Class Numbers Of Quadratic Fields Research Articles

Congruences for odd class numbers of quadratic fields with odd discriminant

For any distinct two primes \(p_1\equiv p_2\equiv 3\) \((\text {mod }4)\), let \(h(-p_1)\), \(h(-p_2)\) and \(h(p_1p_2)\) be the class numbers of the quadratic fields \(\mathbb {Q}(\sqrt{-p_1})\), \(\mathbb {Q}(\sqrt{-p_2})\) and \(\mathbb {Q}(\sqrt{p_1p_2})\), respectively. Let \(\omega _{p_1p_2}:=(1+\sqrt{p_1p_2})/2\) and let \(\varPsi (\omega _{p_1p_2})\) be the Hirzebruch sum of \(\omega _{p_1p_2}\). We show that \(h(-p_1)h(-p_2)\equiv h(p_1p_2)\varPsi (\omega _{p_1p_2})/n\) \((\text {mod }8)\), where \(n=6\) (respectively, \(n=2\)) if \(min {p1, p2} > 3\) (respectively, otherwise). We also consider the real quadratic order with conductor 2 in \(\mathbb {Q}(\sqrt{p_1p_2})\).

Explicit Isogenies of Prime Degree Over Quadratic Fields

Abstract Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This builds on work of David, Larson-Vaintrob, and Momose. Combining this algorithm with work of Bruin–Najman, Özman–Siksek, and most recently Box, we determine the above set of primes for the three quadratic fields, ${\mathbb {Q}}(\sqrt {-10})$, ${\mathbb {Q}}(\sqrt {5})$, and ${\mathbb {Q}}(\sqrt {7})$, providing the first such examples after Mazur’s 1978 determination for $K = {\mathbb {Q}}$. The termination of the algorithm relies on the Generalised Riemann Hypothesis.

Diophantine approximation with prime restriction in real quadratic number fields

The distribution of $$\alpha p$$ modulo one, where p runs over the rational primes and $$\alpha $$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $$\nu >0$$ one can establish the infinitude of primes p satisfying $$||\alpha p||\le p^{-\nu }$$ . The latest record in this regard is Kaisa Matomäki’s landmark result $$\nu =1/3-\varepsilon $$ which presents the limit of currently known technology. Recently, Glyn Harman, and, jointly, Marc Technau and the first-named author, investigated the same problem in the context of imaginary quadratic fields. Glyn Harman obtained an analog for $$\mathbb {Q}(i)$$ of his result in the context of $$\mathbb {Q}$$ , which yields an exponent of $$\nu =7/22$$ . Marc Technau and the first-named author produced an analogue of Bob Vaughan’s result $$\nu =1/4-\varepsilon $$ for all imaginary quadratic number fields of class number 1. In the present article, we establish an analog of the last-mentioned result for real quadratic fields of class number 1 under a certain Diophantine restriction. This setting involves the additional complication of an infinite group of units in the ring of integers. Moreover, although the basic sieve approach remains the same (we use an ideal version of Harman’s sieve), the problem takes a different flavor since it becomes truly 2-dimensional. We reduce it eventually to a counting problem which is, interestingly, related to roots of quadratic congruences. To approximate them, we use an approach by Christopher Hooley based on the theory of binary quadratic forms.

Class Numbers of Quadratic Fields

We present a survey of some recent results regarding the class numbers of quadratic fields

On fundamental units of real quadratic fields of class number 1

In this paper, we give a nontrivial lower bound for the fundamental unit of norm $$-1$$ of a real quadratic field of class number 1.

Torsion of elliptic curves over imaginary quadratic fields of class number 1

Filip Najman examined the possibilities for the group of torsion points on elliptic curves over the number fields $\mathbb {Q}(\sqrt {-1})$ and $\mathbb {Q}(\sqrt {-3})$ in Najman (2011, 2010). In this article, we study the possible torsion structures of elliptic curves over the remaining imaginary quadratic fields of class numbers $1$, i.e., over the fields $\mathbb {Q}(\sqrt {-2})$, $\mathbb {Q}(\sqrt {-7})$, $\mathbb {Q}(\sqrt {-11})$, $\mathbb {Q}(\sqrt {-19})$, $\mathbb {Q}(\sqrt {-43})$, $\mathbb {Q}(\sqrt {-67})$ and $\mathbb {Q}(\sqrt {-163})$.

