Motivated by the recent work of Zhi-Wei Sun [‘Problems and results on determinants involving Legendre symbols’, Preprint, arXiv:2405.03626], we study some matrices concerning subgroups of finite fields. For example, let $q\equiv 3\pmod 4$ be an odd prime power and let $\phi $ be the unique quadratic multiplicative character of the finite field $\mathbb {F}_q$. If the set $\{s_1,\ldots ,s_{(q-1)/2}\}=\{x^2:\ x\in \mathbb {F}_q\setminus \{0\}\}$, then we prove that

$$ \begin{align*}\det[t+\phi(s_i+s_j)+\phi(s_i-s_j)]_{1\le i,j\le (q-1)/2}=\bigg(\frac{q-1}{2}t-1\bigg)q^{{(q-3)}/{4}}.\end{align*} $$

This confirms a conjecture of Zhi-Wei Sun.

Let $\ell $ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell $-towers of multigraphs. In this context, growth patterns are realized by certain analogs of Iwasawa invariants, which depend on the prime $\ell $ and the abelian $\ell $-tower of multigraphs. We formulate and study statistical questions about the behavior of the Iwasawa $\mu $ and $\lambda $ invariants.

The Euler–Mascheroni constant $\gamma =0.5772\ldots \!$ is the $K={\mathbb Q}$ example of an Euler–Kronecker constant $\gamma _K$ of a number field $K.$ In this note, we consider the size of the $\gamma _q=\gamma _{K_q}$ for cyclotomic fields $K_q:={\mathbb Q}(\zeta _q).$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that

$$ \begin{align*} \frac{1}{Q}\sum_{Q<q\le 2Q} |{\gamma_q - \log q}| = o(\log Q).\end{align*} $$

In other words, under EH, the $\gamma _q /\!\log q$ in these ranges converge to the one point distribution at $1$ . This theorem refines and extends a previous result of Ford, Luca and Moree for prime $q.$ The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.

Let f be an elliptic modular form and p an odd prime that is coprime to the level of f. We study the link between divisors of the characteristic ideal of the p-primary fine Selmer group of f over the cyclotomic $\mathbb {Z}_p$ extension of $\mathbb {Q}$ and the greatest common divisor of signed Selmer groups attached to f defined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at p to the context of modular forms.

Let $p=3n+1$ be a prime with $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and let $g\in \mathbb {Z}$ be a primitive root modulo p. Let $0<a_1<\cdots <a_n<p$ be all the cubic residues modulo p in the interval $(0,p)$ . Then clearly the sequence $a_1 \bmod p,\, a_2 \bmod p,\ldots , a_n \bmod p$ is a permutation of the sequence $g^3 \bmod p,\,g^6 \bmod p,\ldots , g^{3n} \bmod p$ . We determine the sign of this permutation.

In the mid 80’s Conner and Perlis showed that for cyclic number fields of prime degree p the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of arbitrary degree. Furthermore, for such fields, we give an explicit description of a Gram matrix of the integral trace in terms of the discriminant of the field.

A Ducci sequence is a sequence of integer $n$-tuples generated by iterating the map

$$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$

$P(n)$

$n$

$n$

$p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$

$2$

$x^{2}-py^{2}=-4$

$x$

$y$

We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal.

We study the values of finite multiple harmonic $q$-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetric multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we prove the duality formula for these values, as an example of linear relations, which induce those among FMZVs and SMZVs simultaneously. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic $q$-series at a primitive root of unity of sufficiently large degree.

We study the asymptotic behaviour of the Bloch–Kato–Shafarevich–Tate group of a modular form $f$ over the cyclotomic ${{\mathbb{Z}}_{p}}$-extension of $\mathbb{Q}$ under the assumption that $f$ is non-ordinary at $p$. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using $p$-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara, and Sprung for supersingular elliptic curves.

In this paper, we provide an explicit upper bound for the Loxton–Kedlaya rank of the maximal abelian extension of ℚ.

We give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson's adelic beta functions, in a manner similar to Ihara's definition of $\ell$-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve.

Since its introduction in 2010 by Lyubashevsky, Peikert and Regev, the ring learning with errors problem (ring-LWE) has become a popular building block for cryptographic primitives, due to its great versatility and its hardness proof consisting of a (quantum) reduction from ideal lattice problems. But, for a given modulus $q$ and degree $n$ number field $K$, generating ring-LWE samples can be perceived as cumbersome, because the secret keys have to be taken from the reduction mod $q$ of a certain fractional ideal ${\mathcal{O}}_{K}^{\vee }\subset K$ called the codifferent or ‘dual’, rather than from the ring of integers ${\mathcal{O}}_{K}$ itself. This has led to various non-dual variants of ring-LWE, in which one compensates for the non-duality by scaling up the errors. We give a comparison of these versions, and revisit some unfortunate choices that have been made in the recent literature, one of which is scaling up by ${|\unicode[STIX]{x1D6E5}_{K}|}^{1/2n}$ with $\unicode[STIX]{x1D6E5}_{K}$ the discriminant of $K$. As a main result, we provide, for any $\unicode[STIX]{x1D700}>0$, a family of number fields $K$ for which this variant of ring-LWE can be broken easily as soon as the errors are scaled up by ${|\unicode[STIX]{x1D6E5}_{K}|}^{(1-\unicode[STIX]{x1D700})/n}$.

In this paper, we give a general explicit form of Cassels’ $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.

We give an explicit formula for the resultant of Chebyshev polynomials of the first, second, third, and fourth kinds. We also compute the resultant of modified cyclotomic polynomials.

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.

Until recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.

However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.

In this paper, we study in particular the cyclotomic fields of composite conductor.

Given a number field F and a prime number p; let Fn denote the cyclotomic extension with [Fn : F] = pn; and let denote its ring of integers. We establish an analogue of the classical Iwasawa theorem for the orders of K2i (){p}.

Let be a commutative algebraic group defined over a number field K. For a prime ℘ in K where has good reduction, let N℘,n be the number of n-torsion points of the reduction of modulo ℘ where n is a positive integer. When is of dimension one and n is relatively prime to a fixed finite set of primes depending on , we determine the average values of N℘,n as the prime ℘ varies. This average value as a function of n always agrees with a divisor function.

Let $p$ be a prime, and let ${{\zeta }_{p}}$ be a primitive $p$-th root of unity. The lattices in Craig's family are $(p\,-\,1)$-dimensional and are geometrical representations of the integral $\mathbb{Z}[{{\zeta }_{p}}]$-ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p\,-\,1$ where $149\,\le \,p\,\le \,3001$, Craig's lattices are the densest packings known. Motivated by this, we construct $(p\,-\,1)(q\,-\,1)$-dimensional lattices from the integral $\mathbb{Z}[{{\zeta }_{pq}}]$-ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}{{\left\langle 1\,-\,{{\zeta }_{q}} \right\rangle }^{j}}$, where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.