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 numerator are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra.
We also present tools developed in SAGE for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus.
Particular attention is given to the case when $[K:K^+]=2$ and $K$ is a CM-field. This case corresponds to tori in $\mathrm{GSp}_{2n}$, and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.
We prove that any integer power of $2$ can be realized as the Tamagawa number of a torus attached to a CM algebra considered by Guo Sheu Yu and Liang Yang Yu. Such a result is obtained by Liang Yang Yu under assuming "a generalized Landau conjecture". The main contribution of this paper is to remove the above assumption.
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.
Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a rational prime and let r be a non-positive integer. By examining the structure of the p-adic group ring Z_p[G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h^0(Spec(L)(r),Z[G])). The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic K-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a wide class of interesting extension…
The equivariant Tamagawa number conjecture (hereinafter called the eTNC) predicts close relationships between algebraic and analytic aspects of motives. In this paper, we prove a lot of new cases of the minus component of the eTNC for $\mathbb{G}_m$ and for CM abelian extensions. One of the main results states that the $p$-component of the eTNC is true when there exists at least one $p$-adic prime that is tamely ramified. The fundamental strategy is inspired by the work of Dasgupta and Kakde on the Brumer-Stark conjecture.
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…
We show that all of the nice behavior for Tamagawa numbers, Tate-Shafarevich sets, and other arithmetic invariants of pseudo-reductive groups over global function fields proved in \cite{rospred} fails in general for non-commutative unipotent groups. We also give some positive results which show that Tamagawa numbers do exhibit some reasonable behavior for arbitrary connected linear algebraic groups over global function fields.
In this article, we classify the characters associated to algebraic points on Shimura curves of $Γ_0(p)$-type, and over a quadratic field we show that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number $p$. This is an analogue of the study of rational points or points over a quadratic field on the modular curve $X_0(p)$ by Mazur and one of the author (Momose). We also apply the result to a finiteness conjecture on abelian varieties with constrained prime power torsion by Rasmussen-Tamagawa.
Let $T$ be an algebraic torus defined over a global field $K$. For any $K$-torsor $X$ under $T$, we relate the Brauer group of $X$ to the adéle class group of $T$ as well as to the Shafarevich Tate group of $T$.
A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper we generalize Chow's theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori.
Let $T$ be an algebraic torus defined over a global field $K$ and split over a finite cyclic extension. In this paper, we determine the Herbrand quotient of the adéle class group of $T$. Our result can be seen as an extension of the so-called first inequality of global class field theory.
We extend the results by R.P. Langlands on representations of (connected) abelian algebraic groups. This is done by considering characters into any divisible abelian topological group. With this we can then prove what is known as the abelian case of the $p$-adic Langlands program.
In this paper, we prove that a smooth hyperbolic projective curve over a finite field can be recovered from L-functions associated to the Hilbert class field of the curve and its constant field extensions. As a consequence, we give a new proof of a result of Mochizuki and Tamagawa that two such curves with isomorphic fundamental groups are themselves isomorphic.
We give a formula for the class number of an arbitrary CM algebraic torus over $\mathbb{Q}$. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.
The main emphasis will be on height upper bounds in the algebraic torus G^{n}_{m}. By height we will mean the absolute logarithmic Weil height. Section 3.2 contains a precise definition of this and other more general height functions. The first appendix gives a short overview of known results in the abelian case. The second appendix contains a few height bounds in Shimura varieties.
We give a criterion for two l-adic Galois representations of an algebraic number field to be isomorphic when restricted to a decomposition group, in terms of the global representations mod l. This is applied to prove a generalization of a conjecture of Rasmussen-Tamagawa under a semistablity condition, extending some results of one of the authors. It is also applied to prove a congruence result on the Fourier coefficients of modular forms.
Given an elliptic curve E over Q with complex multiplication having good reduction at 2, we investigate the 2-adic valuation of the algebraic part of the L-value at 1 for a family of quadratic twists. In particular, we prove a lower bound for this valuation in terms of the Tamagawa number in a form predicted by the conjecture of Birch and Swinnerton-Dyer.
Assuming specific instances of two general conjectures in arithmetic algebraic geometry (bijectivity of $p$-adic regulator maps, injectivity of $p$-adic Abel-Jacobi maps), we prove several cases of the $p$-part of the Tamagawa number conjecture ($p$-TNC) of Bloch-Kato and Fontaine-Perrin-Riou for (homological) motives of modular forms of even weight $\geq4$ in analytic rank $1$. More precisely, we prove our results for a large class of newforms $f$ and prime numbers $p$ that are ordinary for $f$ and such that the weight of $f$ is congruent to $2$ modulo $2(p-1)$. Inspired by work of W. Zhang i…
We obtain bounds on the number of maximal torsion cosets for algebraic subvarieties of n-tori, defined over the rationals, using model theoretic methods.
Given an octonion algebra over a field k, its automorphism group G is an algebraic semisimple k-group of type G_2. We study the maximal tori of G in terms of the algebra C.
