There is an algorithm of Schoof for finding divisors of class numbers of real cyclotomic fields of prime conductor. In this paper we introduce an improvement of the elliptic analogue of this algorithm by using a subgroup of elliptic units given by Weierstrass forms. These elliptic units which can be expressed in terms of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x$-coordinates of points on elliptic curves enable us to use the fast arithmetic of elliptic curves over finite fields.

We propose a fast method of calculating the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$-part of the class numbers in certain non-cyclotomic $\mathbb{Z}_p$-extensions of an imaginary quadratic field using elliptic units constructed by Siegel functions. We carried out practical calculations for $p=3$ and determined $\lambda $-invariants of such $\mathbb{Z}_3$-extensions which were not known in our previous paper.

A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields $K$ for the quartic del Pezzo surface $S$ of singularity type ${\boldsymbol{A}}_{3}$ with five lines given in ${\mathbb{P}}_{K}^{4}$ by the equations ${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$.

We find all quadratic post-critically finite (PCF) rational functions defined over $\mathbb{Q}$, up to conjugation by elements of $\mathop{\rm PGL}_2(\overline{\mathbb{Q}})$. We describe an algorithm to search for possibly PCF functions. Using the algorithm, we eliminate all but 12 rational functions, all of which are verified to be PCF. We also give a complete description of all possible rational preperiodic structures for quadratic PCF functions defined over $\mathbb{Q}$.

We prove modularity of some two-dimensional, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$-adic Galois representations over a totally real field that are nearly ordinary at all places above $2$ and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the $2$-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above $2$.

Boyd showed that the beta expansion of Salem numbers of degree 4 were always eventually periodic. Based on an heuristic argument, Boyd had conjectured that the same is true for Salem numbers of degree 6 but not for Salem numbers of degree 8. This paper examines Salem numbers of degree 8 and collects experimental evidence in support of Boyd’s conjecture.

Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar’s method. Computationally, one of the key challenges in the application of Stauduhar’s method is to find, for a given pair of groups $H<G$ , a $G$ -relative $H$ -invariant, that is a multivariate polynomial $F$ that is $H$ -invariant, but not $G$ -invariant. While generic, theoretical methods are known to find such $F$ , in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.

In this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on ${\rm GL}_2$ over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

When the branch character has root number $- 1$ , the corresponding anticyclotomic Katz $p$ -adic $L$ -function vanishes identically. For this case, we determine the $\mu $ -invariant of the cyclotomic derivative of the Katz $p$ -adic $L$ -function. The result proves, as an application, the non-vanishing of the anticyclotomic regulator of a self-dual CM modular form with root number $- 1$ . The result also plays a crucial role in the recent work of Hsieh on the Eisenstein ideal approach to a one-sided divisibility of the CM main conjecture.

We study lens space surgeries along two different families of 2-component links, denoted by ${A}_{m, n} $ and ${B}_{p, q} $, related with the rational homology $4$-ball used in J. Park’s (generalized) rational blow down. We determine which coefficient $r$ of the knotted component of the link yields a lens space by Dehn surgery. The link ${A}_{m, n} $ yields a lens space only by the known surgery with $r= mn$ and unexpectedly with $r= 7$ for $(m, n)= (2, 3)$. On the other hand, ${B}_{p, q} $ yields a lens space by infinitely many $r$. Our main tool for the proof are the Reidemeister-Turaev torsions, that is, Reidemeister torsions with combinatorial Euler structures. Our results can be extended to the links whose Alexander polynomials are same as those of ${A}_{m, n} $ and ${B}_{p, q} $.

Using the $\ell $ -invariant constructed in our previous paper we prove a Mazur–Tate–Teitelbaum-style formula for derivatives of $p$ -adic $L$ -functions of modular forms at trivial zeros. The novelty of this result is to cover the near-central point case. In the central point case our formula coincides with the Mazur–Tate–Teitelbaum conjecture proved by Greenberg and Stevens and by Kato, Kurihara and Tsuji at the end of the 1990s.

A monic polynomial in ${\mathbf{F} }_{q} [t] $ of degree $n$ over a finite field ${\mathbf{F} }_{q} $ of odd characteristic can be written as the sum of two irreducible monic elements in ${\mathbf{F} }_{q} [t] $ of degrees $n$ and $n- 1$ if $q$ is larger than a bound depending only on $n$. The main tool is a sufficient condition for simultaneous primality of two polynomials in one variable $x$ with coefficients in ${\mathbf{F} }_{q} [t] $.

