For every positive integer d, we show that there must exist an absolute constant $c \gt 0$ such that the following holds: for any integer $n \geqslant cd^{7}$ and any red-blue colouring of the one-dimensional subspaces of $\mathbb{F}_{2}^{n}$, there must exist either a d-dimensional subspace for which all of its one-dimensional subspaces get coloured red or a 2-dimensional subspace for which all of its one-dimensional subspaces get coloured blue. This answers recent questions of Nelson and Nomoto, and confirms that for any even plane binary matroid N, the class of N-free, claw-free binary matroids is polynomially $\chi$-bounded.

Our argument will proceed via a reduction to a well-studied additive combinatorics problem, originally posed by Green: given a set $A \subset \mathbb{F}_{2}^{n}$ with density $\alpha \in [0,1]$, what is the largest subspace that we can find in $A+A$? Our main contribution to the story is a new result for this problem in the regime where $1/\alpha$ is large with respect to n, which utilises ideas from the recent breakthrough paper of Kelley and Meka on sets of integers without three-term arithmetic progressions.

Let E be an elliptic curve defined over ${{\mathbb{Q}}}$ which has good ordinary reduction at the prime p. Let K be a number field with at least one complex prime which we assume to be totally imaginary if $p=2$. We prove several equivalent criteria for the validity of the $\mathfrak{M}_H(G)$-property for ${{\mathbb{Z}}}_p$-extensions other than the cyclotomic extension inside a fixed ${{\mathbb{Z}}}_p^2$-extension $K_\infty/K$. The equivalent conditions involve the growth of $\mu$-invariants of the Selmer groups over intermediate shifted ${{\mathbb{Z}}}_p$-extensions in $K_\infty$, and the boundedness of $\lambda$-invariants as one runs over ${{\mathbb{Z}}}_p$-extensions of K inside of $K_\infty$.

Using these criteria we also derive several applications. For example, we can bound the number of ${{\mathbb{Z}}}_p$-extensions of K inside $K_\infty$ over which the Mordell–Weil rank of E is not bounded, thereby proving special cases of a conjecture of Mazur. Moreover, we show that the validity of the $\mathfrak{M}_H(G)$-property sometimes can be shifted to a larger base field K′.

The cyclicity and Koblitz conjectures ask about the distribution of primes of cyclic and prime-order reduction, respectively, for elliptic curves over $\mathbb {Q}$. In 1976, Serre gave a conditional proof of the cyclicity conjecture, but the Koblitz conjecture (refined by Zywina in 2011) remains open. The conjectures are now known unconditionally “on average” due to work of Banks–Shparlinski and Balog–Cojocaru–David. Recently, there has been a growing interest in the cyclicity conjecture for primes in arithmetic progressions (AP), with relevant work by Akbal–Güloğlu and Wong. In this article, we adapt Zywina’s method to formulate the Koblitz conjecture for AP and refine a theorem of Jones to establish results on the moments of the constants in both the cyclicity and Koblitz conjectures for AP. In doing so, we uncover a somewhat counterintuitive phenomenon: On average, these two constants are oppositely biased over congruence classes. Finally, in an accompanying repository, we give Magma code for computing the constants discussed in this article.

We prove a functional transcendence theorem for the integrals of algebraic forms in families of algebraic varieties. This allows us to prove a geometric version of André’s generalization of the Grothendieck period conjecture, which we state using the formalism of Nori motives.

More precisely, we prove a version of the Ax–Schanuel conjecture for the comparison between the flat and algebraic coordinates of an arbitrary admissible graded polarizable variation of integral mixed Hodge structures. This can be seen as a generalization of the recent Ax–Schanuel theorems of [13, 18] for mixed period maps.

Given an elliptic curve $ E $ over $ \mathbb {Q} $ of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character $ \chi $ of order $ q $, and in many cases its associated special L-value $ \mathscr {L}(E, \chi ) $ is known to be integral after normalizing by certain periods. This article determines the precise value of $ \mathscr {L}(E, \chi ) $ in terms of Birch–Swinnerton-Dyer invariants when $ q = 3 $, classifies their asymptotic densities modulo $ 3 $ by fixing $ E $ and varying $ \chi $, and presents a lower bound on the $ 3 $-adic valuation of $ \mathscr {L}(E, 1) $, all of which arise from a congruence of modular symbols. These results also explain some phenomena observed by Dokchitser–Evans–Wiersema and by Kisilevsky–Nam.

We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan, Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number $H_n$ is divisible by a prime p. The conjectures state that: (i) $J_p$ is always finite and of the order $O(p^2(\log \log p)^{2+\epsilon })$; (ii) the set of primes for which $J_p$ is minimal (called harmonic primes) has density $e^{-1}$ among all primes; (iii) no harmonic number is divisible by $p^4$. We prove parts (i) and (iii) for all $p\leq 16843$ with at most one exception, and enumerate harmonic primes up to $50\times 10^5$, finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of approximately $30$ and $50$, respectively.

