We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous Mertens function, expanding upon work of Ng. Finally, we explore properties of the generalised Mertens function of certain dicyclic number fields as consequences of Artin factorisation.

We study the counts of smooth permutations and smooth polynomials over finite fields. For both counts we prove an estimate with an error term that matches the error term found in the integer setting by de Bruijn more than 70 years ago. The main term is the usual Dickman $\rho$ function, but with its argument shifted.

We determine the order of magnitude of $\log(p_{n,m}/\rho(n/m))$ where $p_{n,m}$ is the probability that a permutation on n elements, chosen uniformly at random, is m-smooth.

We uncover a phase transition in the polynomial setting: the probability that a polynomial of degree n in $\mathbb{F}_q$ is m-smooth changes its behaviour at $m\approx (3/2)\log_q n$.

In this paper, we investigate extensions between graded Verma modules in the Bernstein–Gelfand–Gelfand category $\mathcal{O}$. In particular, we determine exactly which information about extensions between graded Verma modules is given by the coefficients of the R-polynomials. We also give some upper bounds for the dimensions of graded extensions between Verma modules in terms of Kazhdan–Lusztig combinatorics. We completely determine all extensions between Verma module in the regular block of category $\mathcal{O}$ for $\mathfrak{sl}_4$ and construct various “unexpected” higher extensions between Verma modules.

A long standing conjecture states that the ropelength of any alternating knot is at least proportional to its crossing number. In this paper we prove that this conjecture is true. That is, there exists a constant $b_0 \gt 0$ such that $R(K)\ge b_0Cr(K)$ for any alternating knot K, where R(K) is the ropelength of K and Cr(K) is the crossing number of K. In this paper, we prove that this conjecture is true and establish that $b_0 \gt 1/56$.

We investigate the joint distribution of L-functions on the line $ \sigma= {1}/{2} + {1}/{G(T)}$ and $ t \in [ T, 2T]$, where $ \log \log T \leq G(T) \leq { \log T}/{ ( \log \log T)^2 } $. We obtain an upper bound on the discrepancy between the joint distribution of L-functions and that of their random models. As an application we prove an asymptotic expansion of a multi-dimensional version of Selberg’s central limit theorem for L-functions on $ \sigma= 1/2 + 1/{G(T)}$ and $ t \in [ T, 2T]$, where $ ( \log T)^\varepsilon \leq G(T) \leq { \log T}/{ ( \log \log T)^{2+\varepsilon } } $ for $ \varepsilon > 0$.

In this note we investigate the centraliser of a linearly growing element of $\mathrm{Out}(F_n)$ (that is, a root of a Dehn twist automorphism), and show that it has a finite index subgroup mapping onto a direct product of certain “equivariant McCool groups” with kernel a finitely generated free abelian group. In particular, this allows us to show it is VF and hence finitely presented.

We show that for an oriented 4-dimensional Poincaré complex X with finite fundamental group, whose 2-Sylow subgroup is abelian with at most 2 generators, the homotopy type of X is determined by its quadratic 2-type.

Let $h : \mathbb{R}^2 \to \mathbb{R}^2$ be an orientation preserving homeomorphism of the plane. For any bounded orbit $\mathcal{O}(x)=\{h^n(x):n\in\mathbb{Z}\}$ there exists a fixed point $p\in\mathbb{R}^2$ of h linked to $\mathcal{O}(x)$ in the sense of Gambaudo: one cannot find a Jordan curve $C\subseteq\mathbb{R}^2$ around $\mathcal{O}(x)$, separating it from p, that is isotopic to h(C) in $\mathbb{R}^2\setminus\left(\mathcal{O}(x)\cup\{p\}\right)$.

We show that if F is $\mathbb{Q}$ or a multiquadratic number field, $p\in\left\{{2,3,5}\right\}$, and $K/F$ is a Galois extension of degree a power of p, then for elliptic curves $E/\mathbb{Q}$ ordered by height, the average dimension of the p-Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K-rank of elliptic curves $E/\mathbb{Q}$ for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F.