Let $N/ F$ be an odd-degree Galois extension of number fields with Galois group $G$ and rings of integers ${\mathfrak{O}}_{N} $ and ${\mathfrak{O}}_{F} = \mathfrak{O}$. Let $ \mathcal{A} $ be the unique fractional ${\mathfrak{O}}_{N} $-ideal with square equal to the inverse different of $N/ F$. B. Erez showed that $ \mathcal{A} $ is a locally free $\mathfrak{O}[G] $-module if and only if $N/ F$ is a so-called weakly ramified extension. Although a number of results have been proved regarding the freeness of $ \mathcal{A} $ as a $ \mathbb{Z} [G] $-module, the question remains open. In this paper we prove that $ \mathcal{A} $ is free as a $ \mathbb{Z} [G] $-module provided that $N/ F$ is weakly ramified and under the hypothesis that for every prime $\wp $ of $\mathfrak{O}$ which ramifies wildly in $N/ F$, the decomposition group is abelian, the ramification group is cyclic and $\wp $ is unramified in $F/ \mathbb{Q} $. We make crucial use of a construction due to the first author which uses Dwork’s exponential power series to describe self-dual integral normal bases in Lubin–Tate extensions of local fields. This yields a new and striking relationship between the local norm-resolvent and the Galois Gauss sum involved. Our results generalise work of the second author concerning the case of base field $ \mathbb{Q} $.

For each solvable Galois group which appears in degree $9$ and each allowable signature, we find polynomials which define the fields of minimum absolute discriminant.

We give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field, using the theory of Hall algebras and the Langlands correspondence for function fields and ${\mathrm{GL} }_{n} $. As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied by Burban and Schiffmann  [On the Hall algebra of an elliptic curve I, Preprint (2005), arXiv:math/0505148 [math.AG]] and Schiffmann and Vasserot [The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compositio Math. 147 (2011), 188–234]).

Let $K$ be a number field of degree $n$, and let $d_K$ be its discriminant. Then, under the Artin conjecture, the generalized Riemann hypothesis and a certain zero-density hypothesis, we show that the upper and lower bounds of the logarithmic derivatives of Artin $L$-functions attached to $K$ at $s=1$ are $\log \log |d_K|$ and $-(n-1) \log \log |d_K|$, respectively. Unconditionally, we show that there are infinitely many number fields with the extreme logarithmic derivatives; they are families of number fields whose Galois closures have the Galois group $C_n$ for $n=2,3,4,6$, $D_n$ for $n=3,4,5$, $S_4$ or $A_5$.

We prove upper bounds for Hecke–Laplace eigenfunctions on certain Riemannian manifolds $X$ of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are $d$-fold products of $2$-spheres or $3$-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a (‘Weyl-type’) saving of $\mathrm{vol} \hspace{0.167em} (X)^{- 1/ 6+ \varepsilon } $.

Let π(f) be a nearly ordinary automorphic representation of the multiplicative group of an indefinite quaternion algebra B over a totally real field F with associated Galois representation ρf. Let K be a totally complex quadratic extension of F embedding in B. Using families of CM points on towers of Shimura curves attached to B and K, we construct an Euler system for ρf. We prove that it extends to p-adic families of Galois representations coming from Hida theory and dihedral ℤdp-extensions. When this Euler system is non-trivial, we prove divisibilities of characteristic ideals for the main conjecture in dihedral and modular Iwasawa theory.

We construct a family of ideals representing ideal classes of order two in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.

The chromatic polynomial P(G,λ) gives the number of ways a graph G can be properly coloured in at most λ colours. This polynomial has been extensively studied in both combinatorics and statistical physics, but there has been little work on its algebraic properties. This paper reports a systematic study of the Galois groups of chromatic polynomials. We give a summary of the Galois groups of all chromatic polynomials of strongly non-clique-separable graphs of order at most 10 and all chromatic polynomials of non-clique-separable θ-graphs of order at most 19. Most of these chromatic polynomials have symmetric Galois groups. We give five infinite families of graphs: one of these families has chromatic polynomials with a dihedral Galois group and two of these families have chromatic polynomials with cyclic Galois groups. This includes the first known infinite family of graphs that have chromatic polynomials with the cyclic Galois group of order 3.