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 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 $\mathbb{Q}[G]$.

Let F be a subfield of the complex numbers and $f(x)=x^6+ax^5+bx^4+cx^3+bx^2+ax+1 \in F[x]$ an irreducible polynomial. We give an elementary characterisation of the Galois group of $f(x)$ as a transitive subgroup of $S_6$. The method involves determining whether three expressions involving a, b and c are perfect squares in F and whether a related quartic polynomial has a linear factor. As an application, we produce one-parameter families of reciprocal sextic polynomials with a specified Galois group.

We investigate a novel geometric Iwasawa theory for ${\mathbf Z}_p$ -extensions of function fields over a perfect field k of characteristic $p>0$ by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over k associated with a ${\mathbf Z}_p$ -extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty $ . By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$ -module structure of the space $M_n:=H^0(X_n, \Omega ^1_{X_n/k})$ of global regular differential forms as $n\rightarrow \infty .$ For example, for each tower in a basic class of ${\mathbf Z}_p$ -towers, we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + \lambda _r n + c_r(n)$ for all n sufficiently large, where $a_r, \lambda _r$ are rational constants and $c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$ is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on ${\mathbf Z}_p$ -towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$ .

Let K be a number field, let A be a finite-dimensional K-algebra, let $\operatorname {\mathrm {J}}(A)$ denote the Jacobson radical of A and let $\Lambda $ be an $\mathcal {O}_{K}$ -order in A. Suppose that each simple component of the semisimple K-algebra $A/{\operatorname {\mathrm {J}}(A)}$ is isomorphic to a matrix ring over a field. Under this hypothesis on A, we give an algorithm that, given two $\Lambda $ -lattices X and Y, determines whether X and Y are isomorphic and, if so, computes an explicit isomorphism $X \rightarrow Y$ . This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: Given a number field K, a positive integer n and two matrices $A,B \in \mathrm {Mat}_{n}(\mathcal {O}_{K})$ , determine whether A and B are similar over $\mathcal {O}_{K}$ , and if so, return a matrix $C \in \mathrm {GL}_{n}(\mathcal {O}_{K})$ such that $B= CAC^{-1}$ . We give explicit examples that show that the implementation of the latter algorithm for $\mathcal {O}_{K}=\mathbb {Z}$ vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.

By analogy with the trace of an algebraic integer $\alpha $ with conjugates $\alpha _1=\alpha , \ldots , \alpha _d$ , we define the G-measure $ {\mathrm {G}} (\alpha )= \sum _{i=1}^d ( |\alpha _i| + 1/ | \alpha _i | )$ and the absolute ${\mathrm G}$ -measure ${\mathrm {g}}(\alpha )={\mathrm {G}}(\alpha )/d$ . We establish an analogue of the Schur–Siegel–Smyth trace problem for totally positive algebraic integers. Then we consider the case where $\alpha $ has all its conjugates in a sector $| \arg z | \leq \theta $ , $0 < \theta < 90^{\circ }$ . We compute the greatest lower bound $c(\theta )$ of the absolute G-measure of $\alpha $ , for $\alpha $ belonging to $11$ consecutive subintervals of $]0, 90 [$ . This phenomenon appears here for the first time, conforming to a conjecture of Rhin and Smyth on the nature of the function $c(\theta )$ . All computations are done by the method of explicit auxiliary functions.

Let $\alpha $ be a totally positive algebraic integer of degree d, with conjugates $\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$ . The absolute $S_k$ -measure of $\alpha $ is defined by $s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$ . We compute the lower bounds $\upsilon _k$ of $s_k(\alpha )$ for each integer in the range $2\leq k \leq 15$ and give a conjecture on the results for integers $k>15$ . Then we derive the lower bounds of $s_k(\alpha )$ for all real numbers $k>2$ . Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.

Let $f(x)=x^{6}+ax^{4}+bx^{2}+c$ be an irreducible sextic polynomial with coefficients from a field $F$ of characteristic $\neq 2$, and let $g(x)=x^{3}+ax^{2}+bx+c$. We show how to identify the conjugacy class in $S_{6}$ of the Galois group of $f$ over $F$ using only the discriminants of $f$ and $g$ and the reducibility of a related sextic polynomial. We demonstrate that our method is useful for producing one-parameter families of even sextic polynomials with a specified Galois group.

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over $\mathbb{Q}$ with good reduction away from 3, up to $\mathbb{Q}$-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.

In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in $\mathbb{F}_{p^{6}}$ with the number field sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special-$\mathfrak{q}$ strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various polynomial selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special-$\mathfrak{q}$ and various enumeration strategies.

