We show that for $5/6$-th of all primes p, Hilbert’s 10th problem is unsolvable for the ring of integers of $\mathbb {Q}(\zeta _3, \sqrt [3]{p})$. We also show that there is an infinite set S of square-free integers such that Hilbert’s 10th problem is unsolvable over the ring of integers of $\mathbb {Q}(\zeta _3, \sqrt {D}, \sqrt [3]{p})$ for every $D \in S$ and for every prime $p \equiv 2, 5\ \pmod 9$. We use the CM elliptic curves $y^2=x^3-432 D^2$ associated with the cube-sum problem, with D varying in suitable congruence class, in our proof.

A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb {Z}[t]$ and which number fields K the Hasse principle holds for the affine equation $f(t) = \mathbf {N}_{K/\mathbb {Q}}(\mathbf {x}) \neq 0$. Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving [27], which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.

Let $K={\mathbb {Q}}(\sqrt {-7})$ and $\mathcal {O}$ the ring of integers in $K$. The prime $2$ splits in $K$, say $2{\mathcal {O}}={\mathfrak {p}}\cdot {\mathfrak {p}}^*$. Let $A$ be an elliptic curve defined over $K$ with complex multiplication by $\mathcal {O}$. Assume that $A$ has good ordinary reduction at both $\mathfrak {p}$ and ${\mathfrak {p}}^*$. Write $K_\infty$ for the field generated by the $2^\infty$–division points of $A$ over $K$ and let ${\mathcal {G}}={\mathrm {Gal}}(K_\infty /K)$. In this paper, by adopting a congruence formula of Yager and De Shalit, we construct the two-variable $2$-adic $L$-function on $\mathcal {G}$. Then by generalizing De Shalit’s local structure theorem to the two-variable setting, we prove a two-variable elliptic analogue of Iwasawa’s theorem on cyclotomic fields. As an application, we prove that every branch of the two-variable measure has Iwasawa $\mu$ invariant zero.

We determine the asymptotic behavior of the coefficients of Hecke polynomials. In particular, this allows us to determine signs of these coefficients when the level or the weight is sufficiently large. In all but finitely many cases, this also verifies a conjecture on the nanvanishing of the coefficients of Hecke polynomials.

We prove the Ramanujan and Sato–Tate conjectures for Bianchi modular forms of weight at least $2$. More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\operatorname {\mathrm {GL}}_2(\mathbf {A}_F)$ of parallel weight, where F is any CM field. We deduce these theorems from a new potential automorphy theorem for the symmetric powers of $2$-dimensional compatible systems of Galois representations of parallel weight.

Let ${ F}/{ F}_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq 2$ with Galois automorphism $\sigma $, and let R be an algebraically closed field of characteristic $\ell \notin \{0,p\}$. We reduce the classification of $\operatorname {GL}_n({ F}_0)$-distinguished cuspidal R-representations of $\operatorname {GL}_n({ F})$ to the level $0$ setting. Moreover, under a parity condition, we give necessary conditions for a $\sigma $-self-dual cuspidal R-representation to be distinguished. Finally, we classify the distinguished cuspidal ${\overline {\mathbb {F}}_{\ell }}$-representations of $\operatorname {GL}_n({ F})$ having a distinguished cuspidal lift to ${\overline {\mathbb {Q}}_\ell }$.

In this article, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine equations. Among many new results, we highlight our main contributions as follows. First, we show that if a nontrivial shift of a multiplicative subgroup G contains a product set $AB$, then $|A||B|$ is essentially bounded by $|G|$, refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of $M_k(n)$, the largest size of a set such that each pairwise product of its elements is n less than a kth power, refining the recent result of Dixit, Kim, and Murty. One main ingredient in our proof is the first non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest. We also make significant progress toward a conjecture of Sárközy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes p, the set $\{x^2-1: x \in {\mathbb F}_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in ${\mathbb F}_p$ non-trivially.

We study deformation theory of mod p Galois representations of p-adic fields with values in generalised tori, such as L-groups of (possibly non-split) tori. We show that the corresponding deformation rings are formally smooth over a group algebra of a finite abelian p-group. We compute their dimension and the set of irreducible components.

Let G be a finite nilpotent group and $n\in \{3,4, 5\}$. Consider $S_n\times G$ as a subgroup of $S_n\times S_{|G|}\subset S_{n|G|}$, where G embeds into the second factor of $S_n\times S_{|G|}$ via the regular representation. Over any number field k, we prove the strong form of Malle’s conjecture (cf. Malle (2002, Journal of Number Theory 92, 315–329)) for $S_n\times G$ viewed as a subgroup of $S_{n|G|}$. Our result requires that G satisfies some mild conditions.

We investigate the sums $(1/\sqrt {H}) \sum _{X < n \leq X+H} \chi (n)$, where $\chi $ is a fixed non-principal Dirichlet character modulo a prime q, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided $H \rightarrow \infty $ and $(\log H)/\log q \rightarrow 0$ as $q \rightarrow \infty $, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $\chi $, even when $H = q/\log ^{A}q$ for any fixed $A> 0$. However, we show it is true for ‘almost all’ characters on the range $q^{1-o(1)} \leq H = o(q)$.

