For a prime p and a rational elliptic curve $E_{/\mathbb {Q}}$, set $K=\mathbb {Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname {ker}\{E\xrightarrow {p} E\}$. The class group $\operatorname {Cl}_K$ is a module over $\operatorname {Gal}(K/\mathbb {Q})$. Given a fixed odd prime number p, we study the average nonvanishing of certain Galois stable quotients of the mod-p class group $\operatorname {Cl}_K/p\operatorname {Cl}_K$. Here, E varies over all rational elliptic curves, ordered according to height. Our results are conditional, since we assume that the p-primary part of the Tate–Shafarevich group is finite. Furthermore, we assume predictions made by Delaunay for the statistical variation of the p-primary parts of Tate–Shafarevich groups. We also prove results in the case when the elliptic curve $E_{/\mathbb {Q}}$ is fixed and the prime p is allowed to vary.

Let $K_n=\mathbb{Q}(\alpha_n)$ be a family of algebraic number fields where $\alpha_n\in \mathbb{C}$ is a root of the nth exponential Taylor polynomial $\frac{x^n}{n!}+ \frac{x^{n-1}}{(n-1)!}+ \cdots +\frac{x^2}{2!}+\frac{x}{1!}+1$, $n\in \mathbb{N}$. In this paper, we give a formula for the exact power of any prime p dividing the discriminant of Kn in terms of the p-adic expansion of n. An explicit p-integral basis of Kn is also given for each prime p. These p-integral bases quickly lead to the construction of an integral basis of Kn.

Let ${\mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field $K = {\mathbb Q}(\theta )$, where $\theta $ is a root of a monic irreducible polynomial $f(x) = x^n + a(bx+c)^m \in {\mathbb {Z}}[x]$, $1\leq m<n$. We say $f(x)$ is monogenic if $\{1, \theta , \ldots , \theta ^{n-1}\}$ is a basis for ${\mathbb {Z}}_K$. We give necessary and sufficient conditions involving only $a, b, c, m, n$ for $f(x)$ to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup ${\mathbb {Z}}[\theta ]$ in ${\mathbb {Z}}_K$. As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group $S_n$, the symmetric group on n letters.

We construct a new family of quintic non-Pólya fields with large Pólya groups. We show that the Pólya number of such a field never exceeds five times the size of its Pólya group. Finally, we show that these non-Pólya fields are nonmonogenic of field index one.

Let $n$ be an integer congruent to $0$ or $3$ modulo $4$. Under the assumption of the ABC conjecture, we prove that, given any integer $m$ fulfilling only a certain coprimeness condition, there exist infinitely many imaginary quadratic fields having an everywhere unramified Galois extension of group $A_n \times C_m$. The same result is obtained unconditionally in special cases.

We study the Galois module structure of the class groups of the Artin–Schreier extensions K over k of extension degree p, where $k:={\mathbb F}_q(T)$ is the rational function field and p is a prime number. The structure of the p-part $Cl_K(p)$ of the ideal class group of K as a finite G-module is determined by the invariant ${\lambda }_n$, where $G:=\operatorname {\mathrm {Gal}}(K/k)=\langle {\sigma } \rangle $ is the Galois group of K over k, and ${\lambda }_n = \dim _{{\mathbb F}_p}(Cl_K(p)^{({\sigma }-1)^{n-1}}/Cl_K(p)^{({\sigma }-1)^{n}})$. We find infinite families of the Artin–Schreier extensions over k whose ideal class groups have guaranteed prescribed ${\lambda }_n$-rank for $1 \leq n \leq 3$. We find an algorithm for computing ${\lambda }_3$-rank of $Cl_K(p)$. Using this algorithm, for a given integer $t \ge 2$, we get infinite families of the Artin–Schreier extensions over k whose ${\lambda }_1$-rank is t, ${\lambda }_2$-rank is $t-1$, and ${\lambda }_3$-rank is $t-2$. In particular, in the case where $p=2$, for a given positive integer $t \ge 2$, we obtain an infinite family of the Artin–Schreier quadratic extensions over k whose $2$-class group rank (resp. $2^2$-class group rank and $2^3$-class group rank) is exactly t (resp. $t-1$ and $t-2$). Furthermore, we also obtain a similar result on the $2^n$-ranks of the divisor class groups of the Artin–Schreier quadratic extensions over k.

For a principal ideal domain $A$, the Latimer–MacDuffee correspondence sets up a bijection between the similarity classes of matrices in $\textrm{M}_{n}(A)$ with irreducible characteristic polynomial $f(x)$ and the ideal classes of the order $A[x]/(f(x))$. We prove that when $A[x]/(f(x))$ is maximal (i.e. integrally closed, i.e. a Dedekind domain), then every similarity class contains a representative that is, in a sense, close to being a companion matrix. The first step in the proof is to show that any similarity class corresponding to an ideal (not necessarily prime) of degree one contains a representative of the desired form. The second step is a previously unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is maximal, every ideal class contains an ideal of degree one.