Let $K/\mathbb{Q}$ be Galois and let $\eta \,\in \,{{K}^{\times }}$ be such that $\text{Re}{{\text{g}}_{\infty }}\left( \eta \right)\,\ne \,0$. We define the local $\theta $-regulator $\Delta _{p}^{\theta }\left( \eta \right)\,\in \,{{\mathbb{F}}_{p}}$ for the ${{\mathbb{Q}}_{p}}$-irreducible characters $\theta $ of $G=\,\text{Gal}\left( K/\mathbb{Q} \right)$. Let ${{V}_{\theta }}$ be the $\theta $-irreducible representation. A linear representation ${{\mathfrak{L}}^{\theta }}\,\simeq \,\delta \,{{V}_{\theta }}$ is associated with $\Delta _{p}^{\theta }\left( \eta \right)$ whose nullity is equivalent to $\delta \,\ge \,1$. Each $\Delta _{p}^{\theta }\left( \eta \right)$ yields $\text{R}eg_{p}^{\theta }\left( \eta \right)$ modulo $p$ in the factorization ${{\Pi }_{\theta }}{{\left( \text{Reg}_{p}^{\theta }\left( \eta \right) \right)}^{\phi \left( 1 \right)}}$ of $\text{Reg}_{p}^{G}\,\left( \eta \right)\,:=\frac{\text{Re}{{\text{g}}_{p}}\left( \eta \right)}{_{p}[K\,:\,\mathbb{Q}]}$ (normalized $p$-adic regulator). From Prob $\left( \Delta _{p}^{\theta }\left( \eta \right)=0\,\text{and}\,{{\mathfrak{L}}^{\theta }}\simeq \delta {{V}_{\theta }} \right)\,\le {{p}^{-f{{\delta }^{2}}}}$ ($f\,\ge \,1$ is a residue degree) and the Borel-Cantelli heuristic, we conjecture that for $p$ large enough, $\text{Reg}_{p}^{G}\left( \eta \right)$ is a $p$-adic unit or ${{p}^{\phi \left( 1 \right)}}\,||\,\text{Reg}_{p}^{G}\left( \eta \right)$ (a single $\theta $ with $f\,=\,\delta \,=\,1$); this obstruction may be led assuming the existence of a binomial probability law confirmed through numerical studies (groups ${{C}_{3,}}\,{{C}_{5}},\,{{D}_{6}}$) is conjecture would imply that for all $p$ large enough, Fermat quotients, normalized $p$-adic regulators are $p$-adic units and that number fields are $p$-rational.We recall some deep cohomological results that may strengthen such conjectures.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f\in S_2(\Gamma _0(N))$ be a normalized newform such that the abelian variety $A_f$ attached by Shimura to $f$ is the Jacobian of a genus-two curve. We give an efficient algorithm for computing Galois representations associated to such newforms.

Let α be a totally positive algebraic integer of degree d≥2 and α1=α,α2,…,αd be all its conjugates. We use explicit auxiliary functions to improve the known lower bounds of Sk/d, where Sk=∑ di=1αki and k=1,2,3. These improvements have consequences for the search of Salem numbers with negative traces.

For p=3 and p=5, we exhibit a finite nonsolvable extension of ℚ which is ramified only at p, proving in the affirmative a conjecture of Gross. Our construction involves explicit computations with Hilbert modular forms.

In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-$\tau $ expansion of integers in the number fields $\mathbb{Q}\left( \sqrt{-3} \right)$ and $\mathbb{Q}\left( \sqrt{-7} \right)$. The (window) nonadjacent form of $\tau $ -expansion of integers in $\mathbb{Q}\left( \sqrt{-7} \right)$ was first investigated by Solinas. For integers in $\mathbb{Q}\left( \sqrt{-3} \right)$, the nonadjacent form and the window nonadjacent form of the $\tau $ -expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-$\tau $ expansions for integers in all Euclidean imaginary quadratic number fields.

Let $K$ be a real quadratic field, and $p$ a rational prime which is inert in $K$. Let $\alpha $ be a modular unit on ${{\Gamma }_{0}}(N)$. In an earlier joint article with Henri Darmon, we presented the definition of an element $u\left( \alpha ,\,\text{ }\!\!\tau\!\!\text{ } \right)\,\in \,K_{P}^{\times }$ attached to $\alpha $ and each $\tau \,\in \,K$. We conjectured that the $p$-adic number $u(\alpha ,\,\tau )$ lies in a specific ring class extension of $K$ depending on $\tau $, and proposed a “Shimura reciprocity law” describing the permutation action of Galois on the set of $u(\alpha ,\,\tau )$. This article provides computational evidence for these conjectures. We present an efficient algorithm for computing $u(\alpha ,\,\tau )$, and implement this algorithm with the modular unit $\alpha (z)\,=\,\Delta {{(z)}^{2}}\,\Delta (4z)\,/\,\Delta {{(2z)}^{3}}$. Using $p\,=\,3,\,5,\,7\,and\,11$, and all real quadratic fields $K$ with discriminant $D\,<\,500$ such that 2 splits in $K$ and $K$ contains no unit of negative norm, we obtain results supporting our conjectures. One of the theoretical results in this paper is that a certain measure used to define $u(\alpha ,\,\tau )$ is shown to be $\mathbf{Z}$-valued rather than only ${{\mathbf{Z}}_{P}}\,\cap \,\mathbf{Q}-$valued; this is an improvement over our previous result and allows for a precise definition of $u(\alpha ,\,\tau )$, instead of only up to a root of unity.

We prove, for a field $K$ which is cyclic of odd prime power degree over the rationals, that the annihilator of the quotient of the units of $K$ by a suitable large subgroup (constructed from circular units) annihilates what we call the non-genus part of the class group. This leads to stronger annihilation results for the whole class group than a routine application of the Rubin–Thaine method would produce, since the part of the class group determined by genus theory has an obvious large annihilator which is not detected by that method; this is our reason for concentrating on the non-genus part. The present work builds on and strengthens previous work of the authors; the proofs are more conceptual now, and we are also able to construct an example which demonstrates that our results cannot be easily sharpened further.