Let $r_s(n)$ denote the number of representations of $n$ as a sum of $s$ squares. Hurwitz established eleven identities expressing the generating function of $r_3(an+b)$ as a simple infinite product. Cooper and Hirschhorn (Discrete Math 274 (1-3):9–24, 2004) proved that for any $k\geq0$, the generating functions $\sum_{n=0}^\infty r_3\big(3^{2k}n\big)q^n$ and $\sum_{n=0}^\infty r_3\big(3^{2k+1}n\big)q^n$ can be written as linear combinations of two specified generalized eta-quotients. In this paper, we substantially extend these results to high dimensions. Specifically, we prove that for any $k\geq0$ and $3\leq s\leq 100$, the generating functions $\sum_{n=0}^\infty r_s\big(3^{2k+1}n\big)q^n$ and $\sum_{n=0}^\infty r_s\big(3^{2k+2}n\big)q^n$ can also be expressed as linear combinations of certain generalized eta-quotients. Motivated by these results, we conjecture that this phenomenon holds for $r_s(n)$ for all $s\geq3$, and further that $r_s(n)$ satisfies an infinite family of internal congruences modulo high powers of $3$.

We study the congruence classes attained by positive integers D with a prescribed period for the continued fraction of $\sqrt D$. As an application, we refine the results on large ranks of universal quadratic forms over real quadratic fields by imposing congruence conditions on their discriminants.

We initiate a systematic study of nonnegative polynomials P such that $P^k$ is not a sum of squares for any odd $k\geq 1$, calling such P stubborn. We develop a new invariant of a real isolated zero of a nonnegative polynomial in the plane, that we call the SOS-invariant, and relate it to the well-known delta invariant of a plane curve singularity. Using the SOS-invariant we show that any polynomial that spans an extreme ray of the convex cone of nonnegative ternary forms of degree 6 is stubborn. We also show how to use the SOS-invariant to prove stubbornness of ternary forms in higher degree. Furthermore, we prove that in a given degree and number of variables, nonnegative polynomials that are not stubborn form a convex cone, whose interior consists of all strictly positive polynomials.

Let p and q be two primes with $(p,q)\equiv (1,5)$ or $(7,3) \pmod 8$. Lagrange [‘Nombres congruents et courbes elliptiques’, Séminaire Delange-Pisot-Poitou. Théorie des Nombres 16(1) (1974–1975), Article no. 16] and Qin [‘Congruent numbers, quadratic forms and $K_2$’, Math. Ann. 383(3–4) (2022), 1647–1686] showed that if $(\frac {q}{p})=-1$, then $2pq$ is not a congruent number. By using Qin’s method, we prove that if $(p,q)\equiv (1,5) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-pq)\not \equiv p-1 \pmod {16}$, then $2pq$ is not a congruent number; if $(p,q)\equiv (7,3) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-2pq)\not \equiv p+1 \pmod {16}$, then $2pq$ is not a congruent number. Here, $h(-d)$ denotes the class number of the imaginary quadratic field $\mathbb {Q}(\sqrt {-d})$.

We study sums of squares of integers except for a fixed one. For any nonnegative integer n, we find the minimum number of squares of integers except for n whose sums represent all positive integers that are represented by a sum of squares except for n. This problem could be considered as a generalisation of the result of Dubouis [‘Solution of a problem of J. Tannery’, Intermédiaire Math. 18 (1911), 55–56] for the case when $n=0$.

We consider a class of sequence $c(n)$ with the following asymptotic form:

$$ \begin{align*}c(n) \sim \frac C{n^\kappa} \exp\left(\sum_{\lambda\in\mathcal S}A_\lambda n^\lambda \right) \sum_{\mu\in\mathcal T} \frac{\beta_\mu}{n^\mu} \qquad (n \to \infty).\end{align*} $$

$c(n)$

$c(n)$

$\ell $

$S_n$

$\mathfrak {su}(3)$

$\mathfrak {so}(5),$

We introduced positive cones in an earlier paper as a notion of ordering on central simple algebras with involution that corresponds to signatures of hermitian forms. In the current article, we describe signatures of hermitian forms directly out of positive cones, and also use this approach to rectify a problem that affected some results in the previously mentioned paper.

We determine explicit generators for the ring of modular forms associated with the moduli spaces of K3 surfaces with automorphism group $(\mathbb {Z}/2\mathbb {Z})^2$ and of Picard rank 13 and higher. The K3 surfaces in question carry a canonical Jacobian elliptic fibration and the modular form generators appear as coefficients in the Weierstrass-type equations describing these fibrations.

Pursuing ideas in [6], we determine the isometry classes of unimodular lattices of rank $28$, as well as the isometry classes of unimodular lattices of rank $29$ without nonzero vectors of norm $\leq 2$. We also provide some invariant that allows to distinguish these lattices and to independently check with a computer that our lists are complete.

