We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group $A_4, S_4$ and $S_5$. We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying zero density result of Kowalski-Michel, we choose subfamilies of $L$-functions which are zero free close to 1. For these subfamilies, the $L$-functions have the extremal value at $s=1$, and by the class number formula, we obtain the extreme class numbers.

We fix a mistake in the proof of Theorem 1.6 in the paper in the title.

We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the ``obvious'' limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

The fractional Galois ideal is a conjectural improvement on the higher Stickelberger ideals defined at negative integers, and is expected to provide non-trivial annihilators for higher $K$-groups of rings of integers of number fields. In this article, we extend the definition of the fractional Galois ideal to arbitrary (possibly infinite and non-abelian) Galois extensions of number fields under the assumption of Stark's conjectures and prove naturality properties under canonical changes of extension. We discuss applications of this to the construction of ideals in non-commutative Iwasawa algebras.

Let $E$ be an elliptic curve, and let $L_n$ be the Kummer extension generated by a primitive $p^n$-th root of unity and a $p^n$-th root of $a$ for a fixed $a\in \mathbb{Q}^\times-\{\pm 1\}$. A detailed case study by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these authors to predict unbounded and strikingly regular growth for the rank of $E$ over $L_n$ in certain cases. The aim of this note is to explain how some of these predictions might be accounted for by Heegner points arising from a varying collection of Shimura curve parametrisations.

This paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the Sierpiński sieve. These gaskets are given by iterated function systems that do not satisfy the open set condition. We use the methods of Ngai and Wang to compute the dimension of these gaskets.

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

Let $H$ be the Hilbert class field of a CM number field $K$ with maximal totally real subfield $F$ of degree $n$ over $\mathbb{Q}$. We evaluate the second term in the Taylor expansion at $s=0$ of the Galois-equivariant $L$-function $\Theta_{S_{\infty}}(s)$ associated to the unramified abelian characters of $\operatorname{Gal}(H/K)$. This is an identity in the group ring $\mathbb{C}[\operatorname{Gal}(H/K)]$ expressing $\Theta^{(n)}_{S_{\infty}}(0)$ as essentially a linear combination of logarithms of special values $\{\Psi(z_{\sigma})\}$, where $\Psi\colon \mathbb{H}^{n} \rightarrow \mathbb{R}$ is a Hilbert modular function for a congruence subgroup of $SL_{2}(\mathcal{O}_{F})$ and $\{z_{\sigma}: \sigma \in \operatorname{Gal}(H/K)\}$ are CM points on a universal Hilbert modular variety. We apply this result to express the relative class number $h_{H}/h_{K}$ as a rational multiple of the determinant of an $(h_{K}-1) \times (h_{K}-1)$ matrix of logarithms of ratios of special values $\Psi(z_{\sigma})$, thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for $\Psi(z_{\sigma})$ in terms of exponentials of special values of $L$-functions.

Let $E/F$ be a quadratic extension of number fields. In this paper, we show that the genus formula for Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the $2$-rank of the Hilbert kernel of $E$ provided that the $2$-primary Hilbert kernel of $F$ is trivial. However, since the original genus formula is not explicit enough in a very particular case, we first develop a refinement of this formula in order to employ it in the calculation of the $2$-rank of $E$ whenever $F$ is totally real with trivial $2$-primary Hilbert kernel. Finally, we apply our results to quadratic, bi-quadratic, and tri-quadratic fields which include a complete $2$-rank formula for the family of fields $\Q(\sqrt{2},\sqrt{\delta})$ where $\delta$ is a squarefree integer.

Iwasawa's classical asymptotical formula relates the orders of the $p$-parts $X_n$ of the ideal class groups along a $\mathbb{Z}_p$-extension $F_\infty/F$ of a number field $F$ to Iwasawa structural invariants $\la$ and $\mu$ attached to the inverse limit $X_\infty=\varprojlim X_n$. It relies on ``good" descent properties satisfied by $X_n$. If $F$ is abelian and $F_\infty$ is cyclotomic, it is known that the $p$-parts of the orders of the global units modulo circular units $U_n/C_n$ are asymptotically equivalent to the $p$-parts of the ideal class numbers. This suggests that these quotients $U_n/C_n$, so to speak unit class groups, also satisfy good descent properties. We show this directly, \emph{i.e.,} without using Iwasawa's Main Conjecture.

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\geq 0$, we conjecture that $L_n^{(-1-n-r)}(x) = \sum_{j=0}^n \binom{n-j+r}{n-j}x^j/j!$ is a $\Q$-irreducible polynomial whose Galois group contains the alternating group on $n$ letters. That this is so for $r=n$ was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows from recent work of Hajir and Wong that the conjecture is true when $r$ is large with respect to $n\geq 5$. Here we verify it in three situations: i) when $n$ is large with respect to $r$, ii) when $r \leq 8$, and iii) when $n\leq 4$. The main tool is the theory of $p$-adic Newton Polygons.

