Let $C$ be a smooth projective absolutely irreducible curve of genus $g\ge 2$ over a number field $K$ of degree $d$, and let $J$ denote its Jacobian. Let $r$ denote the Mordell–Weil rank of $J\left(K\right)$. We give an explicit and practical Chabauty-style criterion for showing that a given subset $\mathcal{K}\subseteq C\left(K\right)$ is in fact equal to $C\left(K\right)$. This criterion is likely to be successful if $r\le d\left(g-1\right)$. We also show that the only solution to the equation $x^2+y^3=z^{10}$ in coprime nonzero integers is $\left(x,y,z\right)=\left(\pm 3,-2,\pm 1\right)$. This is achieved by reducing the problem to the determination of $K$-rational points on several genus-$2$ curves where $K=\mathbb{Q}$ or $\mathbb{Q}\left(\sqrt[3]{2}\right)$ and applying the method of this paper.

The naïve blowup algebras developed by Keeler, Rogalski, and Stafford, after examples of Rogalski, are the first known class of connected graded algebras that are noetherian but not strongly noetherian. This failure of the strong noetherian property is intimately related to the failure of the point modules over such algebras to behave well in families: puzzlingly, there is no fine moduli scheme for such modules although point modules correspond bijectively with the points of a projective variety $X$. We give a geometric structure to this bijection and prove that the variety $X$ is a coarse moduli space for point modules. We also describe the natural moduli stack $X_{\infty }$ for embedded point modules — an analog of a “Hilbert scheme of one point” — as an infinite blowup of $X$ and establish good properties of $X_{\infty }$. The natural map $X_{\infty }\to X$ is thus a kind of “Hilbert–Chow morphism of one point" for the naïve blowup algebra.

Let $V$ be a nonsingular projective surface defined over $\mathbb{Q}$ and having at least two elliptic fibrations defined over $\mathbb{Q}$; the most interesting case, though not the only one, is when $V$ is a K3 surface with these properties. We also assume that $V\left(\mathbb{Q}\right)$ is not empty. The object of this paper is to prove, under a weak hypothesis, the Zariski density of $V\left(\mathbb{Q}\right)$ and to study the closure of $V\left(\mathbb{Q}\right)$ under the real and the $p$-adic topologies. The first object is achieved by the following theorem:

Let $V$ be a nonsingular surface defined over $\mathbb{Q}$ and having at least two distinct elliptic fibrations. There is an explicitly computable Zariski closed proper subset $X$ of $V$ defined over $\mathbb{Q}$ such that if there is a point $P_0$ of $V\left(\mathbb{Q}\right)$ not in $X$ then $V\left(\mathbb{Q}\right)$ is Zariski dense in $V$.

The methods employed to study the closure of $V\left(\mathbb{Q}\right)$ in the real or $p$-adic topology demand an almost complete knowledge of $V$; a typical example of what they can achieve is as follows. Let $V_c$ be

$$V_c:X_0^4+c{X}_1^4=X_2^4+c{X}_3^{4}forc=2,4\text{or }8;$$

then $V_c\left(\mathbb{Q}\right)$ is dense in $V_c\left({\mathbb{Q}}_2\right)$ for $c=2,4,8$.

Let $X$ be a smooth proper variety over a perfect field $k$ of arbitrary characteristic. Let $D$ be an effective divisor on $X$ with multiplicity. We introduce an Albanese variety $Alb\left(X,D\right)$ of $X$ of modulus $D$ as a higher-dimensional analogue of the generalized Jacobian of Rosenlicht and Serre with modulus for smooth proper curves. Basing on duality of 1-motives with unipotent part (which are introduced here), we obtain explicit and functorial descriptions of these generalized Albanese varieties and their dual functors.

We define a relative Chow group of zero cycles ${CH}_0\left(X,D\right)$ of modulus $D$ and show that $Alb\left(X,D\right)$ can be viewed as a universal quotient of ${CH}_0{\left(X,D\right)}^0$.

As an application we can rephrase Lang’s class field theory of function fields of varieties over finite fields in explicit terms.

We prove that a conjecture of Chai on the additivity of the base change conductor for semiabelian varieties over a discretely valued field is equivalent to a Fubini property for the dimensions of certain motivic integrals. We prove this Fubini property when the valued field has characteristic zero.

This paper presents three results on $F$-singularities. First, we give a new proof of Eisenstein’s restriction theorem for adjoint ideal sheaves using the theory of $F$-singularities. Second, we show that a conjecture of Mustaţă and Srinivas implies a conjectural correspondence of $F$-purity and log canonicity. Finally, we prove this correspondence when the defining equations of the variety are very general.

We develop the algebra of exponential fields and their extensions. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, we define finitely presented extensions, show that finitely generated strong extensions are finitely presented, and classify these extensions. We give an algebraic construction of Zilber’s pseudoexponential fields. As applications of the general results and methods of the paper, we show that Zilber’s fields are not model-complete, answering a question of Macintyre, and we give a precise statement explaining how Schanuel’s conjecture answers all transcendence questions about exponentials and logarithms. We discuss connections with the Kontsevich–Zagier, Grothendieck, and André transcendence conjectures on periods, and suggest open problems.

For coprime integers $g$ and $n$, let ${\ell }_g\left(n\right)$ denote the multiplicative order of $g$ modulo $n$. Motivated by a conjecture of Arnold, we study the average of ${\ell }_g\left(n\right)$ as $n\le x$ ranges over integers coprime to $g$, and $x$ tending to infinity. Assuming the generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of ${\ell }_g\left(p\right)$ as $p\le x$ ranges over primes.

In this paper, we consider congruences of Hilbert modular forms. Sturm showed that mod $\ell$ elliptic modular forms of weight $k$ and level ${\Gamma }_1\left(N\right)$ are determined by the first $\left(k∕12\right)\left[{\Gamma }_1\left(1\right):{\Gamma }_1\left(N\right)\right]mod\ell$ Fourier coefficients. We prove an analogue of Sturm’s result for Hilbert modular forms associated to totally real number fields. The proof uses the positivity of ample line bundles on toroidal compactifications of Hilbert modular varieties.