The tame Gras–Munnier Theorem gives a criterion for the existence of a $ {\mathbb Z}/p{\mathbb Z} $-extension of a number field K ramified at exactly a tame set S of places of K, the finite $v \in S$ necessarily having norm $1$ mod p. The criterion is the existence of a nontrivial dependence relation on the Frobenius elements of these places in a certain governing extension. We give a short new proof which extends the theorem by showing the subset of elements of $H^1(G_S,{\mathbb {Z}}/p{\mathbb {Z}})$ giving rise to such extensions of K has the same cardinality as the set of these dependence relations. We then reprove the key Proposition 2.2 using the more sophisticated Greenberg–Wiles formula based on global duality.

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 $n\equiv11\pmod{24}$.

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 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.

Let K/k be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for K/k and the defect of weak approximation for the norm one torus \[R_{K/k}^1{\mathbb{G}_m}\] . We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of K/k has symmetric or alternating Galois group.

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 $x^2-dy^2=-1$ .

We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.

Let K be an imaginary quadratic field different from $\open{Q}(\sqrt {-1})$ and $\open{Q}(\sqrt {-3})$. For a positive integer N, let KN be the ray class field of K modulo $N {\cal O}_K$. By using the congruence subgroup ± Γ1(N) of SL2(ℤ), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal(KN/K). We also present an algorithm to find all distinct form classes and show how to multiply two form classes. As an application, we describe Gal(KNab/K) in terms of these extended form class groups for which KNab is the maximal abelian extension of K unramified outside prime ideals dividing $N{\cal O}_K$.

A number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.

We use higher ideles and duality theorems to develop a universal approach to higher dimensional class field theory.

In order to study $p$-adic étale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$. Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wildly ramified class field theory for the open subvariety $U$.

For a finite abelian $p$-group $A$ of rank $d\,=\,\dim\,A/pA$, let ${{\mathbb{M}}_{A}}\,:=\,\text{lo}{{\text{g}}_{p}}\,{{\left| A \right|}^{1/d}}$ be its (logarithmic) mean exponent. We study the behavior of the mean exponent of $p$-class groups in pro-$p$ towers $\text{L/K}$ of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-$p$ towers in which the mean exponent of $p$-class groups remains bounded. Several explicit examples are given with $p\,=\,2$. Turning to group theory, we introduce an invariant $\underline{\mathbb{M}}\left( G \right)$ attached to a finitely generated pro-$p$ group $G$; when $G\,=\,\text{Gal}\left( \text{L/K} \right)$, where $L$ is the Hilbert $p$-class field tower of a number field $K$, $\underline{\mathbb{M}}\left( G \right)$ measures the asymptotic behavior of the mean exponent of $p$-class groups inside $\text{L/K}$. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

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.

We show that every modular form on Γ0(2n) (n ⩾ 2) can be expressed as a sum of eta-quotients, which is a partial answer to Ono's problem. Furthermore, we construct a primitive generator of the ring class field of the order of conductor 4N (N ⩾ 1) in an imaginary quadratic field in terms of the special value of a certain eta-quotient.

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.

In this article we explain how the results in our previous article on ‘algebraic Hecke characters and compatible systems of mod $p$ Galois representations over global fields’ allow one to attach a Hecke character to every cuspidal Drinfeld modular eigenform from its associated crystal that was constructed in earlier work of the author. On the technical side, we prove along the way a number of results on endomorphism rings of ${\it\tau}$-sheaves and crystals. These are needed to exhibit the close relation between Hecke operators as endomorphisms of crystals on the one side and Frobenius automorphisms acting on étale sheaves associated to crystals on the other. We also present some partial results on the ramification of Hecke characters associated to Drinfeld modular eigenforms. An important phenomenon absent from the case of classical modular forms is that ramification can also result from places of modular curves of good but non-ordinary reduction. In an appendix, jointly with Centeleghe we prove some basic results on $p$-adic Galois representations attached to $\text{GL}_{2}$-type cuspidal automorphic forms over global fields of characteristic $p$.

This paper presents an algorithm to construct cryptographically strong genus $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ curves and their Kummer surfaces via Rosenhain invariants and related Kummer parameters. The most common version of the complex multiplication (CM) algorithm for constructing cryptographic curves in genus 2 relies on the well-studied Igusa invariants and Mestre’s algorithm for reconstructing the curve. On the other hand, the Rosenhain invariants typically have much smaller height, so computing them requires less precision, and in addition, the Rosenhain model for the curve can be written down directly given the Rosenhain invariants. Similarly, the parameters for a Kummer surface can be expressed directly in terms of rational functions of theta constants. CM-values of these functions are algebraic numbers, and when computed to high enough precision, LLL can recognize their minimal polynomials. Motivated by fast cryptography on Kummer surfaces, we investigate a variant of the CM method for computing cryptographically strong Rosenhain models of curves (as well as their associated Kummer surfaces) and use it to generate several example curves at different security levels that are suitable for use in cryptography.

Let E be an elliptic curve defined over a number field F, and let Σ be a finite set of finite places of F. Let L(s, E, ψ) be the L-function of E twisted by a finite-order Hecke character ψ of F. It is conjectured that L(s, E, ψ) has a meromorphic continuation to the entire complex plane and satisfies a functional equation s ↔ 2 − s. Then one can define the so called minimal order of vanishing ats = 1 of L(s, E, ψ), denoted by m(E, ψ) (see Section 2 for the definition).

In this paper we compute and continue meromorphically to the entire complex plane the zeta functions of twisted quaternionic Shimura varieties of dimension 3. The twist of the quaternionic Shimura varieties is done by a mod ℘ representation of the absolute Galois group.