We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.

For a subset S of nonnegative integers and a vector $\mathbf {a}=(a_1,\ldots ,a_k)$ of positive integers, define the set $V^{\prime }_S(\mathbf {a})=\{ a_1s_1+\cdots +a_ks_k : s_i\in S\}-\{0\}$ . For a positive integer n, let $\mathcal T(n)$ be the set of integers greater than or equal to n. We consider the problem of finding all vectors $\mathbf {a}$ satisfying $V^{\prime }_S(\mathbf {a})=\mathcal T(n)$ when S is the set of (generalised) m-gonal numbers and n is a positive integer. In particular, we completely resolve the case when S is the set of triangular numbers.

The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$, for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

Z.-W. Sun [‘Refining Lagrange’s four-square theorem’, J. Number Theory175 (2017), 169–190] conjectured that every positive integer n can be written as $ x^2+y^2+z^2+w^2\ (x,y,z,w\in \mathbb {N}=\{0,1,\ldots \})$ with $x+3y$ a square and also as $n=x^2+y^2+z^2+w^2\ (x,y,z,w \in \mathbb {Z})$ with $x+3y\in \{4^k:k\in \mathbb {N}\}$ . In this paper, we confirm these conjectures via the arithmetic theory of ternary quadratic forms.

We find and prove a class of congruences modulo 4 for eta-products associated with certain ternary quadratic forms. This study was inspired by similar conjectured congruences modulo 4 for certain mock theta functions.

A (positive definite and integral) quadratic form is said to be prime-universal if it represents all primes. Recently, Doyle and Williams [‘Prime-universal quadratic forms $ax^2+by^2+cz^2$ and $ax^2+by^2+cz^2+dw^2$ ’, Bull. Aust. Math. Soc.101 (2020), 1–12] classified all prime-universal diagonal ternary quadratic forms and all prime-universal diagonal quaternary quadratic forms under two conjectures. We classify all prime-universal diagonal quadratic forms regardless of rank, and prove the so-called 67-theorem for a diagonal quadratic form to be prime-universal.

We study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

Let $\mathbb{Z}$ and $\mathbb{Z}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{Z}^{+}$, let $t(a,b,c,d;n)$ be the number of representations of $n$ by $\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$ with $x,y,z,w\in \mathbb{Z}$. Using theta function identities we prove 13 transformation formulas for $t(a,b,c,d;n)$ and evaluate $t(2,3,3,8;n)$, $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$.

Extending the notion of regularity introduced by Dickson in 1939, a positive definite ternary integral quadratic form is said to be spinor regular if it represents all the positive integers represented by its spinor genus (that is, all positive integers represented by any form in its spinor genus). Jagy conducted an extensive computer search for primitive ternary quadratic forms that are spinor regular, but not regular, resulting in a list of 29 such forms. In this paper, we will prove that there are no additional forms with this property.

Let $\eta \left( z \right)\,\left( z\,\in \,\mathbb{C},\,\operatorname{Im}\left( z \right)\,>\,0 \right)$ denote the Dedekind eta function. We use a recent product-to-sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotients

$$f\left( z \right)\,:=\,{{\eta }^{a\left( {{m}_{1}} \right)}}\,\left( {{m}_{1}}z \right)\,.\,.\,.\,{{\eta }^{a\left( {{m}_{r}} \right)}}\,\left( {{m}_{r}}z \right)\,=\,\sum\limits_{n=1}^{\infty }{c\left( n \right){{e}^{2\pi inz}},\,\,\,z\,\in \,\mathbb{C},\,\operatorname{Im}\left( z \right)\,>\,0,}$$

such that the Fourier coefficients $c\left( n \right)$ vanish for all positive integers $n$ in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if $f\left( z \right)\,=\,{{\eta }^{4}}\left( z \right){{\eta }^{9}}\left( 4z \right){{\eta }^{-2}}\left( 8z \right)$ we have $c\left( n \right)\,=\,0$ for all $n$ in each of the arithmetic progressions ${{\{16k\,+\,14\}}_{k\ge 0}},\,{{\{64k\,+\,56\}}_{k\ge 0}},\,{{\{256k,\,224\}}_{k\ge 0}},\,{{\{1024k\,+\,869\}}_{k\ge 0}},\,.\,.\,.$

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{O}$ be a maximal order in the quaternion algebra $B_p$ over $\mathbb{Q}$ ramified at $p$ and $\infty $. The paper is about the computational problem: construct a supersingular elliptic curve $E$ over $\mathbb{F}_p$ such that ${\rm End}(E) \cong \mathcal{O}$. We present an algorithm that solves this problem by taking gcds of the reductions modulo $p$ of Hilbert class polynomials.

New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice $\mathcal{O}^T$ of $\mathcal{O}$, the order $\mathcal{O}$ is effectively characterized by the three successive minima and two other short vectors of $\mathcal{O}^T\! .$ The desired conditions turn out to hold whenever the $j$-invariant $j(E)$, of the elliptic curve with ${\rm End}(E) \cong \mathcal{O}$, lies in $\mathbb{F}_p$. We can then prove that our algorithm terminates with running time $O(p^{1+\varepsilon })$ under the aforementioned conditions.

As a further application we present an algorithm to simultaneously match all maximal order types with their associated $j$-invariants. Our algorithm has running time $O(p^{2.5 + \varepsilon })$ operations and is more efficient than Cerviño’s algorithm for the same problem.

We prove upper bounds for Hecke–Laplace eigenfunctions on certain Riemannian manifolds $X$ of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are $d$-fold products of $2$-spheres or $3$-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a (‘Weyl-type’) saving of $\mathrm{vol} \hspace{0.167em} (X)^{- 1/ 6+ \varepsilon } $.

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

The convolution sum $\sum{_{m<n/16}\,\sigma (m)\sigma (n\,}-16m)$ is evaluated for all $n\,\in \,\mathbb{N}$. This evaluation is used to determine the number of representations of $n$ by the quadratic form $x_{1}^{2}\,+\,x_{2}^{2}\,+\,x_{3}^{2}\,+\,x_{4}^{2}\,+\,4x_{5}^{2}\,+\,4x_{6}^{2}\,+\,4x_{7}^{2}\,+\,4x_{8}^{2}$.

We consider quadratic diophantine equations of the shape

$$ Q(x_1, \ldots, x_s) = 0 \qquad (1)$$

for a polynomial $Q(X_1, \ldots, X_s) \in \mathbf{Z}[X_1, \ldots, X_s]$ of degree $2$. Let $H$ be an upper bound for the absolute values of the coefficients of $Q$, and assume that the homogeneous quadratic part of $Q$ is non-singular. We prove, for all $s \ge 3$, the existence of a polynomial bound $\Lambda_s(H)$ with the following property: if equation (1) has a solution $\mathbf{x} \in \mathbf{Z}^s$ at all, then it has one satisfying

$$ |x_i| \le \Lambda_s(H) \quad (1 \le i \le s). $$

For $s = 3$ and $s = 4$ no polynomial bounds $\Lambda_s(H)$ were previously known, and for $s \ge 5$ we have been able to improve existing bounds quite significantly.

In this paper we prove that every positive definite n-ary integral quadratic form with 12 < n < 13 (respectively 14 ≦ n ≤ 20) that can be represented by a sum of squares of integral linear forms is represented by a sum of 2 · 3n + n + 6 (respectively 3 · 4n + n + 3) squares. We also prove that every positive definite 7-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 25 squares.