A partition is called a t-core if none of its hook lengths is a multiple of t. Let $a_t(n)$ denote the number of t-core partitions of n. Garvan, Kim and Stanton proved that for any $n\geq1$ and $m\geq1$, $a_t\big(t^mn-(t^2-1)/24\big)\equiv0\pmod{t^m}$, where $t\in\{5,7,11\}$. Let $A_{t,k}(n)$ denote the number of partition k-tuples of n with t-cores. Several scholars have been subsequently investigated congruence properties modulo high powers of 5 for $A_{5,k}(n)$ with $k\in\{2,3,4\}$. In this paper, by utilizing a recurrence related to the modular equation of fifth order, we establish dozens of congruence families modulo high powers of 5 satisfied by $A_{5,k}(n)$, where $4\leq k\leq25$. Moreover, we deduce an infinite family of internal congruences modulo high powers of 5 for $A_{5,4}(n)$. In particular, we generalize greatly a recent result on a congruence family modulo high powers of 5 enjoyed by $A_{5,4}(n)$, which was proved by Saikia, Sarma and Talukdar (Indian J. Pure Appl. Math., 2024). Finally, we conjecture that there exists a similar phenomenon for $A_{5,k}(n)$ with $k\geq26$.

We study optimal dimensionless inequalities

\begin{equation*} \|f\|_{\textrm{U}^k} \leqslant \|f\|_{\ell^{p_{k,n}}} \end{equation*}

that hold for all functions $f\colon\mathbb{Z}^d\to\mathbb{C}$ supported in $\{0,1,\ldots,n-1\}^d$ and estimates

\begin{equation*} \|\mathbb{1}_A\|_{\textrm{U}^k}^{2^k}\leqslant |A|^{t_{k,n}} \end{equation*}

that hold for all subsets A of the same discrete cubes. A general theory, analogous to the work of de Dios Pont, Greenfeld, Ivanisvili, and Madrid, is developed to show that the critical exponents are related by $p_{k,n} t_{k,n} = 2^k$. This is used to prove the three main results of the article:

• • an explicit formula for $t_{k,2}$, which generalizes a theorem by Kane and Tao,

• • two-sided asymptotic estimates for $t_{k,n}$ as $n\to\infty$ for a fixed $k\geqslant2$, which generalize a theorem by Shao, and

• • a precise asymptotic formula for $t_{k,n}$ as $k\to\infty$ for a fixed $n\geqslant2$.

In one-dimensional Diophantine approximation, the Diophantine properties of a real number are characterized by its partial quotients, especially the growth of its large partial quotients. Notably, Kleinbock and Wadleigh [Proc. Amer. Math. Soc. 146(5) (2018), 1833–1844] made a seminal contribution by linking the improvability of Dirichlet’s theorem to the growth of the product of consecutive partial quotients. In this paper, we extend the concept of Dirichlet non-improvable sets within the framework of shrinking target problems. Specifically, consider the dynamical system $([0,1), T)$ of continued fractions. Let $\{z_n\}_{n \ge 1}$ be a sequence of real numbers in $[0,1]$ and let $B> 1$. We determine the Hausdorff dimension of the following set: $ \{x\in [0,1):|T^nx-z_n||T^{n+1}x-Tz_n|<B^{-n}\text { infinitely often}\}. $

Let $K = \mathbf {R}$ or $\mathbf {C}$. An n-element subset A of K is a $B_h$-set if every element of K has at most one representation as the sum of h not necessarily distinct elements of A. Associated with the $B_h$-set $A = \{a_1,\ldots , a_n\}$ are the $B_h$-vectors $\mathbf {a} = (a_1,\ldots , a_n)$ in $K^n$. This article proves that “almost all” n-element subsets of K are $B_h$-sets in the sense that the set of all $B_h$-vectors is a dense open subset of $K^n$.

We explore the relationship between (3-isogeny induced) Selmer group of an elliptic curve and the (3 part of) the ideal class group, over certain non-abelian number fields.

The well-known $abc$-conjecture concerns triples $(a,b,c)$ of nonzero integers that are coprime and satisfy ${a+b+c=0}$. The strong n-conjecture is a generalisation to n summands where integer solutions of the equation ${a_1 + \cdots + a_n = 0}$ are considered such that the $a_i$ are pairwise coprime and satisfy a certain subsum condition. Ramaekers studied a variant of this conjecture with a slightly different set of conditions. He conjectured that in this setting the limit superior of the so-called qualities of the admissible solutions equals $1$ for any n. In this paper, we follow results of Konyagin and Browkin. We restrict to a smaller, and thus more demanding, set of solutions, and improve the known lower bounds on the limit superior: for ${n \geq 6}$ we achieve a lower bound of $\frac 54$; for odd $n \geq 5$ we even achieve $\frac 53$. In particular, Ramaekers’ conjecture is false for every ${n \ge 5}$.

