Relative Class Number Research Articles

Arithmetic of Hecke L-functions of quadratic extensions of totally real fields

Deep work by Shintani in the 1970's describes Hecke L-functions associated to narrow ray class group characters of totally real fields F in terms of what are now known as Shintani zeta functions. However, for [F:Q]=n≥3, Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of F on R+n, so-called Shintani sets. These difficulties were recently resolved in independent work of Charollois, Dasgupta, and Greenberg and Diaz y Diaz and Friedman. For those narrow ray class group characters whose conductor is an inert rational prime in a totally real field F with narrow class number 1, we obtain a natural combinatorial description of these sets, allowing us to obtain a simple description of the associated Hecke L-functions. As a consequence, we generalize earlier work of Girstmair, Hirzebruch, and Zagier, that offer combinatorial class number formulas for imaginary quadratic fields, to real and imaginary quadratic extensions of totally real number fields F with narrow class number 1. For CM quadratic extensions of F, our work may be viewed as an effective affirmative answer to Hecke's Conjecture that the relative class number has an elementary arithmetic expression in terms of the relative discriminant.

Mean square value of $L$-functions at $s=1$ for non-primitive characters, Dedekind sumsand bounds on relative class numbers

Let $d_0$ be a given square-free integer. We give an explicit formula $M_{d_0}(p) =A(d_0)(1+B_{d_0}(p)/p)$ for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the non-primitive odd Dirichlet characters modulo $d_0p$ induced by the odd Dirichlet characters modulo an odd prime $p$. Here $d_0\mapsto A(d_0)$ is an explicit multiplicative arithmetic function and $B_{d_0}(f)$ is a twisted sum over the divisors $d$ of $d_0$ of Dedekind sums $s(h,df)$. To prove that $f\mapsto B_{d_0}(f)$ is $d_0$-periodic, we find a new and closed formula for the Dedekind sums $f\mapsto s(a+bf,c+df)$, for fixed integers $a$, $b$, $c$ and $d$. We deduce explicit upper bounds for relative class numbers of cyclotomic number fields.

Mean square values of L-functions over subgroups for nonprimitive characters, Dedekind sums and bounds on relative class numbers

AbstractAn explicit formula for the mean value of $\vert L(1,\chi )\vert ^2$ is known, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Lately, the authors obtained that the mean value of $\vert L(1,\chi )\vert ^2$ is asymptotic to $\pi ^2/6$ , where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\ \ \pmod {2d}$ which are trivial on a subgroup H of odd order d of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$ , provided that $d\ll \frac {\log p}{\log \log p}$ . Bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Here, for a given integer $d_0>1$ , we consider the same questions for the nonprimitive odd Dirichlet characters $\chi '$ modulo $d_0p$ induced by the odd primitive characters $\chi $ modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi ')\vert ^2$ is asymptotic to $\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$ , where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order $d\ll \frac {\log p}{\log \log p}$ . As a consequence, we improve the previous bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ . Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.

Unification of Chowla’s Problem and Maillet–Demyanenko Determinants

Chowla’s (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of L(1,χ)=∑n=1∞χ(n)n. On the other hand, we refer to determinant expressions for the (relative) class number of a cyclotomic field as the Maillet–Demyanenko determinants (MD). Our aim is to develop the theory of discrete Fourier transforms (DFT) with parity and to unify Chowla’s problem and Maillet–Demyanenko determinants (CPMD) as different-looking expressions of the relative class number via the Dedekind determinant and the base change formula.

The relative class number one problem for function fields, I

We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field mathbb {F}_2 and genus >1; and give a conjectural answer in the remaining cases. The conjecture will be resolved in subsequent papers.

On class number relations and intersections over function fields

The aim of this paper is to study class number relations over function fields and the intersections of Hirzebruch-Zagier type divisors on the Drinfeld-Stuhler modular surfaces. The main bridge is a particular "harmonic" theta series with nebentypus. Using the strong approximation theorem, the Fourier coefficients of this series are expressed in two ways; one comes from modified Hurwitz class numbers and another gives the intersection numbers in question.

Relative class numbers inside the $p$th cyclotomic field

For a prime number $p \equiv 3 \;\mbox{mod $4$}$, we write $p=2n\ell^f+1$ for some power $\ell^f$ of an odd prime number $\ell$ and an odd integer $n$ with $\ell \nmid n$. For $0 \leq t \leq f$, let $K_t$ be the imaginary subfield of ${\Bbb Q}(\zeta_p)$ of degree $2\ell^t$ and let $h_t^-$ be the relative class number of $K_t$. We show that for $n=1$ (resp. $n \geq 3$), a prime number $r$ does not divide the ratio $h_t^-/h_{t-1}^-$ when $r$ is a primitive root modulo $\ell^2$ and $r \geq \ell^{f-t}-1$ (resp. $r \geq (n-2)\ell^{f-t}+1$). In particular, for $n=1$ or $3$, the ratio $h_f^-/h_{f-1}^-$ at the top is not divisible by $r$ whenever $r$ is a primitive root modulo $\ell^2$. Further, we show that the $\ell$-part of $h_t^-/h_{t-1}^-$ stabilizes for ``large $t$ under some assumption.

