Analytic Class Number Formula Research Articles

Tori over number fields and special values at $s=1$

We define a Weil-étale complex with compact support for duals (in the sense of the Bloch dualizing cycles complex \mathbb{Z}^c ) of a large class of \mathbb{Z} -constructible sheaves on an integral1-dimensional proper arithmetic scheme flat over Spec( \mathbb{Z} ). This complex can be thought of as computing Weil-étale homology. For those Z-constructible sheaves that are moreover tamely ramified, we define an “additive” complex which we think of as the Lie algebra of the dual of the \mathbb{Z} -constructible sheaf. The product of the determinants of the additive and Weil-étale complex is called the fundamental line. We prove a duality theorem which implies that the fundamental line has a natural trivialization, giving a multiplicative Euler characteristic. We attach a natural L -function to the dual of a \mathbb{Z} -constructible sheaf; up to a finite number of factors, this L -function is an Artin L -function at s = 1 . Our main theorem contains a vanishing order formula at s = 0 for the L -function and states that, in the tamely ramified case, the special value at s = 0 is given up to sign by the Euler characteristic. This generalizes the analytic class number formula for the special value at s = 1 of the Dedekind zeta function. In the function field case, this is a theorem of Geisser–Suzuki.

Riemann–Roch isometries in the non-compact orbifold setting

We generalize work of Deligne and Gillet–Soulé on a functorial Riemann–Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces of finite type \Gamma\backslash\mathbb H , for \Gamma\subset\mathrm {PSL}_{2}(\mathbb R) a Fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cusps and elliptic fixed points, and the original results of Deligne and Gillet–Soulé do not apply to this setting. Our theorem relates the determinant of cohomology of the trivial sheaf, with an explicit Quillen type metric defined in terms of the Selberg zeta function of \Gamma , to a metrized version of the tautological line bundle in the theory of moduli spaces of pointed orbicurves, and the self-intersection bundle of a suitable twist of the canonical sheaf \omega . We make use of surgery techniques through Mayer–Vietoris formulas for determinants of Laplacians, in order to reduce to explicit evaluations of such for model hyperbolic cusps and cones. We carry out these computations, which are of independent interest for theoretical physics, and can be adapted to other geometries. We go on to derive an arithmetic Riemann–Roch formula in the realm of Arakelov geometry, which applies in particular to integral models of modular curves with elliptic fixed points. This vastly extends previous work of the first author, whose deformation-theoretic methods were limited to torsion free Fuchsian groups. As an application, we treat in detail the case of the modular curve X(1) , and we show the interesting arithmetic content of the metrized \Psi line bundles. From this, we answer the longstanding question of evaluating the Selberg zeta special value Z'(1, \mathrm {PSL}_2 (\mathbb Z)) . The result is expressed in terms of logarithmic derivatives of Dirichlet L functions. In the analogy between Selberg zeta functions and Dedekind zeta functions of number fields, this formula can be seen as the analytic class number formula for Z(s, \mathrm {PSL}_2 (\mathbb Z)) . The methods developed in this article were conceived so that they afford several variants, such as the determinant of cohomology of a flat unitary vector bundle with finite monodromies at cusps. Our work finds its place in the program initiated by Burgos–Kramer–Kühn of extending arithmetic intersection theory to singular Hermitian vector bundles.

The analytic class number formula for 1‐dimensional affine schemes

We derive an analytic class number formula valid for an order in a product of S-integers in global fields, or equivalently for reduced finite-type affine schemes of pure dimension 1 over Z.

An application of the -adic analytic class number formula

We propose an algorithm to verify the$p$-part of the class number for a number field$K$, provided$K$is totally real and an abelian extension of the rational field$\mathbb{Q}$, and$p$is any prime. On fields of degree 4 or higher, this algorithm has been shown heuristically to be faster than classical algorithms that compute the entire class number, with improvement increasing with larger field degrees.

Symmetric powers do not stabilize

We discuss the stabilization of symmetric products Sym^n(X) of a smooth projective variety X in the Grothendieck ring of varieties. For smooth projective surfaces X with non-zero h^0(X, \omega_X), these products do not stabilize; we conditionally show that they do not stabilize in another related sense, in response to a question of R. Vakil and M. Wood. There are analogies between such stabilization, the Dold-Thom theorem, and the analytic class number formula. Finally, we discuss Hodge-theoretic obstructions to the stabilization of symmetric products, and provide evidence for these obstructions in terms of a relationship between the Newton polygon of a certain motivic zeta function associated to a curve, and its Hodge polygon.

The χ-part of the Analytic Class Number Formula, for Global Function Fields

Let $F/k$ be a finite abelian extension of global function fields, totally split at a distinguished place $\infty $ of $k$. We show that a complex Gras conjecture holds for Stark units, and we derive a refined analytic class number formula.

