We consider a certain family of Kudla–Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1, 1), and prove that the arithmetic degrees of these cycles are Fourier coefficients of the central derivative of an Eisenstein series of genus 2. The integral model in question parameterizes abelian surfaces equipped with a non-principal polarization and an action of an imaginary quadratic number ring, and in this setting the cycles are degenerate: they may contain components of positive dimension. This result can be viewed as confirmation, in the degenerate setting and for dimension 2, of conjectures of Kudla and Kudla–Rapoport that predict relations between the intersection numbers of special cycles and the Fourier coefficients of automorphic forms.

We prove that the group of normalized cohomological invariants of degree $3$ modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group $G$ is isomorphic to the torsion part of the Chow group of codimension-$2$ cycles of the respective versal $G$-flag. In particular, if $G$ is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of $G$. As an application, we construct nontrivial cohomological invariants for indecomposable central simple algebras.

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$ are compatible with the usual local constants at all primes not dividing $p$ and in two special cases also at primes dividing $p$. We deduce new cases of the $p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

We construct an Euler system attached to a weight 2 modular form twisted by a Grössencharacter of an imaginary quadratic field $K$, and apply this to bounding Selmer groups.

In this paper we improve the inequalities obtained by Chen in 2009 for the Euler–Mascheroni constant.

The Coleman integral is a $p$ -adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

For an elliptic curve $E/\mathbb{Q}$ without complex multiplication we study the distribution of Atkin and Elkies primes $\ell$ , on average, over all good reductions of $E$ modulo primes $p$ . We show that, under the generalized Riemann hypothesis, for almost all primes $p$ there are enough small Elkies primes $\ell$ to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in $(\log p)^{4+o(1)}$ expected time.

We construct explicit bases for spaces of overconvergent $p$ -adic modular forms when $p=2,3$ and study their interaction with the Atkin operator. This results in an extension of Lauder’s algorithms for overconvergent modular forms. We illustrate these algorithms with computations of slope sequences of some $2$ -adic eigencurves and the construction of Chow–Heegner points on elliptic curves via special values of Rankin triple product L-functions.

Let ${\mathcal{K}}$ be an imaginary quadratic field. Let ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$ be irreducible generic cohomological automorphic representation of $\text{GL}(n)/{\mathcal{K}}$ and $\text{GL}(n-1)/{\mathcal{K}}$ , respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called a Whittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if ${\rm\Pi}$ is cuspidal and the weights of ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$ are in a standard relative position, the critical values of the Rankin–Selberg product $L(s,{\rm\Pi}\times {\rm\Pi}^{\prime })$ are essentially algebraic multiples of the product of the Whittaker periods of ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$ . We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal ${\rm\Pi}$ can be given a motivic interpretation, and can also be related to a critical value of the adjoint $L$ -function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin–Selberg and adjoint $L$ -functions are compatible with Deligne’s conjecture.

We determine the index of five of the seven hypergeometric Calabi–Yau operators that have finite index in $\mathit{Sp}_{4}(\mathbb{Z})$ and in two cases give a complete description of the monodromy group. Furthermore, we find six nonhypergeometric Calabi–Yau operators with finite index in $\mathit{Sp}_{4}(\mathbb{Z})$, most notably a case where the index is one.

In a recent paper, Soundararajan has proved the quantum unique ergodicity conjecture by getting a suitable estimate for the second order moment of the so-called ‘Hecke multiplicative’ functions. In the process of proving this he has developed many beautiful ideas. In this paper we generalize his arguments to a general $k$th power and provide an analogous estimate for the $k$th power moment of the Hecke multiplicative functions. This may be of general interest.

For non-singular intersections of pairs of quadrics in 11 or more variables, we prove an asymptotic for the number of rational points in an expanding box.

We analyse the $\text{mod}~p$ étale cohomology of the Lubin–Tate tower both with compact support and without support. We prove that there are no supersingular representations in the $H_{c}^{1}$ of the Lubin–Tate tower. On the other hand, we show that in $H^{1}$ of the Lubin–Tate tower appears the $\text{mod}~p$ local Langlands correspondence and the $\text{mod}~p$ local Jacquet–Langlands correspondence, which we define in the text. We discuss the local-global compatibility part of the Buzzard–Diamond–Jarvis conjecture which appears naturally in this context.

Let $K$ be a totally real field. By the asymptotic Fermat’s Last Theorem over$K$ we mean the statement that there is a constant $B_{K}$ such that for any prime exponent $p>B_{K}$, the only solutions to the Fermat equation

$$\begin{eqnarray}a^{p}+b^{p}+c^{p}=0,\quad a,b,c\in K\end{eqnarray}$$

$abc=0$

$K$

$K$

$K=\mathbb{Q}(\sqrt{d})$

$d\geqslant 2$

${\textstyle \frac{5}{6}}$

$1$

Let ${\it\mu}_{1},\ldots ,{\it\mu}_{s}$ be real numbers, with ${\it\mu}_{1}$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_{1},\ldots ,x_{s})=(x_{1}-{\it\mu}_{1})^{k}+\cdots +(x_{s}-{\it\mu}_{s})^{k}$. For $k\geqslant 4$, we bound the number of variables needed to ensure that if ${\it\eta}$ is real and ${\it\tau}>0$ is sufficiently large then there exist integers $x_{1}>{\it\mu}_{1},\ldots ,x_{s}>{\it\mu}_{s}$ such that $|\mathfrak{F}(\mathbf{x})-{\it\tau}|<{\it\eta}$. This is a real analogue to Waring’s problem. When $s\geqslant 2k^{2}-2k+3$, we provide an asymptotic formula. We prove similar results for sums of general univariate degree-$k$ polynomials.