Revisiting the average number of divisors of a quadratic polynomial

In this work, we employ a well known result due to Zagier to derive an asymptotic formula for the average number of divisors of a quadratic polynomial of the form $$x^{2}-bx+c$$ with $$b,\,c$$ integers. As simple consequences, we obtain formulas for computing the narrow class number of a quadratic field and relations between the narrow class number and a weighted class number for binary quadratic forms considered by McKee. In the end, we employ the class number relation to compute several examples of the weighted class number.

On a Zeta Function Associated to a Quadratic Order

For a positive integer N, we derive a Kronecker type limit formula for a Dedekind type zeta function $$\zeta _{\mathcal {O}_{K}(N)}(s;[\mathfrak {a}])$$ associated to a wide ideal class of a quadratic order of conductor N of a quadratic field K. As consequences, we establish a Chowla–Selberg type formula for a modular form for $$\Gamma _{0}(N)$$, give a proof to a classical class number relation of a quadratic field, and find an asymptotic formula for the average value of the number of representations by a primitive positive definite quadratic form of discriminant $$N^{2}d$$ with d fundamental. Furthermore, as an application, we obtain a formula for the Petersson norm of a theta function attached to a class character of the ideal group of conductor N of $$\mathcal {O}_{K}$$.

Differential principal factors and Pólya property of pure metacyclic fields

Barrucand and Cohn’s theory of principal factorizations in pure cubic fields [Formula: see text] and their Galois closures [Formula: see text] with [Formula: see text] types is generalized to pure quintic fields [Formula: see text] and pure metacyclic fields [Formula: see text] with [Formula: see text] possible types. The classification is based on the Galois cohomology of the unit group [Formula: see text], viewed as a module over the automorphism group [Formula: see text] of [Formula: see text] over the cyclotomic field [Formula: see text], by making use of theorems by Hasse and Iwasawa on the Herbrand quotient of the unit norm index [Formula: see text] by the number [Formula: see text] of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different [Formula: see text]. The precise structure of the group of differential principal factors is determined with the aid of kernels of norm homomorphisms and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units [Formula: see text]. Generalizing criteria for the Pólya property of Galois closures [Formula: see text] of pure cubic fields [Formula: see text] by Leriche and Zantema, we prove that pure metacyclic fields [Formula: see text] of only one type cannot be Pólya fields. All theoretical results are underpinned by extensive numerical verifications of the [Formula: see text] possible types and their statistical distribution in the range [Formula: see text] of [Formula: see text] normalized radicands.

Overpartition $M2$-rank differences, class number relations, and vector-valued mock Eisenstein series

We prove that the generating function of overpartition $M2$-rank differences is, up to coefficient signs, a component of the vector-valued mock Eisenstein series attached to a certain quadratic form. We use this to compute analogs of the class number relations for $M2$-rank differences. As applications we split the Kronecker-Hurwitz relation into its even and odd parts and calculate sums over Hurwitz class numbers of the form $\sum_{r \in \mathbb{Z}} H(n - 2r^2)$.

Tables of Pure Quintic Fields

By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to pure quintic fields and their pure metacyclic normal fields with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields N having normalized radicands in the range 2≤D3. Our classification is based on the Galois cohomology of the unit group UN, viewed as a module over the automorphism group Gal(N/K) of N over the cyclotomic field K=Q(ξ5), by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index (Uk:NN/K(UN)) by the number #(PN/K/PK) of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different DN/K. The precise structure of the F5-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units (UN:U0). The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.

An elementary proof of the Eichler–Selberg trace formula

Abstract We give a purely algebraic proof of the trace formula for Hecke operators on modular forms for the full modular group SL 2 ⁢ ( ℤ ) {\mathrm{SL}_{2}(\mathbb{Z})} , using the action of Hecke operators on the space of period polynomials. This approach, which can also be applied for congruence subgroups, is more elementary than the classical ones using kernel functions, and avoids the analytic difficulties inherent in the latter (especially in weight two). Our main result is an algebraic property of a special Hecke element that involves neither period polynomials nor modular forms, yet immediately implies both the trace formula and the classical Kronecker–Hurwitz class number relation. This key property can be seen as providing a bridge between the conjugacy classes and the right cosets contained in a given double coset of the modular group.