We construct an fpqc gerbe $\mathcal {E}_{\dot {V}}$ over a global function field F such that for a connected reductive group G over F with finite central subgroup Z, the set of $G_{\mathcal {E}_{\dot {V}}}$-torsors contains a subset $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G)$ which allows one to define a global notion of (Z-)rigid inner forms. There is a localization map $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G) \to H^{1}(\mathcal {E}_{v}, Z \to G)$, where the latter parametrizes local rigid inner forms (cf. [8, 6]) which allows us to organize local rigid inner forms across all places v into coherent families. Doing so enables a construction of (conjectural) global L-packets and a conjectural formula for the multiplicity of an automorphic representation $\pi $ in the discrete spectrum of G in terms of these L-packets. We also show that, for a connected reductive group G over a global function field F, the adelic transfer factor $\Delta _{\mathbb {A}}$ for the ring of adeles $\mathbb {A}$ of F serving an endoscopic datum for G decomposes as the product of the normalized local transfer factors from [6].

We describe the modulo $2$ de Rham-Witt complex of a field of characteristic $2$, in terms of the powers of the augmentation ideal of the $\mathbb {Z}/2$-geometric fixed points of real topological restriction homology ${\mathrm {TRR}}$. This is analogous to the conjecture of Milnor, proved in [Kat82] for fields of characteristic $2$, which describes the modulo $2$ Milnor K-theory in terms of the powers of the augmentation ideal of the Witt group of symmetric forms. Our proof provides a somewhat explicit description of these objects, as well as a calculation of the homotopy groups of the geometric fixed points of ${\mathrm {TRR}}$ and of real topological cyclic homology, for all fields.

We show that for all real biquadratic fields not containing $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7}$ and $\sqrt{13}$, the Pythagoras number of the ring of algebraic integers is at least 6. We also provide an upper bound on the norm and the minimal (codifferent) trace of additively indecomposable integers in some families of these fields.

What proportion of integers $n \leq N$ may be expressed as $x^2 + dy^2$ for some $d \leq \Delta $, with $x,y$ integers? Writing $\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$ for some $\alpha \in (-\infty , \infty )$, we show that the answer is $\Phi (\alpha ) + o(1)$, where $\Phi $ is the Gaussian distribution function $\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$.

A consequence of this is a phase transition: Almost none of the integers $n \leq N$ can be represented by $x^2 + dy^2$ with $d \leq (\log N)^{\log 2 - \varepsilon }$, but almost all of them can be represented by $x^2 + dy^2$ with $d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$.

We demonstrate the existence of K-multimagic squares of order N consisting of $N^2$ distinct integers whenever $N> 2K(K+1)$. This improves our earlier result [D. Flores, ‘A circle method approach to K-multimagic squares’, preprint (2024), arXiv:2406.08161] in which we only required $N+1$ distinct integers. Additionally, we present a direct method by which our analysis of the magic square system may be used to show the existence of $N \times N$ magic squares consisting of distinct kth powers when

$$ \begin{align*}N> \begin{cases} 2^{k+1} & \text{if}\ 2 \leqslant k \leqslant 4, \\ 2 \lceil k(\log k + 4.20032) \rceil & \text{if}\ k \geqslant 5, \end{cases}\end{align*} $$

improving on a recent result by Rome and Yamagishi [‘On the existence of magic squares of powers’, preprint (2024), arxiv:2406.09364].

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying

$$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$

$n\geq k$

$a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$

$a_k\neq 0$

$(x_n)_{n\geq 0}$

$\mathbb {Q}_p$

$\pm \alpha $

$\alpha $

$\mathbb {Q}_p$

$(x_n)_{n\geq 0}$

$\mathbb {Q}_p$

$\mathbb {Q}$

To any k-dimensional subspace of $\mathbb {Q}^n$ one can naturally associate a point in the Grassmannian $\mathrm {Gr}_{n,k}(\mathbb {R})$ and two shapes of lattices of rank k and $n-k$, respectively. These lattices originate by intersecting the k-dimensional subspace and its orthogonal with the lattice $\mathbb {Z}^n$. Using unipotent dynamics, we prove simultaneous equidistribution of all of these objects under congruence conditions when $(k,n) \neq (2,4)$.

We construct pairs of residually finite groups with isomorphic profinite completions such that one has non-vanishing and the other has vanishing real second bounded cohomology. The examples are lattices in different higher-rank simple Lie groups. Using Galois cohomology, we actually show that $\operatorname {SO}^0(n,2)$ for $n \ge 6$ and the exceptional groups $E_{6(-14)}$ and $E_{7(-25)}$ constitute the complete list of higher-rank Lie groups admitting such examples.

The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $\mathbb {Z}$-form) that is universal over K. We prove the nonexistence of universal $\mathbb {Z}$-forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.

In our paper, we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq {\mathbb {Z}}/q{\mathbb {Z}}$ in the case of composite q. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.

We give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves Hermitian lattices over number fields.