On elliptic curves of prime power conductor over imaginary quadratic fields with class number 1

The main result of this paper is to extend from $\Q$ to each of the nine imaginary quadratic fields of class number one a result of Serre (1987) and Mestre-Oesterl\'e (1989), namely that if $E$ is an elliptic curve of prime conductor then either $E$ or a $2$-, $3$- or $5$-isogenous curve has prime discriminant. For four of the nine fields, the theorem holds with no change, while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime. The proof is conditional in two ways: first that the curves are modular, so are associated to suitable Bianchi newforms; and second that a certain level-lowering conjecture holds for Bianchi newforms. We also classify all elliptic curves of prime power conductor and non-trivial torsion over each of the nine fields: in the case of $2$-torsion, we find that such curves either have CM or with a small finite number of exceptions arise from a family analogous to the Setzer-Neumann family over $\Q$.

Spectral Decomposition Formulas for Zeta Functions of Imaginary Quadratic Fields of Class Number One

The squared absolute value of the Dedekind zeta functions of imaginary quadratic fields of class number one on the critical line is expressed in terms of averaged values associated with the spectrum of the automorphic Laplacian with respect to the full modular group.

Spectral Decomposition Formulas for Zeta-Functions of Imaginary Quadratic Fields of Class Number One

Spectral Decomposition Formulas for Zeta-Functions of Imaginary Quadratic Fields of Class Number One

Addendum to “Imaginary quadratic fields of class number 2 and Levy–Hendy quadratics” [J. Number Theory 178 (2017) 40–46

We add a paper of S. Louboutin who was essentially the first to prove Theorem 1.3 of my paper mentioned in the title, based on Hendy's original paper.

A Further Stroll into the Eisenstein Primes

If one imagines the Eisenstein primes to be lily pads in the pond of complex numbers, could a frog hop from the origin to infinity with jumps of bounded size? If the frog was confined to the real number line, the answer is no. Good heuristic arguments exist for it not being possible in the complex plane, but there is still no formal proof for this conjecture.If the frog's journey terminates for a given hop size, it implies that a prime-free “moat” greater than the hop size completely surrounds the origin.In the earlier MONTHLY article “A Stroll Through the Gaussian Primes,” Ellen Gethner, Stan Wagon, and Brian Wick [3] explored this problem in the Gaussian primes and by computational methods proved the existence of a -moat. Additionally they proved that prime-free neighborhoods of arbitrary radius k surrounding a Gaussian prime exist.In their concluding remarks, Gethner et al. note that “Similar questions about walks to infinity may be asked for the finitely many imaginary quadratic fields of class number 1.”This paper takes up that challenge by examining similar questions about walks to infinity in the ring of Eisenstein integers. We prove that prime-free neighborhoods of arbitrary radius k surrounding an imaginary quadratic prime exist by directly generalizing Gethner's proof regarding prime-free neighborhoods surrounding a Gaussian prime. By computational means we establish the existence of a -moat in the Eisenstein primes and provide an estimate for the bounds of any k-moat. A subtle difference between the structure of Eisenstein and Gaussian integers results in our estimate being significantly different from our initial expectation.

Imaginary quadratic fields of class number 2 and Levy–Hendy quadratics

Let d=pq≡3(mod 4) with prime p,q and q<p. We prove a precise bound in Hendy's theorem on imaginary quadratic fields Q(−d) with class number h(−d)=2 which is a full analogue of Frobenius–Rabinowitsch theorem (the case of class number one). Namely it is shown that h(−d)=2 if and only if the Levy–Hendy quadratic Ld(x)=qx2+qx+ℓ0 with ℓ0=p+q4 takes only prime values for integers x in the interval 0≤x≤ℓ0−2. We discuss also two related conjectures in connection with generalizing a result of Chowla et al. ([1]) on the class number and the least prime quadratic residue.