Indivisibility of relative class numbers of totally imaginary quadratic extensions and vanishing of these relative Iwasawa invariants

We study an indivisibility problem of the relative class numbers of CM fields. For prime p>3, Kohnen–Ono gave a lower bound of the number of the imaginary quadratic fields whose class numbers are prime to p by using modular forms of half-integral weight. We generalize their method to Hilbert modular forms and give a lower bound of the number of CM quadratic extensions K/F whose relative class numbers prime to p for totally real number field F which is Galois over Q and sufficiently large prime p. Combining the indivisibility result with the decomposition condition of p, we show a result on vanishing of relative Iwasawa invariants.

Note on the class number of the $p$th cyclotomic field, III

Let $p$ be a prime number of the form $p=2\ell^f+1$ with an odd prime number $\ell$, and $h_p^-$ the relative class number of the $p$th cyclotomic field $K=\mathbb{Q}(\zeta_p)$. When $f=1$, it is conjectured that $h_p^-$ is odd, and there are several results related to this conjecture. In this paper, we deal with the case $f \geq 2$. For $0 \leq t \leq f$, let $h_{p,t}^-$ denote the relative class number of the imaginary subfield $K_t$ of $K$ of degree $2\ell^t$ over $\mathbb{Q}$. We show that the ratio $h_{p,f}^-/h_{p,f-1}^-$ is not divisible by a prime number $r$ if $r$ is a primitive root modulo $\ell^2$. Further, when $r \leq 47$, we give some computational results on the ratio $h_{p,t}^-/h_{p,t-1}^-$ for $1 \leq t \leq f$. In the range of our computation, we find that the ratio is divisible by $r$ only in some exceptional cases.

Class number relations between pure fields and their Galois closures

Let p be an odd prime and let m be a positive p-th power free integer greater than one. Let K be an algebraic number field generated by the positive p-th root of m over the field of rational numbers, L its Galois closure in the field of complex numbers, and k the p-th cyclotomic field. We calculate a class number relation among them, which gives a sufficient condition for the class number of K to be a multiple of p. Moreover, we show that a rational prime q unequal to p is a divisor of the class number of L if and only if q divides that of k or K.

Note on the Class Number of the pth Cyclotomic Field, II

For an odd prime number p, let h−p denote the relative class number of the pth cyclotomic field . It is conjectured that h−p is odd when p is of the form p = 2ℓ + 1 with an odd prime number ℓ, and it is known that the conjecture is valid if 2 is a primitive root modulo ℓ. In this article, we handle a prime number p of the form p = 2e + 1ℓ + 1 with e ⩾ 1 and an odd prime number ℓ. For 1 ⩽ e ⩽ 4, we prove that h−p is odd whenever 2 is a primitive root modulo ℓ with the help of computer. Without the assumption on ℓ, we compute and find that, in the range e ⩾ 1, ℓ < 220 and p < 237, h−p is even only for four exceptional cases, where ℓ happened to be a Mersenne prime number. Further, computing for larger p with a Mersenne prime ℓ, we find one more exception.

Note on Bernoulli numbers associated to some Dirichlet character of prime conductor

We fix an integer \({n \geq 1}\) and a divisor m of n such that n/m is odd. Let p be a prime number of the form \({p=2n\ell+1}\) for some odd prime number \({\ell}\) with \({\ell \nmid m}\). Let \({S=pB_{1,2m\ell}}\) be the p times of the generalised Bernoulli number associated to an odd Dirichlet character of conductor p and order \({2m\ell}\), which is an algebraic integer of the \({2m\ell}\)th cyclotomic field. It is known that \({S \neq 0}\). More strongly, we show that when \({\ell}\) is sufficiently large, the trace of \({\zeta^{-1}S}\) to the \({2m}\)th cyclotomic field does not vanish for any\({\ell}\)th root \({\zeta}\) of unity. We also show a related result on indivisibility of relative class numbers.

A combinatorial refinement of the Kronecker-Hurwitz class number relation

We give a refinement of the Kronecker-Hurwitz class number relation, based on a tesselation of the Euclidean plane into semi-infinite triangles labeled by P S L 2 ( Z ) \mathrm {PSL}_2(\mathbb {Z}) that may be of independent interest.

Eichler–Selberg type identities for mixed mock modular forms

Using holomorphic projection, we work out a parametrization for all relations of products (resp. Rankin–Cohen brackets) of weight 32 mock modular forms with holomorphic shadow and weight 12 modular forms in the spirit of the Kronecker–Hurwitz class number relations. In particular we obtain new proofs for several class number relations among which some are classical, others are relatively new. We also obtain similar results for the mock theta functions.