Composition of binary quadratic forms over number fields

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

A Density of Ramified Primes

Let K be a cyclic number field of odd degree over $${\mathbb {Q}}$$ Q with odd narrow class number, such that 2 is inert in $$K/{\mathbb {Q}}$$ K / Q . We define a family of number fields $$\{K(p)\}_p$$ { K ( p ) } p , depending on K and indexed by the rational primes p that split completely in $$K/{\mathbb {Q}}$$ K / Q , in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in $$K(p)/{\mathbb {Q}}$$ K ( p ) / Q is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension $$K/{\mathbb {Q}}$$ K / Q . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.

Redei reciprocity, governing fields and negative Pell

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation \[{x^2} - d{y^2} = - 1\] .

Sider’s Puzzle and the Mormon Afterlife

There is a puzzle about divine justice stemming from the fact that God seems required to judge on the basis of criteria that are vague. Justice is proportional, however, it seems God violates proportionality by sending those on the borderline of heaven to an eternity in hell. This is Ted Sider’s problem of Hell and Vagueness. On the face of things, this poses a challenge only to a narrow class of classical Christians, those that hold a retributive theory of divine punishment. We show that this puzzle can be extended to the picture of divine judgement and the afterlife found in Mormon theology. This is significant because at first glance, the Mormon picture of the afterlife looks like it fails to co-operate with Sider’s puzzle. In Mormon theology, there are not two afterlife states, but three: a low, a middle, and a high kingdom. There is no afterlife state quite like Hell, and the states that function similarly to Hell aren’t places of eternal suffering. We argue that appearances are misleading. While it may be true that no place in the Mormon afterlife is bad in the sense that its inhabitants suffer eternal bodily harm, it is true that many of the places in the Mormon afterlife are bad in the sense that their inhabitants lack access to significant goods. This allows Sider’s puzzle to re-engage as a puzzle about distributive Justice. After setting out this alternative version of the puzzle, we argue that Mormon theology has sufficient resources to reject proportionality as a constraint on divine judgment by adopting a nuanced version of universalism called escapism.

Higher genus theory

In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

ON IWASAWA THEORY OF RUBIN–STARK UNITS AND NARROW CLASS GROUPS

Let K be a totally real number field of degree r. Let K∞ denote the cyclotomic -extension of K, and let L∞ be a finite extension of K∞, abelian over K. The goal of this paper is to compare the characteristic ideal of the χ-quotient of the projective limit of the narrow class groups to the χ-quotient of the projective limit of the rth exterior power of totally positive units modulo a subgroup of Rubin–Stark units, for some $\overline{\mathbb{Q}_{2}}$-irreducible characters χ of Gal(L∞/K∞).

On the 8-rank of narrow class groups of ℚ(−4pq), ℚ(−8pq), and ℚ(8pq)

Let [Formula: see text]. We study the [Formula: see text]-part of the narrow class group in thin families of quadratic number fields of the form [Formula: see text], where [Formula: see text] are prime numbers, and we prove new lower bounds for the proportion of narrow class groups in these families that have an element of order [Formula: see text]. In the course of our proof, we prove a general double-oscillation estimate for the quadratic residue symbol in quadratic number fields.

Homeric Memnon

This chapter uses Homer to triangulate the relationship between inscriber and statue. Memnon is a ghost from the epic past anchored in the Egyptian present; what better way to honor him than to inscribe Homer’s words on his body? This evocation of Homer is not restricted to a narrow class of visitors. Imperial authors such as Lucian and Philostratus engage with Homer but write specifically for an elite audience. The Memnon inscriptions that echo Homer, however, are created by and for a more diverse public. All the inscriptions participate at some level in reactivating the mythical past, as if the trip to Thebes paralleled an epic trip to the Underworld. The chapter argues that visitors who sought out the statue were hoping precisely for such a “close encounter,” an experience that would connect them with the Homeric past, and that this experience transcended differences in social status and educational background.