Let $q$ be an algebraic integer of degree $d \geq 2$. Consider the rank of the multiplicative subgroup of $\BbC^*$ generated by the conjugates of $q$. We say $q$ is of {\em full rank} if either the rank is $d-1$ and $q$ has norm $\pm 1$, or the rank is $d$. In this paper we study some properties of $\BbZ[q]$ where $q$ is an algebraic integer of full rank. The special cases of when $q$ is a Pisot number and when $q$ is a Pisot-cyclotomic number are also studied. There are four main results. \begin{compactenum}[\rm(1)] \item If $q$ is an algebraic integer of full rank and $n$ is a fixed positive integer, then there are only finitely many $m$ such that $\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$. \item If $q$ and $r$ are algebraic integers of degree $d$ of full rank and $\BbZ[q^n] = \BbZ[r^n]$ for infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$, where $r'$ is some conjugate of $r$ and $\omega$ is some root of unity. \item Let $r$ be an algebraic integer of degree at most $3$. Then there are at most $40$ Pisot numbers $q$ such that $\BbZ[q] = \BbZ[r]$. \item There are only finitely many Pisot-cyclotomic numbers of any fixed order. \end{compactenum}

Let $X$ be a smooth projective geometrically connected curve over a finite field with function field $K$. Let $\G$ be a connected semisimple group scheme over $X$. Under certain hypotheses we prove the equality of two numbers associated with $\G$. The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of $\G$-torsors on $X$. Our results are most useful for studying connected components as much is known about Tamagawa numbers.

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 $\Q(\sqrt{-3})$ and $\Q(\sqrt{-7})$. The (window) nonadjacent form of $\tau$-expansion of integers in $\Q(\sqrt{-7})$ was first investigated by Solinas. For integers in $\Q(\sqrt{-3})$, 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 totally real number field of degree $n$. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained from geometric considerations. This result depends on unfolding an integral over a compact torus.

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 $u(\alpha, \tau) \in K_p^\times$ 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.

We introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraic $K$-groups of rings of integers in number fields. Our conjecture is motivic in the sense that it involves the (transcendental) Borel regulator as well as being related to $l$-adic \'{e}tale cohomology. In addition, the conjecture generalises the well-known Coates--Sinnott conjecture. For example, for a totally real extension when $r = -2, -4, -6, \dotsc$ the Coates--Sinnott conjecture merely predicts that zero annihilates $K_{-2r}$ of the ring of $S$-integers while our conjecture predicts a non-trivial annihilator. By way of supporting evidence, we prove the corresponding (conjecturally equivalent) conjecture for the Galois action on the \'{e}tale cohomology of the cyclotomic extensions of the rationals.

Let $E$ be an elliptic curve defined over $\Q$ and without complex multiplication. Let $K$ be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes $p \leq x$ for which $\Q(\pi_p) = K$, where $\pi_p$ denotes the Frobenius endomorphism of $E$ at $p$. More precisely, under a generalized Riemann hypothesis we show that this number is $O_{E}(x^{\slfrac{17}{18}}\log x)$, and unconditionally we show that this number is $O_{E, K}\bigl(\frac{x(\log \log x)^{\slfrac{13}{12}}} {(\log x)^{\slfrac{25}{24}}}\bigr)$. We also prove that the number of imaginary quadratic fields $K$, with $-\disc K \leq x$ and of the form $K = \Q(\pi_p)$, is $\gg_E\log\log\log x$ for $x\geq x_0(E)$. These results represent progress towards a 1976 Lang--Trotter conjecture.

Let $E_{/\qq}$ be an elliptic curve with good ordinary reduction at a prime $p>2$. It has a well-defined Iwasawa $\mu$-invariant $\mu(E)_p$ which encodes part of the information about the growth of the Selmer group $\sel E{K_n}$ as $K_n$ ranges over the subfields of the cyclotomic $\zzp$-extension $K_\infty/\qq$. Ralph Greenberg has conjectured that any such $E$ is isogenous to a curve $E'$ with $\mu(E')_p=0$. In this paper we prove Greenberg's conjecture for infinitely many curves $E$ with a rational $p$-torsion point, $p=3$ or $5$, no two of our examples having isomorphic $p$-torsion. The core of our strategy is a partial explicit evaluation of the global duality pairing for finite flat group schemes over rings of integers.

To every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the {\it minimum number of wild primes}. Earlier investigations have established lower bounds for this number. In this paper an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered to be an exact formula for the minimum number of wild primes.

Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in $K$. Let $p$ be a rational prime that splits as $\mathfrak{p}_{1}\mathfrak{p}_{2}$ in $K$. Let $E_{p^{n}}$ denote the $p^{n}$-division points on $E$. Assume that $F(E_{p^{n}})$ is abelian over $K$ for all $n\geq 0$. This paper proves that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty}$-Selmer group of $E$ over $F(E_{p^{\infty}})$ is a finitely generated free $\Lambda$-module, where $\Lambda$ is the Iwasawa algebra of $\Gal\bigl(F(E_{p^{\infty}})/ F(E_{\mathfrak{p}_{1}^{\infty}\mathfrak{p}_{2}})\bigr)$. It also gives a simple formula for the rank of the Pontrjagin dual as a $\Lambda$-module.

Let $K$ be a finite Galois extension of the field of rational numbers with unit rank greater than~3. We prove that the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally.

We provide the first unconditional proof that the ring $\mathbb{Z} [\sqrt{14}]$ is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of $\mathbb{Q}$. It is proved that if $K$ is a real quadratic field (modulo the existence of two special primes of $K$) or if $K$ is a cyclotomic extension of $\mathbb{Q}$ then: \begin{center} \emph{% the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain.} \end{center} The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when $K$ is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property.