On Néron class groups of abelian varieties

Let F be a global field, let S ∞ be the set of archimedean primes of F and let S be any nonempty finite set of primes of F containing S ∞ . In this paper we study the Néron S -class group C A , F , S of an abelian variety A defined over F . In the well-known analogy that exists between the Birch and Swinnerton–Dyer conjecture for A over F and the analytic class number formula for the field F (in the number field case), the finite group C A , F , S ∞ (not the Tate–Shafarevich group of A ) is a natural analog of the ideal class group of F .

The discriminant of an algebraic torus

For a torus T defined over a global field K, we revisit an analytic class number formula obtained by Shyr in the 1970s as a generalization of Dirichletʼs class number formula. We prove a local–global presentation of the quasi-discriminant of T, which enters into this formula, in terms of cocharacters of T. This presentation can serve as a more natural definition of this invariant.

Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters

The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the "special values" of L-functions in terms of cohomological data. The main conjecture of Iwasawa theory describes a p-adic L-function in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of both conjectures (up to the prime 2) for L-functions attached to Dirichlet characters. We use the insight of Kato and B. Perrin-Riou that these two conjectures can be seen as incarnations of the same mathematical content. In particular, they imply each other. By a bootstrapping process using the theory of Euler systems and explicit reciprocity laws, both conjectures are reduced to the analytic class number formula. Technical problems with primes dividing the order of the character are avoided by using the correct cohomological formulation of the main conjecture.

Computation of Stark-Tamagawa units

Let K be a totally real number field and let l denote an odd prime number. We design an algorithm which computes strong numerical evidence for the validity of the Equivariant Tamagawa Number Conjecture for the Q[G]-equivariant motive h0(Spec(L)), where L/K is a cyclic extension of degree l and group G. This conjecture is a very deep refinement of the classical analytic class number formula. In the course of the algorithm, we compute a set of special units which must be considered as a generalization of the (conjecturally existing) Stark units associated to first order vanishing Dirichlet L-functions.

Another Look at the Index Formulas of Cyclotomic Number Theory

To the cyclotomic number fieldKgenerated by the roots of unity of orderfwe attach a Galois module which is a hybrid of the Stickelberger ideal and the group of circular units; this Galois module also admits interpretation as the universal punctured distribution of conductorf. We embed our Galois module in another naturally occurring Galois module, and prove that the index is exactly the class number ofK. By avoiding even-odd splittings and the analytic class number formula, we are able to avoid the consideration of {±1}-cohomology groups as well. Cohomological considerations become necessary only when making comparisons to the classical index formulas of Kummer, Hasse, Iwasawa, and Sinnott.

A Demjanenko Matrix for Abelian Fields of Prime Power Conductor

We introduce a generalized Demjanenko matrix associated with an arbitrary abelian field of odd prime power conductor, and exhibit direct connections between this matrix and both the relative class number and the cyclotomic units of the field. Beyond using the analytic class number formula, all arguments are elementary. Combining the two connections yields a simple proof that the relative class number is odd if and only if all the totally positive cyclotomic units are squares of cyclotomic units, which was known by results of Hasse and Garbanati. An interesting feature of our new class number formula is its expression as the determinant of a matrix with relatively small integer entries. Thus we also easily obtain a reasonable upper bound on the relative class number.

𝐿-functions and class numbers of imaginary quadratic fields and of quadratic extensions of an imaginary quadratic field

Starting from the analytic class number formula involving its L-function, we first give an expression for the class number of an imaginary quadratic field which, in the case of large discriminants, provides us with a much more powerful numerical technique than that of counting the number of reduced definite positive binary quadratic forms, as has been used by Buell in order to compute his class number tables. Then, using class field theory, we will construct a periodic character χ \chi , defined on the ring of integers of a field K that is a quadratic extension of a principal imaginary quadratic field k, such that the zeta function of K is the product of the zeta function of k and of the L-function L ( s , χ ) L(s,\chi ) . We will then determine an integral representation of this L-function that enables us to calculate the class number of K numerically, as soon as its regulator is known. It will also provide us with an upper bound for these class numbers, showing that Hua’s bound for the class numbers of imaginary and real quadratic fields is not the best that one could expect. We give statistical results concerning the class numbers of the first 50000 quadratic extensions of Q ( i ) {\mathbf {Q}}(i) with prime relative discriminant (and with K/Q a non-Galois quartic extension). Our analytic calculation improves the algebraic calculation used by Lakein in the same way as the analytic calculation of the class numbers of real quadratic fields made by Williams and Broere improved the algebraic calculation consisting in counting the number of cycles of reduced ideals. Finally, we give upper bounds for class numbers of K that is a quadratic extension of an imaginary quadratic field k which is no longer assumed to be of class number one.