Using Pólya’s Fourier expansion, these results may be reformulated as statements about the distribution of certain Fourier series with number theoretic coefficients. Tools used in the proofs include the existence of characters with large partial sums on short initial segments, and moment estimates for trigonometric polynomials with random multiplicative coefficients.

We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs (i.e., $x,y\in {\mathbb N}$ such that $x^2\pm y^2=z^2$ for some $z\in {\mathbb N}$). We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.

A bielliptic surface (or hyperelliptic surface) is a smooth surface with a numerically trivial canonical divisor such that the Albanese morphism is an elliptic fibration. In the first part of this article, we study the structure of bielliptic surfaces over a field of characteristic different from $2$ and $3$, in order to prove the Shafarevich conjecture for bielliptic surfaces with rational points. Furthermore, we demonstrate that the Shafarevich conjecture does not generally hold for bielliptic surfaces without rational points. In particular, this article completes the study of the Shafarevich conjecture for minimal surfaces of Kodaira dimension $0$. In the second part of this article, we study a Néron model of a bielliptic surface. We establish the potential existence of a Néron model for a bielliptic surface when the residual characteristic is not equal to $2$ or $3$.

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].

Inspired by work of Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb. 23 (2019), 249–254] on the minimal excludant or ‘mex’ of partitions, we define four new classes of minimal excludants for overpartitions and establish relations to certain functions due to Ramanujan.

Let C and W be two integer sets. If $C+W=\mathbb {Z}$, then we say that C is an additive complement to W. If no proper subset of C is an additive complement to W, then we say that C is a minimal additive complement to W. We study the existence of a minimal additive complement to $W=\{w_i\}_{i=1}^{\infty}$ when W is not eventually periodic and $w_{i+1}-w_{i}\in \{2,3\}$ for all i.

We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the Generalized Riemann Hypothesis, we show that the average analytic rank of isomorphism classes of elliptic curves over K is bounded above by $(9\deg (K)+1)/2$, when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.

We determine the cohomology of the closed Drinfeld stratum of p-adic Deligne–Lusztig schemes of Coxeter type attached to arbitrary inner forms of unramified groups over a local non-archimedean field. We prove that the corresponding torus weight spaces are supported in exactly one cohomological degree and are pairwise non-isomorphic irreducible representations of the pro-unipotent radical of the corresponding parahoric subgroup. We also prove that all Moy–Prasad quotients of this stratum are maximal varieties, and we investigate the relation between the resulting representations and Kirillov’s orbit method.

We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi–Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo q, where q varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $\phi(n)$ among the coprime residues to the ‘correct’ moduli q that vary uniformly in a similar range and also quantify the failure of equidistribution of the values of $\phi(n)$ among the coprime residue classes to the ‘incorrect’ moduli.

The triangle removal states that if G contains $\varepsilon n^2$ edge-disjoint triangles, then G contains $\delta (\varepsilon )n^3$ triangles. Unfortunately, there are no sensible bounds on the order of growth of $\delta (\varepsilon )$, and at any rate, it is known that $\delta (\varepsilon )$ is not polynomial in $\varepsilon $. Csaba recently obtained an asymmetric variant of the triangle removal, stating that if G contains $\varepsilon n^2$ edge-disjoint triangles, then G contains $2^{-\operatorname {\mathrm {poly}}(1/\varepsilon )}\cdot n^5$ copies of $C_5$. To this end, he devised a new variant of Szemerédi’s regularity lemma. We obtain the following results:

• • We first give a regularity-free proof of Csaba’s theorem, which improves the number of copies of $C_5$ to the optimal number $\operatorname {\mathrm {poly}}(\varepsilon )\cdot n^5$.

• • We say that H is $K_3$-abundant if every graph containing $\varepsilon n^2$ edge-disjoint triangles has $\operatorname {\mathrm {poly}}(\varepsilon )\cdot n^{\lvert V(H)\rvert }$ copies of H. It is easy to see that a $K_3$-abundant graph must be triangle-free and tripartite. Given our first result, it is natural to ask if all triangle-free tripartite graphs are $K_3$-abundant. Our second result is that assuming a well-known conjecture of Ruzsa in additive number theory, the answer to this question is negative.

Our proofs use a mix of combinatorial, number-theoretic, probabilistic and Ramsey-type arguments.

Let $g(x)=x^3+ax^2+bx+c$ and $f(x)=g(x^3)$ be irreducible polynomials with rational coefficients, and let $ {\mathrm{Gal}}(f)$ be the Galois group of $f(x)$ over $\mathbb {Q}$. We show $ {\mathrm{Gal}}(f)$ is one of 11 possible transitive subgroups of $S_9$, defined up to conjugacy; we use $ {\mathrm{Disc}}(f)$, $ {\mathrm{Disc}}(g)$ and two additional low-degree resolvent polynomials to identify $ {\mathrm{Gal}}(f)$. We further show how our method can be used for determining one-parameter families for a given group. Also included is a related algorithm that, given a field $L/\mathbb {Q}$, determines when L can be defined by an irreducible polynomial of the form $g(x^3)$ and constructs such a polynomial when it exists.