In 1972, Heilbronn introduced the notion of virtual characters and used it to study simple real zeros of Dedekind zeta functions. One of the consequences of his elegant work is the following. Let $\mathrm{K}/ \mathrm{F}$ be a Galois extension of number fields of odd degree. Then any real simple zero of $\zeta_{\mathrm{K}}$ is necessarily a simple zero of $\zeta_{\mathrm{F}}$. The ethos of this paper is to carry out investigation for arbitrary odd order real zeros. While the Riemann zeta function is conjectured to have only simple zeros, the same does not hold for arbitrary Dedekind zeta functions. One of the consequences of our work is that any Galois number field K of odd degree cannot have a non-trivial odd order real zero. Such parity is at least in conformity with extended Riemann hypothesis as the order of vanishing of the Dedekind zeta function $\zeta_{\mathrm{K}}$ at $1/2$ is necessarily even. We also indicate, via a number of illustrative examples (see Remarks 1·1 and 1·2), that in some sense our results are optimal.

We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of multiplicity one type results for the number-theoretic objects attached to L-functions. These results follow from our main result, which has slightly weaker hypotheses than previous multiplicity one theorems for L-functions. Significantly stronger results are available when the L-function is known to be automorphic.

In this paper, we investigate the extension of uniformisation results for Gromov hyperbolic spaces beyond the standard geodesic setting. By establishing a Gehring-Hayman type theorem for conformal deformations of any intrinsic Gromov hyperbolic space, we provide a framework for analysing spaces that do not necessarily admit geodesics. As a primary application, we prove that any complete intrinsic hyperbolic space with at least two points in the Gromov boundary can be uniformised by densities induced by Busemann functions. Furthermore, we establish that there exists a natural identification between the Gromov boundary of the original space and the metric boundary of the deformed space.

Let $N \ge 1$, $k \ge 2$ even, and $\sigma$ denote a sign pattern for N. In this paper, we first determine the exact proportion of forms in $S_k(N)$ and $S_k^{\mathrm{new}}(N)$ with a given Atkin–Lehner sign pattern $\sigma$. Then we study the asymptotic behaviour of the Hecke operators $T_p$ over the subspaces of $S_k(N)$ and $S_k^{{\mathrm{new}}}(N)$ with Atkin–Lehner sign pattern $\sigma$. In particular, for the p-adic Plancherel measure $\mu_p$, we show that the Hecke eigenvalues for $T_p$ over these subspaces are $\mu_p$-equidistributed as $N+k \to \infty$.

Gowers and Hatami initiated the inverse theory for the uniformity norms $U^k$ of matrix-valued functions on non-abelian groups by proving a 1%-inverse theorem for the $U^2$-norm and relating it to stability questions for almost representations. In this paper, we take a step toward an inverse theory for higher-order uniformity norms of matrix-valued functions on arbitrary groups by examining the 99% regime for the $U^k$-norm on perfect groups of bounded commutator width.

This analysis prompts a classification of Leibman’s quadratic maps between non-abelian groups. Our principal contribution is a complete description of these maps via an explicit universal construction. From this classification we deduce several applications: A full classification of quadratic maps on arbitrary abelian groups; a proof that no nontrivial polynomial maps of degree greater than one exist on perfect groups; stability results for approximate polynomial maps.

It is easy to see that every k-edge-colouring of the complete graph on $2^k+1$ vertices contains a monochromatic odd cycle. In 1973, Erdős and Graham asked to estimate the smallest L(k) such that every k-edge-colouring of $K_{2^k+1}$ contains a monochromatic odd cycle of length at most L(k). Recently, Girão and Hunter obtained the first nontrivial upper bound by showing that $L(k)=O({2^k}/({k^{1-o(1)}}))$, which improves the trivial bound by a polynomial factor. We obtain an exponential improvement by proving that $L(k)=O(k^{3/2}2^{k/2})$. Our proof combines tools from algebraic combinatorics and approximation theory.

We show that any finitely presented group with an index two subgroup is realised as the fundamental group of a closed smooth non-orientable four-manifold that admits an exotic smooth structure, which is obtained by performing a Gluck twist. The orientation 2-covers of these four-manifolds are diffeomorphic. These two smooth structures remain inequivalent after adding arbitrarily many copies of the product of a pair of 2-spheres and stabilise after adding a single copy of the complex projective plane.

We establish a fixed-point theorem for the face maps that consist in deleting the ith entry of an ordered set. Furthermore, we show that there exists random finite sets of integers that are almost invariant under such deletions. Consequences for various monoids of order-preserving transformations of $\mathbf{N}$ are discussed in an appendix.

Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of $2\times 2$ matrices with fixed non-zero determinant $n\in\mathbb{N}$, and with coefficients bounded in absolute value by X. In this paper we present a new proof of this result, which also gives an improved error term as $X\rightarrow\infty$. Similar to Afifurrahman’s result, our error term is uniform in both n and X, and our estimates are significant for X as small as $n^{1/2+\delta}$. To complement this, we also demonstrate that the exponent $1/2+\delta$ in this statement cannot be reduced, by establishing a result which gives a different asymptotic main term when n is either a prime or the square of a prime, and when $X=n^{1/2}$.