In this paper, we investigate Frobenius eigenvalues of the compactly supported rigid cohomology of a variety defined over a finite field with q elements, using Dwork’s method. Our study yields several arithmetic consequences. First, we establish that the zeta functions of a set of related affine varieties can reveal all Frobenius eigenvalues of the rigid cohomology of the variety up to a Tate twist. This result does not seem to be known for the $\ell$-adic cohomology. As a second application, we provide several q-divisibility lower bounds for the Frobenius eigenvalues of the rigid cohomology of the variety, in terms of the dimension and multi-degrees of the defining equations. These divisibility bounds for rigid cohomology are generally better than what is suggested from the best known divisibility bounds in $\ell$-adic cohomology, both before and after the middle cohomological degree.

Continuing our work on group-theoretic generalisations of the prime Ax–Katz Theorem, we give a lower bound on the p-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\dots,f_r)$ of r maps $f_j\,{:}\,A\rightarrow B_j$ between arbitrary finite commutative groups A and $B_j$ in terms of the invariant factors of $A, B_1,B_2, \cdots,B_r$ and the functional degrees of the maps $f_1,f_2, \dots,f_r$.

Let p be an odd prime, and suppose that $E_1$ and $E_2$ are two elliptic curves which are congruent modulo p. Fix an Artin representation $\tau\,{:}\,G_{F}\rightarrow \mathrm{GL}_2(\mathbb{C})$ over a totally real field F, induced from a Hecke character over a CM-extension $K/F$. Assuming $E_1$ and $E_2$ are ordinary at p, we compute the variation in the $\mu$- and $\lambda$-invariants for the $\tau$-part of the Iwasawa Main Conjecture, as one switches from $E_1$ to $E_2$. Provided an Euler system exists, it will follow directly that IMC$(E_1,\tau)$ is true if and only if IMC$(E_2,\tau)$ is true.

A real number is simply normal to base b if its base-b expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency at which the digit $1$ appears in the binary expansion of $2^{p/q}$ and a mean value of the Riemann zeta function on vertical arithmetic progressions. In particular, we show that

$$\begin{align*}\lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } \zeta\left(\frac{2 n\pi i}{\log 2}\right) \frac{e^{2n\pi i p/q} }{n} =0 \end{align*}$$

$2^{p/q}$

$2$

Weighted sieves are used to detect numbers with at most S prime factors with $S \in \mathbb{N}$ as small as possible. When one studies problems with two variables in somewhat symmetric roles (such as Chen primes, that is primes p such that $p+2$ has at most two prime factors), one can utilise the switching principle. Here we discuss how different sieve weights work in such a situation, concentrating in particular on detecting a prime along with a product of at most three primes.

As applications, we improve on the works of Yang and Harman concerning Diophantine approximation with a prime and an almost prime, and prove that, in general, one can find a pair $(p, P_3)$ when both the original and the switched problem have level of distribution at least $0.267$.

Let $\pi$ be an irreducible cuspidal automorphic representation of ${\mathrm{GL}}_n(\mathbb{A}_{\mathbb{Q}})$ with associated L-function $L(s, \pi)$. We study the behaviour of the partial Euler product of $L(s, \pi)$ at the centre of the critical strip. Under the assumption of the Generalised Riemann Hypothesis for $L(s, \pi)$ in conjunction with the Ramanujan–Petersson conjecture as necessary, we establish an asymptotic, off a set of finite logarithmic measure, for the partial Euler product at the central point, which confirms a conjecture of Kurokawa (2012). As an application, we obtain results towards Chebyshev’s bias in the recently proposed framework of Aoki–Koyama (2023).

This paper discusses variants of Weber’s class number problem in the spirit of arithmetic topology to connect the results of Sinnott–Kisilevsky and Kionke. Let p be a prime number. We first prove the p-adic convergence of class numbers in a ${\mathbb{Z}_{p}}$-extension of a global field and a similar result in a ${\mathbb{Z}_{p}}$-cover of a compact 3-manifold. Secondly, we establish an explicit formula for the p-adic limit of the p-power-th cyclic resultants of a polynomial using roots of unity of orders prime to p, the p-adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with p-adic limits being in ${\mathbb{Z}}$ and find the cases such that the base p-class numbers are small and $\nu$’s are arbitrarily large.

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.