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.

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren [‘An elementary test for the Galois group of a quartic polynomial’, Amer. Math. Monthly 96(2) (1989), 133–137], to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the literature. Finally, we devote a separate section to the cyclic case.

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We prove that there exist exactly three distinct monogenic trinomials of the form $x^4+bx^2+d$ whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.

For an elliptic curve E defined over a number field K and $L/K$ a Galois extension, we study the possibilities of the Galois group Gal$(L/K)$, when the Mordell–Weil rank of $E(L)$ increases from that of $E(K)$ by a small amount (namely 1, 2, and 3). In relation with the vanishing of corresponding L-functions at $s=1$, we prove several elliptic analogues of classical theorems related to Artin’s holomorphy conjecture. We then apply these to study the analytic minimal subfield, first introduced by Akbary and Murty, for the case when order of vanishing is 2. We also investigate how the order of vanishing changes as rank increases by 1 and vice versa, generalizing a theorem of Kolyvagin.

We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions $K/\mathbb{Q}$ with Galois group isomorphic to $A_4$, $S_4$, $A_5$, and dihedral groups of order $2p^n$ for certain prime powers $p^n$. In particular, when $K/\mathbb{Q}$ is a Galois extension with Galois group $G$ isomorphic to $A_4$, $S_4$ or $A_5$, we give necessary and sufficient conditions for the ring of integers $\mathcal{O}_{K}$ to be free over its associated order in the rational group algebra $\mathbb{Q}[G]$.

Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $\gamma$ of elements of S, one can associate an arboreal representation to $\gamma$, generalising the case of iterating a single polynomial. We study the probability that a random sequence $\gamma$ produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets S defined over $\mathbb{Z}[t]$, and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all S possessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.

In this paper, we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over ${\mathbb {Q}}$ satisfying $\max \{|a_2|, |a_1|, |a_0|\} \leq X$ is bounded from above by $O(X (\log X)^2)$. We also count the number of abelian monic binary cubic forms with integer coefficients up to a natural equivalence relation ordered by the so-called Bhargava–Shankar height. Finally, we prove an assertion characterizing the splitting field of 2-torsion points of semi-stable abelian elliptic curves.

We give a family of real quadratic fields such that the 2-class field towers over their cyclotomic $\mathbb Z_2$ -extensions have metabelian Galois groups of abelian invariants $[2,2,2]$ . We also consider the boundedness of the Galois groups in relation to Greenberg’s conjecture, and calculate their class-2 quotients with an explicit example.

We investigate unramified extensions of number fields with prescribed solvable Galois group G and certain extra conditions. In particular, we are interested in the minimal degree of a number field K, Galois over $\mathbb {Q}$ , such that K possesses an unramified G-extension. We improve the best known bounds for the degree of such number fields K for certain classes of solvable groups, in particular for nilpotent groups.

We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.

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.

It is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.

Let $p$ be an odd prime. For a number field $K$, we let $K_{\infty }$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\text{Gal}(K_{\infty }/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite $p$-group occurs as the $p$-class tower group of an imaginary quadratic field. Here we do the same for an arbitrary real quadratic field $K$ as base. As before, the action of $\text{Gal}(K/\mathbb{Q})$ on the $p$-class tower group of $K$ plays a crucial role; however, the presence of units of infinite order in the ground field significantly complicates the possibilities for the groups that can occur. We also sharpen our results in the imaginary quadratic field case by removing a certain hypothesis, using ideas of Boston and Wood. In the appendix, we show how the probabilities introduced for finite $p$-groups can be extended in a consistent way to the infinite pro-$p$ groups which can arise in both the real and imaginary quadratic settings.

We provide evidence for this conclusion: given a finite Galois cover $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We show that this holds if the number of branch points of $f$ is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of $\mathbb{Q}$ of given group and bounded discriminant. This widely extends a result of Granville on the lack of $\mathbb{Q}$-rational points on quadratic twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the abc-conjecture (a diophantine reformulation of the case $G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a few finite groups $G$ for which the above conclusion holds unconditionally for almost all covers of $\mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$. We also introduce a local–global principle for specializations of Galois covers $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ and show that it often fails if $f$ has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local–global conclusion underscores the ‘smallness’ of the specialization set of a Galois cover of $\mathbb{P}_{\mathbb{Q}}^{1}$. On the other hand, it allows to generate conditionally ‘many’ curves over $\mathbb{Q}$ failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

Let $k$ be an imaginary quadratic field with $\operatorname{Cl}_{2}(k)\simeq V_{4}$. It is known that the length of the Hilbert $2$-class field tower is at least $2$. Gerth (On 2-class field towers for quadratic number fields with$2$-class group of type$(2,2)$, Glasgow Math. J. 40(1) (1998), 63–69) calculated the density of $k$ where the length of the tower is $1$; that is, the maximal unramified $2$-extension is a $V_{4}$-extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.

For any integer $m\neq 0$, we prove that $f(x)=x^{9}+9mx^{6}+192m^{3}$ is irreducible over $\mathbb{Q}$ and that the Galois group of $f(x)$ over $\mathbb{Q}$ is the dihedral group of order 18. Moreover, we show that for infinitely many values of $m$, the splitting fields for $f(x)$ are distinct.

We classify all polynomials $P(X)\in \mathbb{Q}[X]$ with rational coefficients having the property that the quotient $(\unicode[STIX]{x1D706}_{i}-\unicode[STIX]{x1D706}_{j})/(\unicode[STIX]{x1D706}_{k}-\unicode[STIX]{x1D706}_{\ell })$ is a rational number for all quadruples of roots $(\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D706}_{j},\unicode[STIX]{x1D706}_{k},\unicode[STIX]{x1D706}_{\ell })$ with $\unicode[STIX]{x1D706}_{k}\neq \unicode[STIX]{x1D706}_{\ell }$.

Given a faithful action of a finite group $G$ on an algebraic curve $X$ of genus $gx\,\ge \,2$, we give explicit criteria for the induced action of $G$ on the Riemann–Roch space ${{H}^{0}}\left( X,\,{{\mathcal{O}}_{X}}\left( D \right) \right)$ to be faithful, where $D$ is a $G$-invariant divisor on $X$ of degree at least ${{2}_{gX}}\,-\,2$. This leads to a concise answer to the question of when the action of $G$ on the space ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we provide an explicit basis of ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$. Finally, we give applications in deformation theory and in coding theory and discuss the analogous problem for the action of $G$ on the first homology ${{H}_{1}}\left( X,\,\mathbb{Z}/m\mathbb{Z} \right)$ if $X$ is a Riemann surface.

In this article we develop a test to determine whether a sextic polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic to the alternating group ${{A}_{4}}$. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

Following the analogies between 3-manifolds and number rings in arithmetic topology, we study the homology of branched covers of 3-manifolds. In particular, we show some analogues of Iwasawa’s theorems on ideal class groups and unit groups, Hilbert’s Satz 90, and some genus-theory–type results in the context of 3-dimensional topology. We also prove that the 2-cycles valued Tate cohomology of branched Galois covers is a topological invariant, and we give a new insight into the analogy between 2-cycle groups and unit groups.