The study of the birational properties of algebraic $k$-tori began in the sixties and seventies with work of Voskresenkii, Endo, Miyata, Colliot-Thélène and Sansuc. There was particular interest in determining the rationality of a given algebraic $k$-tori. As rationality problems for algebraic varieties are in general difficult, it is natural to consider relaxed notions such as stable rationality, or even retract rationality. Work of the above authors and later Saltman in the eighties determined necessary and sufficient conditions to determine when an algebraic torus is stably rational, respec…
If T is an algebraic torus defined over a discretely valued field K with perfect residue field k, we relate the K-cohomology of T to the k-cohomology of certain objects associated to T. When k has cohomological dimension <= 1, our results have a particularly simple form and yield, more generally, isomorphisms between the abelian K-cohomology of a reductive group G over K and the k-cohomology of a certain quotient of the algebraic fundamental group of G.
This is an expanded version of my talk given at the International Conference ``Algebra and Number Theory'' dedicated to the 80th anniversary of V. E. Voskresenskii, which was held at the Samara State University in May 2007. The goal is to give an overview of results of V. E. Voskresenskii on arithmetic and birational properties of algebraic tori which culminated in his monograph "Algebraic Tori" published in Russian 30 years ago. I shall try to put these results and ideas into somehow broader context and also to give a brief digest of the relevant activity related to the period after the Engli…
Let $p$ be an odd prime number, $D_p$ be the dihedral group of order $2p$, $h_p$ and $h^+_p$ be the class numbers of $\bm{Q}(ζ_p)$ and $\bm{Q}(ζ_p+ ζ_p^{-1})$ respectively. Theorem. $h_p^+=1$ if and only if, for any field $k$ admitting a $D_p$-extension, all the algebraic $D_p$-tori over $k$ are stably rational. A similar result for $h_p=1$ and $C_p$-tori is valid also.
We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.
Let k be a finite field, a global field or a local non-archimedean field. Let H_1 and H_2 be two split, connected, semisimple algebraic groups defined over k. We prove that if H_1 and H_2 share the same set of maximal k-tori up to k-isomorphism, then the Weyl groups W(H_1) and W(H_2) are isomorphic, and hence the algebraic groups modulo their centers are isomorphic except for a switch of a certain number of factors of type B_n and C_n. We remark that due to a recent result of Philippe Gille, above result holds for fields which admit arbitrary cyclic extensions.
We calculate the formal group law which represents the completion of the Néron model of an algebraic torus over the rationals that splits in a tamely ramified abelian extension. As a tools in the proof, we define and give criterions to compute the Weil restriction of a formal group law and the analog of the fixed part of a formal group law with respect to the action of a (finite) group.
We consider the covering map $π:\mathbb{C}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq \mathbb{C}^n$ we describe the topological closure of $π(X)$ in $\mathbb T$. We obtain a similar description when $\mathbb{T}$ is a real torus and $X\subseteq \mathbb{R}^n$ is a set definable in an o-minimal structure over the reals.
Let $k$ be a field and $T$ be an algebraic $k$-torus. In 1969, over a global field $k$, Voskresenskii proved that there exists an exact sequence $0\to A(T)\to H^1(k,{\rm Pic}\,\overline{X})^\vee\to Sha(T)\to 0$ where $A(T)$ is the kernel of the weak approximation of $T$, $Sha(T)$ is the Shafarevich-Tate group of $T$, $X$ is a smooth $k$-compactification of $T$, $\overline{X}=X\times_k\overline{k}$, ${\rm Pic}\,\overline{X}$ is the Picard group of $\overline{X}$ and $\vee$ stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus $T=R^{(1)}_{K/k}(G_m)$ o…
We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_2n over a global field, which completes some of the main results from the paper "Weakly commensurable arithmetic groups and isospectral locally symmetric spaces" (Pub. Math. IHES, 2009) by Prasad and Rapinchuk and gives a new proof of a result from another paper by the same authors.
Langlands has described the irreducible admissible representations of $T$, when $T$ is the group of points of an algebraic torus over a local field. Also, Langlands described the automorphic representations of $T_{\mathbb A}$ when $T_{\mathbb A}$ is the group of adelic points of an algebraic torus over a global field $F$. We describe irreducible (in the local setting) and automorphic (in the global setting) $ε$-genuine representations for covers of tori, also known as metaplectic tori, which arise from a framework of Brylinski and Deligne. In particular, our results include a description of sp…
Let $K/k$ be a finite Galois extension and $π= \fn{Gal}(K/k)$. An algebraic torus $T$ defined over $k$ is called a $π$-torus if $T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n$ for some integer $n$. The set of all algebraic $π$-tori defined over $k$ under the stably isomorphism form a semigroup, denoted by $T(π)$. We will give a complete proof of the following theorem due to Endo and Miyata \cite{EM5}. Theorem. Let $π$ be a finite group. Then $T(π)\simeq C(Ω_{\bm{Z}π})$ where $Ω_{\bm{Z}π}$ is a maximal $\bm{Z}$-order in $\bm{Q}π$ containing $\bm{Z}π$ and $C(Ω_{\bm{Z}π})$ is the local…
In this paper we introduce a method to obtain algebraic information using arithmetic one in the study of tori and their principal homogeneous spaces. In particular, using some results of the authors with Tingyu Lee, we determine the unramified Brauer groups of some norm one tori, and their torsors.
We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We ma…