In this article, we provide an explicit upper bound for $h_K \mathcal {R}_K d_K^{-1/2}$ which depends on an effective constant in the error term of the Ideal Theorem.

We study the discriminants of the minimal polynomials $\mathcal {P}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive $n\equiv 11\pmod {24}$. We show that $\Delta (\mathcal {P}_n)$ divides $\Delta (H_n)$, where $H_n$ is the ring class polynomial, with quotient a perfect square and determine the sign of $\Delta (\mathcal {P}_n)$ based on the ideal class group structure of the order of discriminant $-n$. We also show that the discriminant of the number field generated by $j({(-1+\sqrt {-n})}/{2})$, where j is the j-invariant, divides $\Delta (\mathcal {P}_n)$. Moreover, using Ye’s computation of $\log|\Delta(H_n)|$ [‘Revisiting the Gross–Zagier discriminant formula’, Math. Nachr. 293 (2020), 1801–1826], we show that 3 never divides $\Delta(H_n)$, and thus $\Delta(\mathcal{P}_n)$, for all squarefree $n\equiv11\pmod{24}$.

An explicit formula forthe mean value of $\vert L(1,\chi )\vert ^2$ is known, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Lately, the authors obtained that the mean value of $\vert L(1,\chi )\vert ^2$ is asymptotic to $\pi ^2/6$, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\ \ \pmod {2d}$ which are trivial on a subgroup H of odd order d of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$, provided that $d\ll \frac {\log p}{\log \log p}$. Bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Here, for a given integer $d_0>1$, we consider the same questions for the nonprimitive odd Dirichlet characters $\chi '$ modulo $d_0p$ induced by the odd primitive characters $\chi $ modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi ')\vert ^2$ is asymptotic to $\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order $d\ll \frac {\log p}{\log \log p}$. As a consequence, we improve the previous bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$. Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.

Let $m>1$ and $\mathfrak {d} \neq 0$ be integers such that $v_{p}(\mathfrak {d}) \neq m$ for any prime p. We construct a matrix $A(\mathfrak {d})$ of size $(m-1) \times (m-1)$ depending on only of $\mathfrak {d}$ with the following property: For any tame $ \mathbb {Z}/m \mathbb {Z}$ -number field K of discriminant $\mathfrak {d}$ , the matrix $A(\mathfrak {d})$ represents the Gram matrix of the integral trace-zero form of K. In particular, we have that the integral trace-zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant.

Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let $\mathcal {D}$ denote the set of positive squarefree integers having no prime factors congruent to $3$ modulo $4$ . Stevenhagen [19] conjectured that the density of d in $\mathcal {D}$ such that the negative Pell equation $x^2-dy^2=-1$ is solvable with $x, y \in \mathbb {Z}$ is $58.1\%$ , to the nearest tenth of a percent. By studying the distribution of the $8$ -rank of narrow class groups $\operatorname {\mathrm {Cl}}^+(d)$ of $\mathbb {Q}(\sqrt {d})$ , we prove that the infimum of this density is at least $53.8\%$ .

We use circulant matrices and hyperelliptic curves over finite fields to study some arithmetic properties of certain determinants involving Legendre symbols and kth power residues.

Stark conjectured that for any $h\in \Bbb {N}$ , there are only finitely many CM-fields with class number h. Let $\mathcal {C}$ be the class of number fields L for which L has an almost normal subfield K such that $L/K$ has solvable Galois closure. We prove Stark’s conjecture for $L\in \mathcal {C}$ of degree greater than or equal to 6. Moreover, we show that the generalised Brauer–Siegel conjecture is true for asymptotically good towers of number fields $L\in \mathcal {C}$ and asymptotically bad families of $L\in \mathcal {C}$ .

In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient$\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$. Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$-Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$. The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

For any odd prime p, we construct an infinite family of imaginary quadratic fields whose class numbers are divisible by p. We give a corollary that settles Iizuka’s conjecture for the case n=1 and p>2.

We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

We show that for $100\%$ of the odd, square free integers $n> 0$ , the $4$ -rank of $\text {Cl}(\mathbb{Q} (i, \sqrt {n}))$ is equal to $\omega _3(n) - 1$ , where $\omega _3$ is the number of prime divisors of n that are $3$ modulo $4$ .

We give a formula for the class number of an arbitrary complex mutliplication (CM) algebraic torus over $\mathbb {Q}$ . This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.

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.