We provide an introduction to enumerating and constructing invariants of group representations via character methods. The problem is contextualized via two case studies, arising from our recent work: entanglement invariants for characterizing the structure of state spaces for composite quantum systems; and Markov invariants, a robust alternative to parameter-estimation intensive methods of statistical inference in molecular phylogenetics.

We estimate the number of solutions to $|y^{2}-x^{3}|\leqslant X$ with $N\leqslant y\leqslant 2N$, in terms of both $N$ and $X$.

The celebrated Smith–Minkowski–Siegel mass formula expresses the mass of a quadratic lattice $(L,Q)$ as a product of local factors, called the local densities of $(L,Q)$. This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme $\underline{G}$ over $\mathbb{Z}_{2}$ with generic fiber $\text{Aut}_{\mathbb{Q}_{2}}(L,Q)$, which satisfies $\underline{G}(\mathbb{Z}_{2})=\text{Aut}_{\mathbb{Z}_{2}}(L,Q)$. Our method works for any unramified finite extension of $\mathbb{Q}_{2}$. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unramified finite extensions of $\mathbb{Q}_{2}$. As an example, we give the mass formula for the integral quadratic form $Q_{n}(x_{1},\dots ,x_{n})=x_{1}^{2}+\cdots +x_{n}^{2}$ associated to a number field $k$ which is totally real and such that the ideal $(2)$ is unramified over $k$.

By establishing an improved level of distribution we study almost-primes of the form $f(p,n)$ where $f$ is an irreducible binary form over $\mathbb{Z}$.

We describe an online database of number fields which accompanies this paper. The database centers on complete lists of number fields with prescribed invariants. Our description here focuses on summarizing tables and connections to theoretical issues of current interest.

In this paper, we study the structure of the local components of the (shallow, i.e. without $U_{p}$) Hecke algebras acting on the space of modular forms modulo $p$ of level $1$, and relate them to pseudo-deformation rings. In many cases, we prove that those local components are regular complete local algebras of dimension $2$, generalizing a recent result of Nicolas and Serre for the case $p=2$.

We generalize the work of Bertolini and Darmon on the anticyclotomic main conjecture for elliptic curves to modular forms of higher weight.

We explain which Weierstrass ${\wp}$ -functions are locally definable from other ${\wp}$ -functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way.

We construct $p$-adic families of Klingen–Eisenstein series and $L$-functions for cusp forms (not necessarily ordinary) unramified at an odd prime $p$ on definite unitary groups of signature $(r,0)$ (for any positive integer $r$) for a quadratic imaginary field ${\mathcal{K}}$ split at $p$. When $r=2$, we show that the constant term of the Klingen–Eisenstein family is divisible by a certain $p$-adic $L$-function.

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}(H)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each pair of factorizations $z,z^{\prime }$ of $a$ , there exist factorizations $z=z_{0},\dots ,z_{k}=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$ , $z_{i}$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases, suppose that $|G|\geq 3$ . Then the catenary degree depends only on the class group $G$ and we have $\mathsf{c}(H)\in [3,\mathsf{D}(G)]$ , where $\mathsf{D}(G)$ denotes the Davenport constant of $G$ . The cases when $\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$ have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldinger et al. [‘The catenary degree of Krull monoids I’, J. Théor. Nombres Bordeaux23 (2011), 137–169], we determine the class groups satisfying $\mathsf{c}(H)=\mathsf{D}(G)-1$ . Apart from the extremal cases mentioned, the precise value of $\mathsf{c}(H)$ is known for no further class groups.

Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on $\text{GL}_{n+1}\times \text{GL}_{n}$ along the diagonal subgroup $\text{GL}_{n}$ and express it in terms of the Rankin–Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This extends the theory of Rankin–Selberg integrals to all automorphic forms on $\text{GL}_{n+1}\times \text{GL}_{n}$.

Let $p\geqslant 5$ be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its Galois representation contains, up to finite error, a principal congruence subgroup ${\rm\Gamma}(L)$ of $\text{SL}_{2}(\mathbb{Z}_{p}[[T]])$ for a principal ideal $(L)\neq 0$ of $\mathbb{Z}_{p}[[T]]$ for the canonical ‘weight’ variable $t=1+T$. If $L\notin {\rm\Lambda}^{\times }$, the power series $L$ is proven to be a factor of the Kubota–Leopoldt $p$-adic $L$-function or of the square of the anticyclotomic Katz $p$-adic $L$-function or a power of $(t^{p^{m}}-1)$.

