil-étale cohomology and the equivariant Tamagawa number conjecture for constructible sheaves in characteristic $p$

本文针对有限域上簇的可构造 étale 层，利用 Weil-étale 上同调证明了相关非交换 L-函数在负整数处的特殊值公式，从而给出等变玉川数猜想的几何类比并推广了若干已有结果。

On Bloch-Kato's Tamagawa number conjecture for Hecke characters of imaginary quadratic number fields

This is essentially the author's thesis submited to The University of Chicago (May 1997). I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain Hecke characters. I study the exponential map and local Tamagawa number for all odd primes (both ordinary and supersingular), using Kato's explicit reciprocity law for one dimensional Lubin-Tate formal group. I also study p-part of Shafarevich-Tate group for motives associated to Hecke characters, using Rubin's Main Conjecture in Iwasawa theory. Interestingly a congruence property between p-adic periods of elliptic curve and weig…

Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties

We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In view of the Raynaud parametrisation, this translates into a purely algebraic problem concerning the number of $H$-invariant points on a quotient of $C_n$-lattices $Λ/eΛ'$ for varying subgroups $H$ of $C_n$ and integers $e$. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions (corresponding to varying $e$) and one that computes Tamagawa numbers up to rational squares in general extensions. As an application…

塔玛加瓦数与椭圆曲线的正秩

Tamagawa numbers and positive rank of elliptic curves

本文探讨无需寻找无限阶点或计算L函数即可预测椭圆曲线正秩的方法，通过塔玛加瓦数乘积值预测正秩，并证明其与奇偶性猜想的关系。

模 $q - 1$ 下的塔马加瓦因子

Hearing Tamagawa Factors modulo $q - 1$

该研究将阿贝尔簇的塔马加瓦因子与 $p$ 进拉普拉斯积分算子的小波特征值联系，通过局部 $L$ 因子、Serre 不变量的相关结果，证明了二者模 $q - 1$ 的同余关系。

BSD猜想中的局部因子：一个统一的统计视角

Local Factors in the BSD Conjecture: A Unified Statistical View

本文为短Weierstrass形式的椭圆曲线推导了编码Tamagawa积的Euler积生成函数，计算了四个统计量的极限分布与矩界，并证明多数曲线Tamagawa积平凡，结果由AxiomProver在Lean中自动形式化。

On the local Tamagawa number conjecture for Tate motives over tamely ramified fields

The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic $L$-functions with the functional equation. The local conjecture was proven for Tate motives over finite unramified extensions $K/\mathbb{Q}_p$ by Bloch and Kato. We use the theory of $(ϕ, Γ_K)$-modules and a reciprocity law due to Cherbonnier and Colmez to provide a new proof in the case of unramified extensions, and to prove the conjecture for the motive $\mathbb{Q}_p(2)$ over certain tamely ramified extensions.

A building-theoretic approach to relative Tamagawa numbers of quasi-split semisimple groups over global function fields

Let G be a semisimple quasi-split group defined over a global function field. We express the relative Tamagawa number of G in terms of local data including the number of types of special vertices in one orbit of the Bruhat-Tits building over some completion with respect to a place and the class number of G at that place.

Rasmussen-Tamagawa 猜想的基变换版本

A Base Change Version of Rasmussen-Tamagawa Conjecture

本文证明了Shafarevich猜想的一致版本，并作为推论，针对数域上处处具有潜在好约化的特定阿贝尔簇类证明了Rasmussen-Tamagawa猜想。

On the relative K-group in the ETNC, Part II

In a previous paper we showed, under some assumptions, that the relative K-group in the Burns-Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a K-group of locally compact equivariant modules. This viewpoint, as well as the usual one, come with generator-relator presentations (due to Bass-Swan and Nenashev) and in this paper we provide an explicit map.

Explicit Tamagawa numbers for certain algebraic tori over number fields

Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of the corresponding Galois group of prime order $p$, we build an algebraic torus over $k$ whose rational points are elements of $K^\times$ sent to $k^\times$ via the norm map $N_{K/K^+}$. The goal is to compute the Tamagawa number of that torus explicitly via Ono's formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when $K/k$ is Galois. Partial results including the numerat…

Equivariant epsilon constant conjectures for weakly ramified extensions

We study the local epsilon constant conjecture as formulated by Breuning. This conjecture fits into the general framework of the equivariant Tamagawa number conjecture (ETNC) and should be interpreted as a consequence of the expected compatibility of the ETNC with the functional equation of Artin-L-functions. Let K / Q_p be unramified. Under some mild technical assumption we prove Breuning's conjecture for weakly ramified abelian extensions N / K with cyclic ramification group. As a consequence of Breuning's local-global principle we obtain the validity of the global epsilon constant conjectur…

The refined Tamagawa number conjectures for $\mathrm{GL}_2$

Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that the associated $p$-adic Galois representation has large image. We establish a new and refined "Birch and Swinnerton-Dyer type" formula for Bloch-Kato Selmer groups of the central critical twist of $f$ via Kolyvagin derivatives of $L$-values instead of complex analytic or $p$-adic variation of $L$-values only under the Iwasawa main conjecture localized at the augmentation ideal. Our formula determines the exact rank and module structure of the Selmer groups and is insensitive to weight, the local behavior of $f$ at $p$, and analytic…

