Let M be a closed oriented 3-manifold equipped with an Euler structure e and an acyclic representation of its fundamental group. We define a twisted self-linking homology class of the diagonal of the two-point configuration space of M with respect to e. This twisted self-linking homology class appears as an obstruction in the Chern–Simons perturbation theory. When the representation is the maximal free abelian representation $\rho_0$, we prove that our self-linking class is a properly defined “logarithmic derivative” of the Reidemeister–Turaev torsion of $(M,\rho_0,e)$ equipped with the given Euler structure.

In their celebrated paper [CLR10], Caputo, Liggett and Richthammer proved Aldous’ conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying random walk. A crucial ingredient in the proof was the Octopus Inequality — a certain inequality of operators in the group ring $\mathbb{R}\left[{\mathrm{Sym}}_{n}\right]$ of the symmetric group. Here we generalise the Octopus Inequality and apply it to generalising the Caputo–Liggett–Richthammer Theorem to certain hypergraphs, proving some cases of a conjecture of Caputo.

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.

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).

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 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.

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 $f_1,f_2, \dots,f_r$.

If all of the entries of a large $S_n$ character table are covered up and you are allowed to uncover one entry at a time, then how can you quickly identify all of the indexing characters and conjugacy classes? We present a fast algorithmic solution that works even when n is so large that almost none of the entries of the character table can be computed. The fraction of the character table that needs to be uncovered is $O( n^2 \exp({-}2\pi\sqrt{n/6}))$, and for many of these entries we are only interested in whether the entry is zero.

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′.

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.