On S-class number relations of algebraic tori in Galois extensions of global fields

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

On the class number of the lpth cyclotomic number field

The first factor of the class number of a cyclotomic number field can be obtainable by the analytic class number formula and there are some tables which show the decompositions of the first factors into primes. But, using just the analytic formula, we cannot tell what kinds of primes will appear as the factors of the class number of a given cyclotomic number field, except for those of the genus number, or the irregular primes. It is significant to find in advance the prime factors, particularly those prime to the degree of the field. For instance, in the table of the first factors we can pick out some pairs (l, p) of two odd primes l and p such that the class number of each lpth cyclotomic number field is divisible by l even if p 1 (mod l). If p ≡ (mod l) for l ≥ 5 or p ≡ 1 (mod 32) for l = 3, then it is easy from the outset to achieve our intention of finding the factor l using the genus number formula. Otherwise it seems to be difficult. We wish to make it clear algebraically why the class number has the prime factor l.

On the computation of the class number of an algebraic number field

It is shown how the analytic class number formula can be used to produce an algorithm which efficiently computes the class number h of an algebraic number field F. The method assumes the truth of the Generalized Riemann Hypothesis in order to estimate the residue of the Dedekind zeta function of F at s = 1 s = 1 sufficiently well that h can be determined unambiguously. Given the regulator R of F and a known divisor h ∗ {h^ \ast } of h, it is shown that this technique will produce the value of h in O ( | d F | 1 + ε / ( h ∗ R ) 2 ) O(|{d_F}{|^{1 + \varepsilon }}/{({h^ \ast }R)^2}) elementary operations, where d F {d_F} is the discriminant of F. Thus, if h > | d F | 1 / 8 h > |{d_F}{|^{1/8}} , then the complexity of computing h (with h ∗ = 1 {h^ \ast } = 1 ) is O ( | d F | 1 / 4 + ε ) O(|{d_F}{|^{1/4 + \varepsilon }}) .

Congruences dyadiques entre nombres de classes de corps quadratiques

We prove here a general mod 16 congruence between the class numbers of the two fields $$\mathbb{Q}(\sqrt d )$$ and $$\mathbb{Q}(\sqrt { - d} )$$ , for any square free positive integer d. Similarly we obtain congruences mod 64 relating the class numbers of the quadratic extensions corresponding to the integers d, 2d, -d, -2d. We use the p-adic analytic class number formula, which leads to study the difference of the values at 0 and 1 of the KUBOTA-LEOPOLDT's p-adic L-function (for p=2, and for the quadratic character associated to the real quadratic field). We use then IWASAWA's series, conveniently described for our purpose by Barsky. This gives a new proof for mod 16 congruences formally or simultaneously obtained by various authors. Moreover we get new results mod 16 and mod 64. Part A) presents the method and results, Part B) the proofs. All the details can be found in [4].

A note on Iwasawa invariants and the main conjecture

We fix a rational prime p, possibly 2, and a CM field K. Let A K ∞ − denote the minus component of the p-primary class group of K ∞, the basic Z p -extension of K. The Pontryagin dual ( A K ∞ −) p is a noetherian, torsion Z p [[ T]]-module whose characteristic polynomial we denote by f( T). Iwasawa's Main Conjecture relates the algebraically defined f( T) to an analytically defined power series F( T) given by a p-adic L-function. Using the analytic class number formula, we give evidence for it based on the Iwasawa invariants of f( T) and F( T). I would like to thank Benedict H. Gross for suggesting this approach.

Local Units Modulo Circular Units

In his paper, Some Modules in the Theory of Cyclotomic Fields [2], Iwasawa obtained the remarkable theorem that the quotient of the $p$-adic cyclotomic units by the completion of the circular units is isomorphic to the quotient of the group ring by the Stickleberger ideal. He then used this to deduce some interesting global results, the most striking of which is an explanation of the plus part of the analytic class number formula under the assumption that the class group at the first layer is cyclic, together with a regularity assumption. In this note, we will show how with the results in our paper, Division Values in Local Fields [1], it is now possible to give a substantially simpler proof of the above theorem. We also describe, briefly, how one can obtain various global results of Iwasawa from this Theorem, which are not included in either Lang’s [3], or Washington’s [4] books.

Local units modulo circular units

In his paper, Some Modules in the Theory of Cyclotomic Fields [2], Iwasawa obtained the remarkable theorem that the quotient of the p p -adic cyclotomic units by the completion of the circular units is isomorphic to the quotient of the group ring by the Stickleberger ideal. He then used this to deduce some interesting global results, the most striking of which is an explanation of the plus part of the analytic class number formula under the assumption that the class group at the first layer is cyclic, together with a regularity assumption. In this note, we will show how with the results in our paper, Division Values in Local Fields [1], it is now possible to give a substantially simpler proof of the above theorem. We also describe, briefly, how one can obtain various global results of Iwasawa from this Theorem, which are not included in either Lang’s [3], or Washington’s [4] books.