Tamagawa Number Research Articles

Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves

We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over Q \mathbb {Q} of analytic ranks 0 0 and 1 1 . We apply our techniques to show that if E E is a non-CM elliptic curve over Q \mathbb {Q} of conductor ≤ 1000 \leq 1000 and rank 0 0 or 1 1 , then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the L L -series is true for E E , up to odd primes that divide either Tamagawa numbers of E E or the degree of some rational cyclic isogeny with domain E E . Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 0 or 1 1 , this completely verifies the full conjecture for these curves up to the primes excluded above.

On the density of unnormalized Tamagawa numbers of orthogonal groups, II

This is the second part of a series of three papers. In this series of papers, we determine the density of unnormalized Tamagawa numbers of projective special orthogonal groups defined over a fixed number field. In this part, we determine local constants at infinite places which appear in the density theorem and prove the density theorem when the number of variables $n$ is odd. For that purpose we make a Dirichlet series whose coefficients are unnormalized Tamagawa numbers of projective special orthogonal groups and prove that it is rather simple when $n$ is odd.

Regulator constants and the parity conjecture

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p ∞-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 ∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of $\mathop{\mathrm{Gal}}(K^{\infty}/{\mathbb{Q}})$ . 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 between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.

Self-duality of Selmer groups

AbstractThe first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the $\Q_p$G-representation naturally associated to the p∞-Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.

Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers

Abstract.LetXbe a smooth projective geometrically connected curve over a finite field with function fieldK. Let Gbe a connected semisimple group scheme overX. Under certain hypotheses we prove the equality of two numbers associated with G. The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of Gtorsors onX. Our results are most useful for studying connected components as much is known about Tamagawa numbers.

Iwasawa Theory and Motivic L-functions

We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture.

Multiples of integral points on elliptic curves

If E is a minimal elliptic curve defined over Z , we obtain a bound C, depending only on the global Tamagawa number of E, such that for any point P ∈ E ( Q ) , nP is integral for at most one value of n > C . As a corollary, we show that if E / Q is a fixed elliptic curve, then for all twists E ′ of E of sufficient height, and all torsion-free, rank-one subgroups Γ ⊆ E ′ ( Q ) , Γ contains at most 6 integral points. Explicit computations for congruent number curves are included.

Tamagawa defect of Euler systems

As remarked by Mazur and Rubin [B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (799) (2004)] one does not expect the Kolyvagin system obtained from an Euler system for a p-adic Galois representation T to be primitive (in the sense of the above mentioned reference) if p divides a Tamagawa number at a prime ℓ ≠ p ; thus fails to compute the correct size of the relevant Selmer module. In this paper we obtain a lower bound for the size of the cokernel of the Euler system to Kolyvagin system map in terms of the local Tamagawa numbers of T, refining a result of [B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (799) (2004)]. We show how this partially accounts for the missing Tamagawa factors in Kato's calculations with his Euler system.

Global divisibility of Heegner points and Tamagawa numbers

AbstractWe improve Kolyvagin’s upper bound on the order of the p-primary part of the Shafarevich–Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely that predicted by the Birch and Swinnerton-Dyer conjectural formula.

On the Density of Unnormalized Tamagawa Numbers of Orthogonal Groups I

(1.1) G = GL(1)×GL(n), V = SymAff. We regard V as the space of quadratic forms in n ≥ 1 variables. In these papers we mainly consider the case n ≥ 3, but need to consider all positive integers n ∈ Z>0 for technical reasons. Let V ss k = {x ∈ Vk | detx = 0} . For x ∈ V ss k , we define the special orthogonal group SO(x) in the well-known manner. We define PSO(x) to be SO(x) modulo its center, and call it the projective special orthogonal group of x. Then

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Atiyah and Bott used equivariant Morse theory applied to the Yang–Mills functional to calculate the Betti numbers of moduli spaces of vector bundles over a Riemann surface, rederiving inductive formulae obtained from an arithmetic approach which involved the Tamagawa number of SLn. This article attempts to survey and extend our understanding of this link between Yang–Mills theory and Tamagawa numbers, and to explain how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth projective curve over ℂ can be adapted to the setting of 𝔸1-homotopy theory to study the motivic cohomology of these moduli spaces over an algebraically closed field.

Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture

AbstractWe assume the validity of the equivariant Tamagawa number conjecture for a certain motive attached to an abelian extension K/k of number fields, and we calculate the Fitting ideal of the dual of clK− as a Galois module, under mild extra hypotheses on K/k. This builds on concepts and results of Tate, Burns, Ritter and Weiss. If k is the field of rational numbers, our results are unconditional.

Explicit units and the equivariant Tamagawa number conjecture, II

We show that a special case of the equivariant Tamagawa number conjecture implies explicit restrictions on the structure of ideal class groups that are in general much finer than those which can be obtained by using the method of Thaine.

On the motivic class of the stack of bundles

Let G be a split connected semisimple group over a field. We give a conjectural formula for the motivic class of the stack of G-bundles over a curve C, in terms of special values of the motivic zeta function of C. The formula is true if C = P 1 or G = SL n . If k = C , upon applying the Poincaré or called the Serre characteristic by some authors the formula reduces to results of Teleman and Atiyah–Bott on the gauge group. If k = F q , upon applying the counting measure, it reduces to the fact that the Tamagawa number of G over the function field of C is | π 1 ( G ) | .

On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SLn

AbstractWe compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SLn is one.

On congruence relations between the fundamental units of biquadratic fields

Given any distinct prime numbers p , q , and r satisfying certain simple congruence conditions, we display a congruence relation between the fundamental units for the biquadratic field K = Q ( p r , q ) , modulo a certain prime ideal of O K . This congruence in particular implies the validity of the equivariant Tamagawa number conjecture formulated by Burns and Flach for the pair ( h 0 ( Spec K ) , Z [ Gal ( K / Q ) ] ) .

On the equivariant Tamagawa number conjecture for [formula omitted]-extensions of number fields

For a finite Galois extension K / k of number fields, with Galois group G, the equivariant Tamagawa number conjecture of Burns and Flach relates the leading coefficients of Artin L-functions to an element of K 0 ( Z [ G ] , R ) arising from the Tate sequence. This conjecture is known to be true for certain non-abelian Galois extensions over Q with Galois group being the dihedral or quaternion group. In this article, we shall verify the conjecture for an A 4 -extension over Q , by explicitly constructing the Tate sequence using Chinburg's methods.

Volumes of symmetric spaces via lattice points

We show how to use elementary methods to compute the volume of Slk R/ Slk Z. We compute the volumes of certain unbounded regions in Euclidean space by counting lattice points and then appeal to the machinery of Dirichlet series to get estimates of the growth rate of the number of lattice points appearing in the region as the lattice spacing decreases. We also present a proof of the closely related result that the Tamagawa number is 1.

On the equivariant main conjecture of Iwasawa theory

Recently, D. Burns and C. Greither (Invent. Math., 2003) deduced an equivariant version of the main conjecture for abelian number fields. This was the key to their proof of the equivariant Tamagawa number conjecture. A. Huber and G. Kings (Duke Math. J., 2003) also use a variant of the Iwasawa main conjecture to prove the Tamagawa number conjecture for Dirichlet motives. We use the result of the second pair of authors and the Theorem of Ferrero-Washington to reprove the equivariant main conjecture in a slightly more general form. The main idea of the proof is essentially the same as in the paper of D. Burns and C. Greither, but we can replace complicated considerations of Iwasawa $mu$-invariants by a considerably simpler argument.

On the equivariant Tamagawa number conjecture for Abelian extensions of a quadratic imaginary field

Let k be a quadratic imaginary field, p a prime which splits in k/\mathbb Q and does not divide the class number h_k of k . Let L denote a finite abelian extension of k and let K be a subextension of L/k . In this article we prove the p -part of the Equivariant Tamagawa Number Conjecture for the pair (h^0(\operatorname{Spec}(L)), \mathbb Z[\operatorname{Gal}(L/K)]) .