Let $A$ be an abelian variety over a global field $K$ of characteristic $p\geqslant 0$. If $A$ has nontrivial (respectively full) $K$-rational $l$-torsion for a prime $l\neq p$, we exploit the fppf cohomological interpretation of the $l$-Selmer group $\text{Sel}_{l}\,A$ to bound $\#\text{Sel}_{l}\,A$ from below (respectively above) in terms of the cardinality of the $l$-torsion subgroup of the ideal class group of $K$. Applied over families of finite extensions of $K$, the bounds relate the growth of Selmer groups and class groups. For function fields, this technique proves the unboundedness of $l$-ranks of class groups of quadratic extensions of every $K$ containing a fixed finite field $\mathbb{F}_{p^{n}}$ (depending on $l$). For number fields, it suggests a new approach to the Iwasawa ${\it\mu}=0$ conjecture through inequalities, valid when $A(K)[l]\neq 0$, between Iwasawa invariants governing the growth of Selmer groups and class groups in a $\mathbb{Z}_{l}$-extension.

This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$ , where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.

We extend results of Andrews and Bressoud [‘Vanishing coefficients in infinite product expansions’, J. Aust. Math. Soc. Ser. A27(2) (1979), 199–202] on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if

$$\begin{eqnarray}\frac{(q^{r-tk},q^{mk-(r-tk)};q^{mk})_{\infty }}{(\pm q^{r},\pm q^{mk-r};q^{mk})_{\infty }}=:\mathop{\sum }_{n=0}^{\infty }c_{n}q^{n}\end{eqnarray}$$

$k$

$m$

$s$

$t$

$r=sm+t$

$c_{kn-rs}$

Extending the idea of Dabrowski [‘On the proportion of rank 0 twists of elliptic curves’, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 483–486] and using the 2-descent method, we provide three general families of elliptic curves over $\mathbb{Q}$ such that a positive proportion of prime-twists of such elliptic curves have rank zero simultaneously.

We give a new proof of Vinogradov’s three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on the theory of $L$ -functions, either explicitly or implicitly. Our proof is sieve theoretical and uses a transference principle, the idea of which was first developed by Green [Ann. of Math. (2) 161 (3) (2005), 1609–1636] and used in the proof of Green and Tao’s theorem [Ann. of Math. (2) 167 (2) (2008), 481–547]. To make our argument work, we also develop an additive combinatorial result concerning popular sums, which may be of independent interest.

One way to study certain classes of polynomials is by considering examples that are attached to combinatorial objects. Any graph $G$ has an associated reciprocal polynomial $R_{G}$ , and with two particular classes of reciprocal polynomials in mind one can ask the questions: (a) when is $R_{G}$ a product of cyclotomic polynomials (giving the cyclotomic graphs)? (b) when does $R_{G}$ have the minimal polynomial of a Salem number as its only non-cyclotomic factor (the non-trivial Salem graphs)? Cyclotomic graphs were classified by Smith (Combinatorial structures and their applications, Proceedings of Calgary International Conference, Calgary, AB, 1969 (eds R. Guy, H. Hanani, H. Saver and J. Schönheim; Gordon and Breach, New York, 1970) 403–406); the maximal connected ones are known as Smith graphs. Salem graphs are ‘spectrally close’ to being cyclotomic, in that nearly all their eigenvalues are in the critical interval $[-2,2]$ . On the other hand, Salem graphs do not need to be ‘combinatorially close’ to being cyclotomic: the largest cyclotomic induced subgraph might be comparatively tiny.

We define an $m$ -Salem graph to be a connected Salem graph $G$ for which $m$ is minimal such that there exists an induced cyclotomic subgraph of $G$ that has $m$ fewer vertices than $G$ . The $1$ -Salem subgraphs are both spectrally close and combinatorially close to being cyclotomic. Moreover, every Salem graph contains a $1$ -Salem graph as an induced subgraph, so these $1$ -Salem graphs provide some necessary substructure of all Salem graphs. The main result of this paper is a complete combinatorial description of all $1$ -Salem graphs: in the non-bipartite case there are $25$ infinite families and $383$ sporadic examples.

We consider classes of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least s. We provide a Frostman-type lemma to determine if a limsup set is in such a class. Suppose that E = lim sup En ⊂ [0, 1], and that μn are probability measures with support in En. If there exists a constant C such that

for all n, then, under suitable conditions on the limit measure of the sequence (μn), we prove that the set E is in the class .

As an application we prove that, for α > 1 and almost all λ ∈ (½, 1), the set

where and ak ∈ {0, 1}}, belongs to the class . This improves one of our previously published results.

We classify generically transitive actions of semi-direct products on ℙ2. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for semi-direct products .