The averaged distance structure of one-dimensional regular model sets is determined via their pair-correlation functions. The latter lead to covariograms and cross covariograms of the windows, which give continuous functions in internal space. While they are simple tent-shaped, piecewise linear functions for intervals, the typical case for inflation systems leads to convolutions of Rauzy fractals, which are difficult to compute. In the presence of an inflation structure, an alternative path is possible via the exact renormalisation structures of the pair-correlation functions. We introduce this approach and derive two concrete examples, which display unexpectedly complex and wild behaviour.

In this paper, we study the variation of Selmer groups in families of modular Galois representations that are congruent modulo a fixed prime $p \geq 5$. Motivated by analogies with Goldfeld’s conjecture on ranks in quadratic twist families of elliptic curves, we investigate the stability of Selmer groups defined over $\mathbb{Q}$ via Greenberg’s local conditions under congruences of residual Galois representations. Let X be a positive real number. Fix a residual representation $\bar{\rho}$ and a corresponding modular form f of weight 2 and optimal level. We count the number of level-raising modular forms g of weight 2 that are congruent to f modulo p, with level $N_g\leq X$, such that the p-rank of the Selmer groups of g equals that of f. Under some mild assumptions on $\bar{\rho}$, we prove that this count grows at least as fast as $X (\log X)^{\alpha - 1}$ as $X \to \infty$, for an explicit constant $\alpha \gt 0$. The main result is a partial generalisation of theorems of Ono and Skinner on rank-zero quadratic twists to the setting of modular forms and Selmer groups.

We study the representation of non-weakly compact operators between $AL$-spaces. In this setting, we show that every operator admits a best approximant in the ideal of weakly compact operators. Using duality arguments, we extend this result to operators between $C(L)$-spaces where $L$ is extremally disconnected. We also characterize the weak essential norm for operators between $AL$-spaces in terms of factorizations of the identity on $\ell_1$. As a consequence, we deduce that the weak Calkin algebra $\mathscr{B}(E)/\mathscr{W}(E)$ admits a unique algebra norm for every $AL$-space $E$. By duality, similar results are obtained for $C(K)$-spaces. In particular, we prove that for operators $T: L_{\infty}[0,1] \to L_{\infty}[0,1]$ the weak essential norm, the residuum norm, and the De Blasi measure of weak compactness coincide, answering a question of González, Saksman and Tylli.

The notion of weighted $\alpha $-composition was introduced by Ruhan Zhao in the 1990s. In this paper, we study several analytic function spaces that are closely related to weighted $\alpha $-composition. These include $\alpha $-Bloch spaces, $F(p,q,s)$ spaces, and Campanato spaces. We obtain derivative-free characterizations for $\alpha $-Bloch spaces and $F(p,q,s)$ spaces, which improve some previous results in the literature. We also obtain a certain version of Carleson measures for Campanato spaces and $F(p,q,s)$ spaces.

We classify all Polish semigroup topologies on the symmetric inverse monoid $I_{\mathbb N}$ on the natural numbers $\mathbb N$. This result answers a question of Elliott et al. There are countably infinitely many such topologies. Under containment, these Polish semigroup topologies form a join-semilattice with infinite descending chains, no infinite ascending chains, and arbitrarily large finite anti-chains. Also, we show that the monoid $I_{\mathbb N}$ endowed with any second countable $T_1$ semigroup topology is homeomorphic to the Baire space $\mathbb N^{\mathbb N}$.

We investigate heat kernel-based and other p-energy norms ($1\lt p\lt\infty$) on bounded and unbounded metric measure spaces, in particular, on nested fractals and their blow-ups. With the weak-monotonicity properties for these semi-norms, we generalise the celebrated Bourgain–Brezis-Mironescu (BBM) type characterisation for $p\neq2$. When the underlying space admits a heat kernel satisfying the sub-Gaussian estimates, we establish the equivalence of various p-energy semi-norms and weak-monotonicity properties, and show that these weak-monotonicity properties hold when $p=2$ (that is the case of Dirichlet form). Our paper’s key results concern the equivalence and verification of various weak-monotonicity properties on fractals. Consequently, many classical results on p-energy norms hold on nested fractals and their blow-ups, including the BBM type characterisation and Gagliardo-Nirenberg inequality.

We provide four equivalent combinatorial conditions for a simple assembly graph (rigid vertex graph where all vertices are of degree 1 or 4) to have the largest number of Hamiltonian sets of polygonal paths relative to its size. These conditions serve to prove the conjecture that such a maximum, which is equal to $F_{2n+1}-1$, where $F_k$ denotes the $k$th Fibonacci number, is achieved only for special assembly graphs, called tangled cords.

Let $\{a_n(x)\}_{n\geq 1}$ be the sequence of digits of $x\in (0,1)$ in an infinite iterated function system with polynomial decay of the derivative. We first study the multifractal spectrum of the convergence exponent, defined by the sequence of digits $\{a_n(x)\}_{n\geq 1}$ and the weighted products of distinct digits with finite numbers, and then calculate the Hausdorff dimensions of the intersections of sets defined by the convergence exponent of the weighted product of distinct digits with finite numbers and sets of points whose digits are nondecreasing in such iterated function systems.

The compactly supported $\mathbb {A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm {K}_0(\mathrm {Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm {GW}(k)$ of the base field k. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on $\mathrm {GW}(k)$ and show that the compactly supported $\mathbb {A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently étale k-algebras. In this paper, we define the class $\mathrm {Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb {A}^1$-Euler characteristic respects the power structures and study the algebraic properties of the subring $\mathrm {K}_0(\mathrm {Sym}_k)$ of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb {A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.