An integer $n$ is said to be $y$-friable if its largest prime factor $P^{+}(n)$ is less than $y$. In this paper, it is shown that the $y$-friable integers less than $x$ have a weak exponent of distribution at least $3/5-{\it\varepsilon}$ when $(\log x)^{c}\leqslant x\leqslant x^{1/c}$ for some $c=c({\it\varepsilon})\geqslant 1$, that is to say, they are well distributed in the residue classes of a fixed integer $a$, on average over moduli ${\leqslant}x^{3/5-{\it\varepsilon}}$ for each fixed $a\neq 0$ and ${\it\varepsilon}>0$. We apply this to the estimation of the sum $\sum _{2\leqslant n\leqslant x,P^{+}(n)\leqslant y}{\it\tau}(n-1)$ when $(\log x)^{c}\leqslant y$. This follows and improves on previous work of Fouvry and Tenenbaum. Our proof combines the dispersion method of Linnik in the setting of Bombieri, Fouvry, Friedlander and Iwaniec with recent work of Harper on friable integers in arithmetic progressions.

Let $K$ be a number field. For any system of semisimple mod $\ell$ Galois representations $\{{\it\phi}_{\ell }:\text{Gal}(\bar{\mathbb{Q}}/K)\rightarrow \text{GL}_{N}(\mathbb{F}_{\ell })\}_{\ell }$ arising from étale cohomology (Definition 1), there exists a finite normal extension $L$ of $K$ such that if we denote ${\it\phi}_{\ell }(\text{Gal}(\bar{\mathbb{Q}}/K))$ and ${\it\phi}_{\ell }(\text{Gal}(\bar{\mathbb{Q}}/L))$ by $\bar{{\rm\Gamma}}_{\ell }$ and $\bar{{\it\gamma}}_{\ell }$, respectively, for all $\ell$ and let $\bar{\mathbf{S}}_{\ell }$ be the $\mathbb{F}_{\ell }$-semisimple subgroup of $\text{GL}_{N,\mathbb{F}_{\ell }}$ associated to $\bar{{\it\gamma}}_{\ell }$ (or $\bar{{\rm\Gamma}}_{\ell }$) by Nori’s theory [On subgroups of$\text{GL}_{n}(\mathbb{F}_{p})$, Invent. Math. 88 (1987), 257–275] for sufficiently large $\ell$, then the following statements hold for all sufficiently large $\ell$.

A(i) The formal character of $\bar{\mathbf{S}}_{\ell }{\hookrightarrow}\text{GL}_{N,\mathbb{F}_{\ell }}$ (Definition 1) is independent of $\ell$ and equal to the formal character of $(\mathbf{G}_{\ell }^{\circ })^{\text{der}}{\hookrightarrow}\text{GL}_{N,\mathbb{Q}_{\ell }}$, where $(\mathbf{G}_{\ell }^{\circ })^{\text{der}}$ is the derived group of the identity component of $\mathbf{G}_{\ell }$, the monodromy group of the corresponding semi-simplified $\ell$-adic Galois representation ${\rm\Phi}_{\ell }^{\text{ss}}$.

A(ii) The non-cyclic composition factors of $\bar{{\it\gamma}}_{\ell }$ and $\bar{\mathbf{S}}_{\ell }(\mathbb{F}_{\ell })$ are identical. Therefore, the composition factors of $\bar{{\it\gamma}}_{\ell }$ are finite simple groups of Lie type of characteristic $\ell$ and are cyclic groups.

B(i) The total $\ell$-rank $\text{rk}_{\ell }\bar{{\rm\Gamma}}_{\ell }$ of $\bar{{\rm\Gamma}}_{\ell }$ (Definition 14) is equal to the rank of $\bar{\mathbf{S}}_{\ell }$ and is therefore independent of $\ell$.

B(ii) The $A_{n}$-type $\ell$-rank $\text{rk}_{\ell }^{A_{n}}\bar{{\rm\Gamma}}_{\ell }$ of $\bar{{\rm\Gamma}}_{\ell }$ (Definition 14) for $n\in \mathbb{N}\setminus \{1,2,3,4,5,7,8\}$ and the parity of $(\text{rk}_{\ell }^{A_{4}}\bar{{\rm\Gamma}}_{\ell })/4$ are independent of $\ell$.

In this article we discuss three types of mean values of the Euler double zeta-function. To get the results, we introduce three approximate formulas for this function.

We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behaviour near roots of unity.

Let $f$ be a rational function on an algebraic curve over the complex numbers. For a point $p$ and local parameter $x$ we can consider the Taylor series for $f$ in the variable $x$. In this paper we give an upper bound on the frequency with which the terms in the Taylor series have $0$ as their coefficient.