A note on prime valued polynomials quadratic fields of class number one

A note on prime valued polynomials quadratic fields of class number one

A genus two curve related to the class number one problem

We give another solution to the class number one problem via a correspondence of imaginary quadratic fields of class number h(−d)=1 to integral points on a genus 2 curve K3 defined by the equation 8x8−32x6y+40x4y2+64x5−16x2y3−128x3y+y4+48xy2+96x2−32y−24=0. In fact one can find all rational points on K3. Besides the integral points related to the integral points on Heegner's elliptic curve K2:y2=2x(x3−1), the curve K3 has four more (rational) points: (−3,6), (1,2), (−917,6289), (−15579,424866241). Our approach is a further step of exploiting Weber's conjecture (Birch's theorem after [1]) on special values of Schläfli–Weber modular functions. It involves also a genus 9 curve K1 and its sub-covering K1→Ks studied independently by Heegner and Stark.

On the divisibility of class numbers of quadratic fields and the solvability of diophantine equations

In this paper we provide criteria for the insolvability of the Diophantine equation $x^2+D=y^n$. This result is then used to determine the class number of the quadratic field $\mathbb{Q}(\sqrt{-D})$. We also determine some criteria for the divisibility of the class number of the quadratic field $\mathbb{Q}(\sqrt{-D})$ and this result is then used to discuss the solvability of the Diophantine equation $x^2+D=y^n$.

CLASS NUMBER FORMULAS FOR CERTAIN BICYCLIC BIQUADRATIC NUMBER FIELDS AS FINITE SUMS INVOLVING JACOBI ELLIPTIC FUNCTIONS

We give formulas for the class numbers of bicyclic biquadratic number fields containing an imaginary quadratic field of class number one. The class number is expressed as a finite sum in terms of the basic Jacobi elliptic functions, playing a similar role as the trigonometric sine in Dirichlet classical class number formula.

Continued Fractions and Gauss' Class Number Problem for Real Quadratic Fields

The main purpose of this article is to present a numerical data which shows relations between real quadratic fields of class number 1 and a mysterious behavior of the period of simple continued fraction expansion of certain quadratic irrationals. For that purpose, we define a class number, a fundamental unit,a discriminant and a Yokoi invariant for a non-square positive integer, and then see that a generalization of theorems of Siegel and of Yokoi holds. These and a theorem of Friesen and Halter-Koch imply several interesting conjectures for solving Gauss' class number problem for real quadratic fields.

The moments and statistical distribution of class numbers of quadratic fields with prime discriminant

We establish asymptotic formulas for all the moments of class numbers of quadratic fields with prime discriminant. Further, we study the statistical distribution of these class numbers, introducing a certain random Euler product.

Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem

Let Y ns + ( n ) be the open non-cuspidal locus of the modular curve X ns + ( n ) associated to the normalizer of a non-split Cartan subgroup of level n. As Serre pointed out, an imaginary quadratic field of class number one gives rise to an integral point on Y ns + ( n ) for suitably chosen n. In this note, we give a genus formula for the modular curves X ns + ( n ) and we give three new solutions to the class number one problem using the modular curves X ns + ( n ) for n = 16 , 20 , 21 . These are the only such modular curves of genus ⩽2 that had not yet been exploited.

Randomness of the square root of 2 and the Giant Leap, Part 1

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let α be an arbitrary real root of a quadratic equation with integer coefficients; say, α = \( \sqrt 2 \). Given any rational number 0 < x < 1 (say, x = 1/2) and any positive integer n, we count the number of elements of the sequence α, 2α, 3α, …, nα modulo 1 that fall into the subinterval [0, x]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected number” nx from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ n ≤ N. Depending on α and x, we may need an extra additive correction of constant times logarithm of N; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm of N. If N is large, the distribution of this renormalized counting number, as n runs in 1 ≤ n ≤ N, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as N tends to infinity. This is the main result of the paper (see Theorem 1.1). The proof is rather complicated and long; it has many interesting detours and byproducts. For example, the exact determination of the key constant factors (in the additive and multiplicative norming), which depend on α and x, requires surprisingly deep algebraic tools such as Dedeking sums, the class number of quadratic fields, and generalized class number formulas. The crucial property of a quadratic irrational is the periodicity of its continued fraction. Periodicity means self-similarity, which leads us to Markov chains: our basic probabilistic tool to prove the central limit theorem. We also use a lot of Fourier analysis. Finally, I just mention one byproduct of this research: we solve an old problem of Hardy and Littlewood on diophantine sums.