We show that every modular form on Γ0(2n) (n ⩾ 2) can be expressed as a sum of eta-quotients, which is a partial answer to Ono's problem. Furthermore, we construct a primitive generator of the ring class field of the order of conductor 4N (N ⩾ 1) in an imaginary quadratic field in terms of the special value of a certain eta-quotient.

We prove that a nonreal algebraic number $\unicode[STIX]{x1D703}$ with modulus greater than $1$ is a complex Pisot number if and only if there is a nonzero complex number $\unicode[STIX]{x1D706}$ such that the sequence of fractional parts $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ has a finite number of limit points. Also, we characterise those complex Pisot numbers $\unicode[STIX]{x1D703}$ for which there is a convergent sequence of the form $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ for some $\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }$ . These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.

We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$, with degree $2d$ where $d$ is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$.

For any positive integer $M$ we show that there are infinitely many real quadratic fields that do not admit $M$ -ary universal quadratic forms (without any restriction on the parity of their cross coefficients).

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.

We describe an algorithm for finding the coefficients of $F(X)$ modulo powers of $p$ , where $p\neq 2$ is a prime number and $F(X)$ is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic ${\it\lambda}$ -invariants attached to those cubic extensions $K/\mathbb{Q}$ with cyclic Galois group ${\mathcal{A}}_{3}$ (up to field discriminant ${<}10^{7}$ ), and also tabulate the class number of $K(e^{2{\it\pi}i/p})$ for $p=5$ and $p=7$ . If the ${\it\lambda}$ -invariant is greater than zero, we then determine all the zeros for the corresponding branches of the $p$ -adic $L$ -function and deduce ${\rm\Lambda}$ -monogeneity for the class group tower over the cyclotomic $\mathbb{Z}_{p}$ -extension of $K$ .

Supplementary materials are available with this article.

In this paper, we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic approach for obtaining explicit defining equations for some of these towers and, in particular, give a new explicit example of an optimal tower over a quadratic finite field.

In terms of class field theory we give a necessary and sufficient condition for an integer to be representable by the quadratic form $x^{2}+xy+ny^{2}$ ($n\in \mathbb{N}$ arbitrary) under extra conditions $x\equiv 1\;\text{mod}\;m$, $y\equiv 0\;\text{mod}\;m$ on the variables. We also give some examples where their extended ring class numbers are less than or equal to $3$.

Let $K$ be a number field with ring of integers ${\mathcal{O}}$. After introducing a suitable notion of density for subsets of ${\mathcal{O}}$, generalising the natural density for subsets of $\mathbb{Z}$, we show that the density of the set of coprime $m$-tuples of algebraic integers is $1/{\it\zeta}_{K}(m)$, where ${\it\zeta}_{K}$ is the Dedekind zeta function of $K$. This generalises a result found independently by Mertens [‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math. 77 (1874), 289–338] and Cesàro [‘Question 75 (solution)’, Mathesis 3 (1883), 224–225] concerning the density of coprime pairs of integers in $\mathbb{Z}$.

Using the elements of order four in the narrow ideal class group, we construct generators of the maximal elementary $2$-class group of real quadratic number fields with even discriminant which is a sum of two squares and with fundamental unit of positive norm. We then give a characterization of when two of these generators are equal in the narrow sense in terms of norms of Gaussian integers.

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

In previous work, Ohno conjectured, and Nakagawa proved, relations between thecounting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of ‘extra functional equations’ involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper, we generalize their result by proving a similar identity relating certain degree-$\ell$ fields to Galois groups $D_{\ell }$ and $F_{\ell }$, respectively, for any odd prime $\ell$; in particular, we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.

We study the growth of $\unicode[STIX]{x0428}$ and $p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive ${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are $p$-adic and $l$-adic Lie extensions for $l\neq p$, in particular cyclotomic and other $\mathbb{Z}_{l}$-extensions.

In this article we explain how the results in our previous article on ‘algebraic Hecke characters and compatible systems of mod $p$ Galois representations over global fields’ allow one to attach a Hecke character to every cuspidal Drinfeld modular eigenform from its associated crystal that was constructed in earlier work of the author. On the technical side, we prove along the way a number of results on endomorphism rings of ${\it\tau}$-sheaves and crystals. These are needed to exhibit the close relation between Hecke operators as endomorphisms of crystals on the one side and Frobenius automorphisms acting on étale sheaves associated to crystals on the other. We also present some partial results on the ramification of Hecke characters associated to Drinfeld modular eigenforms. An important phenomenon absent from the case of classical modular forms is that ramification can also result from places of modular curves of good but non-ordinary reduction. In an appendix, jointly with Centeleghe we prove some basic results on $p$-adic Galois representations attached to $\text{GL}_{2}$-type cuspidal automorphic forms over global fields of characteristic $p$.

Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf{Q}$ . The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let $p$ be an odd prime split in $K/\mathbf{Q}$ . We prove the non-triviality of the $p$ -adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf{Z}_{p}$ -anticyclotomic extension of  $K$ . The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the $\mathbf{Z}_{p}$ -anticyclotomic extension of  $K$ .

We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behaviour near roots of unity.

Let $f$ be a rational function on an algebraic curve over the complex numbers. For a point $p$ and local parameter $x$ we can consider the Taylor series for $f$ in the variable $x$. In this paper we give an upper bound on the frequency with which the terms in the Taylor series have $0$ as their coefficient.

The discriminant of a trinomial of the form $x^{n}\pm \,x^{m}\pm \,1$ has the form $\pm n^{n}\pm (n-m)^{n-m}m^{m}$ if $n$ and $m$ are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, when $n$ is congruent to 2 (mod 6) we have that $((n^{2}-n+1)/3)^{2}$ always divides $n^{n}-(n-1)^{n-1}$ . In addition, we discover many other square factors of these discriminants that do not fit into these parametric families. The set of primes whose squares can divide these sporadic values as $n$ varies seems to be independent of $m$ , and this set can be seen as a generalization of the Wieferich primes, those primes $p$ such that $2^{p}$ is congruent to 2 (mod $p^{2}$ ). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants.