Integral Binary Quadratic Forms Research Articles

Quadratic forms and Genus Theory: A link with 2‐descent and an application to nontrivial specializations of ideal classes

AbstractGenus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well‐known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any principal ideal domain (PID) . When , we show that the Genus Theory map is the quadratic form version of the 2‐descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of nontrivial specializations has density 1.

On the Simultaneous Sign Changes of Coefficients of Rankin–Selberg L-Functions over a Certain Integral Binary Quadratic Form

In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

Eisenstein series of even weight 𝑘≥2 and integral binary quadratic forms

We prove a conjecture of Matsusaka on the analytic continuation of hyperbolic Eisenstein series in weight 2 2 on the full modular group S L 2 ( Z ) \mathrm {SL}_{2}(\mathbb {Z}) .

Arithmetic properties of Fourier coefficients of meromorphic modular forms

We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no non-trivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we prove that their coefficients are non-vanishing and have either constant or alternating signs. Finally, we obtain a relation between the Fourier coefficients of meromorphic modular forms, the coefficients of the $j$-function, and the partition function.

Signs of Fourier coefficients of cusp forms at integers represented by an integral binary quadratic form

Signs of Fourier coefficients of cusp forms at integers represented by an integral binary quadratic form

Binary quartic forms with bounded invariants and small Galois groups

In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by integral binary quadratic forms $f(x,y)$ of non-zero discriminant. Then, we shall enumerate the $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of all such forms associated to a fixed $f(x,y)$.

Binary quadratic forms represented by sums of squares in an essentially unique way

In this article, we determine all positive definite integral binary quadratic forms that are represented by the sum of k integer squares in an essentially unique way for each integer $$k\ge 4$$.

Composition of Binary Quadratic Forms

In 2004, Bhargava introduced a new way to understand the composition law of integral binary quadratic forms through what he calls the ‘cubes of integers’. The goal of this article is to introduce the reader to Bhargava’s cubes and this new composition law, as well as to relate it to the composition law as it is known classically. We will present a historical exposition of the subject, from Gauss to Bhargava, and see how the different formulations of the composition laws are equivalent.

Arithmetic progressions in binary quadratic forms and norm forms

We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms, this improves significantly upon an earlier result of Dey and Thangadurai.

English

To each indefinite integral binary quadratic form Q, we may associate the geodesic in ℍ through the roots of quadratic equation Q(x, 1). In this paper we study the asymptotic distribution (as discriminant tends to infinity) of the angles between these geodesics and one fixed vertical geodesic which intersects all of them.

Primes represented by positive definite binary quadratic forms

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a constant factor of its asymptotic size: unconditional, conditional on the Generalized Riemann Hypothesis (GRH), and for almost all discriminants. The key feature of these estimates is that they hold whenever $x$ exceeds a small power of $D$ and, in some cases, this range of $x$ is essentially best possible. In particular, if $f$ is reduced then this optimal range of $x$ is achieved for almost all discriminants or by assuming GRH. We also exhibit an upper bound for the number of primes represented by $f$ in a short interval and a lower bound for the number of small integers represented by $f$ which have few prime factors.

A note on the gaps between zeros of Epstein's zeta-functions on the critical line

It is proved that Epstein's zeta-function $\zeta_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + i\gamma$ with $ T \leq \gamma \leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 45--53).

A Chebotarev Variant of the Brun–Titchmarsh Theorem and Bounds for the Lang-Trotter conjectures

We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke L-functions that were recently proved by the authors. Our result produces an improvement for the best unconditional bounds toward two conjectures of Lang and Trotter regarding the distribution of traces of Frobenius for elliptic curves and holomorphic cuspidal modular forms. We also obtain new results on the distribution of primes represented by positive-definite integral binary quadratic forms.

Diagonal quadratic forms representing all binary diagonal quadratic forms

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.

Gauss composition for [formula omitted], and the universal Jacobian of the Hurwitz space of double covers

In this paper, we give an explicit description of the moduli space of line bundles on hyperelliptic curves, including singular curves. We study the universal Jacobian J2,g,n of degree n line bundles over the Hurwitz stack of double covers of P1 by a curve of genus g. Our main results are: the construction of a smooth, irreducible, universally closed (but not separated) moduli compactification J‾bd2,g,n of J2,g,n whose points we describe simply and explicitly as sections of certain vector bundles on P1; a description of the global geometry and moduli properties of these stacks; and a computation of the Picard groups of J‾bd2,g,n and J2,g,n in the cases when n−g is even. An important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms. This parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms, which in particular gives the group law on integral binary quadratic forms first discovered by Gauss.

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.

Rings of small rank over a Dedekind domain and their ideals

In 2001, M. Bhargava stunned the mathematical world by extending Gauss's 200-year-old group law on integral binary quadratic forms, now familiar as the ideal class group of a quadratic ring, to yield group laws on a vast assortment of analogous objects. His method yields parametrizations of rings of degree up to 5 over the integers, as well as aspects of their ideal structure, and can be employed to yield statistical information about such rings and the associated number fields. In this paper, we extend a selection of Bhargava's most striking parametrizations to cases where the base ring is not Z but an arbitrary Dedekind domain R. We find that, once the ideal classes of R are properly included, we readily get bijections parametrizing quadratic, cubic, and quartic rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss composition for which Bhargava is famous. We expect that our results will shed light on the analytic distribution of extensions of degree up to 4 of a fixed number field and their ideal structure.

The order of binary quadratic forms from representation numbers

We investigate the relationship between the numbers of representations of certain integers by a primitive integral binary quadratic form [Formula: see text] of discriminant [Formula: see text] and the order of the class of [Formula: see text] in the form class group of discriminant [Formula: see text], in the case when this order is even. The explicit form of the solutions obtained is used to give a partial answer to a question regarding which multiples of [Formula: see text] can be parameterized in a particular way.

Counting square discriminants

Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by investigating the analytic properties of a certain double Dirichlet series, which is a shifted convolution sum of certain classical automorphic forms.

The Length of an Arithmetic Progression Represented by a Binary Quadratic Form

In this paper we prove that if Q(x,y) = ax2 + bxy+cy2 is an integral binary quadratic form with a nonzero, nonsquare discriminant d and if Q represents an arithmetic progression {kn+ℓ : n = 0, 1,…, R-1}, where k and ℓ are positive integers, then there are absolute constants C1 >0 and L1 >0$ such that R < C1ℓ (k2|d|)L1. Moreover, we prove that every nonzero integral binary quadratic form represents a nontrivial $3$-term arithmetic progression infinitely often.