The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p, as $E/\mathbb{Q}$ varies, the difference in dimension between the Galois fixed space in the p-Selmer group of $E/K$ and the p-Selmer group of $E/F$ has bounded average.

This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus g curves of fixed degree passing through g points. We compute the tropical multiplicity provided by a correspondence theorem due to T. Nishinou and show that it is possible to refine this multiplicity in the style of the Block–Göttsche refined multiplicity to get tropical refined invariants.

We give a vanishing and classification result for holomorphic differential forms on smooth projective models of the moduli spaces of pointed K3 surfaces. We prove that there is no nonzero holomorphic k-form for $0<k<10$ and for even $k>19$. In the remaining cases, we give an isomorphism between the space of holomorphic k-forms with that of vector-valued modular forms ($10\leq k \leq 18$) or scalar-valued cusp forms (odd $k\geq 19$) for the modular group. These results are in fact proved in the generality of lattice-polarisation.

For odd n we construct a path $\rho\;:\;\thinspace \Pi_1(S) \to SL(n\mathbb{R})$ of discrete, faithful, and Zariski dense representations of a surface group such that $\rho_t(\Pi_1(S)) \subset SL(n,\mathbb{Q})$ for every $t\in \mathbb{Q}$.

In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $\mathrm{GL}_2(\mathbb{Z}_p)$-type V in $\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.

We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a “general” ${\Bbb R}$-ideal. We show that the minimal log discrepancy (“mld” for short) of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustaţă–Nakamura’s conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs.

Given any polynomial in two variables of degree at most three with rational integer coefficients, we obtain a new search bound to decide effectively if it has a zero with rational integer coefficients. On the way we encounter a natural problem of estimating singular points. We solve it using elementary invariant theory but an optimal solution would seem to be far from easy even using the full power of the standard Height Machine.

On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $1/2+i\gamma$ of the Riemann zeta function, we show that the sequence

\begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }} )^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\},\end{equation*}

${\gamma }$

$a<b$

$m_\gamma$

$1/2+i{\gamma }$

${\gamma }$

$\gamma (\!\log T)/2\pi$

${\gamma }\in \Gamma_{[a, b]}$

$0<{\gamma }\leq T$

Let $\mathfrak{A}$ be a finite abelian group. In this paper, we classify harmonic $\mathfrak{A}$-covers of a tropical curve $\Gamma$ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on $\Gamma$. We give a realisability criterion for harmonic $\mathfrak{A}$-covers by patching local monodromy data in an extended homology group on $\Gamma$. As an explicit example, we work out the case $\mathfrak{A}=\mathbb{Z}/p\mathbb{Z}$ and explain how realisability for such covers is related to the nowhere-zero flow problem from graph theory.

Let $G= N\rtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 \rtimes {\rm Bohr}(H)$ and $Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of ${\rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N. In the case where N is abelian, we have ${\rm Bohr}(G)\cong A \rtimes {\rm Bohr}(H)$ and ${\rm Prof}(G)\cong B \rtimes {\rm Prof}(H),$ where A (respectively B) is the dual group of the group of unitary characters of N with finite H-orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= \Lambda\wr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${\rm Bohr}(\Lambda\wr H)$ is isomorphic to ${\rm Bohr}(\Lambda^{\rm Ab}\wr H)$ and ${\rm Prof}(\Lambda\wr H)$ is isomorphic to ${\rm Prof}(\Lambda^{\rm Ab} \wr H),$ where $\Lambda^{\rm Ab}=\Lambda/ [\Lambda, \Lambda]$ is the abelianisation of $\Lambda.$ As examples, we compute ${\rm Bohr}(G)$ and ${\rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.

We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of k-differentials of infinite area are affine varieties for all k. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics.

Erdős, Graham and Selfridge considered, for each positive integer n, the least value of $t_n$ so that the integers $n+1, n+2, \dots, n+t_n $ contain a subset the product of whose members with n is a square. An open problem posed by Granville concerns the size of $t_n$, under the assumption of the ABC conjecture. We establish some results on the distribution of $t_n$, and in the process solve Granville’s problem unconditionally.