Abelian L-Functions at s=1 and Explicit Reciprocity for Rubin-Stark Elements

Given an abelian, CM extension K of any totally real number field k, we consider two conjectures `of Stark type'. The `Integrality Conjecture' concerns the image of a p-adic map `\mathfrak{s}_{K/k,S}' determined by the minus-part of the S-truncated equivariant L-function for K/k at s=1. It is connected to the Equivariant Tamagawa Number Conjecture of Burns and Flach. The `Congruence Conjecture' says that \mathfrak{s}_{K/k,S} gives an explicit reciprocity law for the element predicted by the corresponding Rubin-Stark Conjecture for K^+/k. We study the general properties of these conjectures and…

全实域 Iwasawa 主猜想的整等变细化

An Integral Equivariant Refinement of the Iwasawa Main Conjecture for Totally Real Fields

本文在全实数域的 Abel、CM 扩张下，通过计算非空 $T$ 对应的 $X_S^{T,-}$ 的 Fitting ideal，给出了 Wiles 全实域 Iwasawa 主猜想的一个整等变细化，并扩展了多项已有猜想与结果。

Ritter-Weiss 模的等变岩泽理论及其应用

Equivariant Iwasawa Theory for Ritter-Weiss Modules and Applications

本文针对全实数域的有限阿贝尔 CM 扩张构造 Ritter-Weiss 模的岩泽对应物，证明其投射维数、二次呈现及由等变 p 进 L 函数生成的主 Fitting 理想，并应用于岩泽模与 ETNC 负 p 部分。

Abstract $\ell$-adic $1$-motives and Tate's class

In a previous paper we constructed a new class of Iwasawa modules as $\ell$--adic realizations of what we called abstract $\ell$--adic $1$--motives in the number field setting. We proved in loc. cit. that the new Iwasawa modules satisfy an equivariant main conjecture. In this paper we link the new modules to the $\ell$--adified Tate canonical class, defined by Tate in 1960 and give an explicit construction of (the minus part of) $\ell$--adic Tate sequences for any Galois CM extension $K/k$ of an arbitrary totally real number field $k$. These explicit constructions are significant and useful in…

A representative of $RΓ(N,T)$ for higher dimensional twists of $\mathbb Z_p^r(1)$

Let $N/K$ be a Galois extension of $p$-adic number fields and let $V$ be a $p$-adic representation of the absolute Galois group $G_K$ of $K$. The equivariant local $ε$-constant conjecture $C_{EP}^{na}(N/K, V)$ is related to the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin $L$-functions and it can be formulated as the vanishing of a certain element $R_{N/K}$ in $K_0(\mathbb Z_p[G],\mathbb Q_p^c[G])$. One of the main technical difficulties in the computation of $R_{N/K}$ arises from the so-called cohomological term $C_{N/K}$, which requires th…

Dirichlet $L$-series at $s=0$ and the scarcity of Euler systems

We study Euler systems for $\mathbb{G}_m$ over a number field $k$. Motivated by a distribution-theoretic idea of Coleman, we formulate a conjecture regarding the existence of such systems that is elementary to state and yet strictly finer than Kato's equivariant Tamagawa number conjecture for Dirichlet $L$-series at $s=0$. To investigate the conjecture, we develop an abstract theory of `Euler limits' and, in particular, prove the existence of canonical `restriction' and `localisation' sequences in this theory. By using this approach we obtain a variety of new results, ranging from a proof, mod…

Representations of affine Kac-Moody groups over local and global fields: a survey of some recent results

Let G be a reductive algebraic group over a local field K or a global field F. It is well know that there exists a non-trivial and interesting representation theory of the group G(K) as well as the theory of automorphic forms on the corresponding adelic group. The purpose of this paper is to give a survey of some recent constructions and results, which show that there should exist an analog of the above theories in the case when G is replaced by the corresponding affine Kac-Moody group (which is essentially built from the formal loop group G((t)) of G). Specifically we discuss the following to…

Regulator constants and the parity conjecture

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/Q is abelian and K^\infty is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of Gal(K^\infty/Q). We also give analogous results when K/Q is non-abelian, the base field is not Q and E is replaced by an abelian variety. The heart of the paper is a study of relations b…

On Tate--Shafarevich groups of one-dimensional families of commutative group schemes over number fields

Given a smooth geometrically connected curve $C$ over a field $k$ and a smooth commutative group scheme $G$ of finite type over the function field $K$ of $C$ we study the Tate--Shafarevich groups given by elements of $H^1(K,G)$ locally trivial at completions of $K$ associated with closed points of $C$. When $G$ comes from a $k$-group scheme and $k$ is a number field (or $k$ is a finitely generated field and $C$ has a $k$-point) we prove that the Tate--Shafarevich group is finite, generalizing a result of Saïdi and Tamagawa for abelian varieties. We also give examples of nontrivial Tate--Shafar…

Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p

We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore localisation by using Oliver-Taylor's theory upon integral logarithms. This calculation reduces the existence of the non-commutative p-adic zeta function to certain congruence conditions among abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne-Ribet'…

Operations on locally free classgroups

Let G be a finite group and K a number field. We show that the "k-th Adams operation" defined by Cassou-Nogues and Taylor on the locally free class group Cl(Z_K G) is a symmetric power operation, if k is coprime to the order of G. Using the equivariant Adams-Riemann-Roch theorem, we furthermore give a geometric interpretation of a formula established by Burns and Chinburg for these operations.