We consider an elliptic surface $\pi :\mathcal{ℰ}\to {\mathbb{P}}^1$ defined over a number field $k$ and study the problem of comparing the rank of the special fibers over $k$ with that of the generic fiber over $k\left({\mathbb{P}}^1\right)$. We prove, for a large class of rational elliptic surfaces, the existence of infinitely many fibers with rank at least equal to the generic rank plus two.

Let $R$ be a complete discrete valuation ring with algebraically closed residue field $k$ and fraction field $K$. Let $X_K$ be a proper smooth and geometrically connected scheme over $K$. Néron defined a canonical pairing on $X_K$ between $0$-cycles of degree zero and divisors which are algebraically equivalent to zero. When $X_K$ is an abelian variety, and if one restricts to those $0$-cycles supported on $K$-rational points, Néron gave an expression of his pairing involving intersection multiplicities on the Néron model $A$ of $A_K$ over $R$. When $X_K$ is a curve, Gross and Hriljac gave independently an analogous description of Néron’s pairing, but for arbitrary $0$-cycles of degree zero, by means of intersection theory on a proper flat regular $R$-model $X$ of $X_K$.

We show that these intersection computations are valid for an arbitrary scheme $X_K$ as above and arbitrary $0$-cycles of degree zero, by using a proper flat normal and semifactorial model $X$ of $X_K$ over $R$. When $X_K=A_K$ is an abelian variety, and $X=\overline{A}$ is a semifactorial compactification of its Néron model $A$, these computations can be used to study the relative algebraic equivalence on $\overline{A}∕R$. We then obtain an interpretation of Grothendieck’s duality for the Néron model $A$, in terms of the Picard functor of $\overline{A}$ over $R$. Finally, we give an explicit description of Grothendieck’s duality pairing when $A_K$ is the Jacobian of a curve of index one.

Let $K$ be a field extension of an uncountable base field $k$, let $\sigma$ be a $k$-automorphism of $K$, and let $\delta$ be a $k$-derivation of $K$. We show that if $D$ is one of $K\left(x;\sigma \right)$ or $K\left(x;\delta \right)$, then $D$ either contains a free algebra over $k$ on two generators, or every finitely generated subalgebra of $D$ satisfies a polynomial identity. As a corollary, we show that the quotient division ring of any iterated Ore extension of an affine PI domain over $k$ is either again PI, or else it contains a free algebra over its center on two variables.

The spaces of Maass forms of even weight and of arbitrary order are studied. It is shown that, if we allow exponential growth at the cusps, these spaces are as large as algebraic restrictions allow. These results also apply to higher-order holomorphic forms of even weight.

We present a new relation between an invariant of singularities in characteristic zero (the log canonical threshold) and an invariant of singularities defined via the Frobenius morphism in positive characteristic (the $F$-pure threshold). We show that the set of limit points of sequences of the form $\left(c_p\right)$, where $c_p$ is the $F$-pure threshold of an ideal on an $n$-dimensional smooth variety in characteristic $p$, coincides with the set of log canonical thresholds of ideals on $n$-dimensional smooth varieties in characteristic zero. We prove this by combining results of Hara and Yoshida with nonstandard constructions.

We restate the semistable reduction theorem from geometric invariant theory in the context of spaces of morphisms from ${\mathbb{P}}^n$ to itself. For every complete curve $C$ downstairs, we get a ${\mathbb{P}}^n$-bundle on an abstract curve $D$ mapping finite-to-one onto $C$, whose trivializations correspond to not necessarily complete curves upstairs with morphisms corresponding to identifying each fiber with the morphism the point represents. Finding a trivial bundle is equivalent to finding a complete $D$ upstairs mapping finite-to-one onto $C$; we prove that in every space of morphisms, there exists a curve $C$ for which no such $D$ exists. In the case when $D$ exists, we bound the degree of the map from $D$ to $C$ in terms of $C$ for $C$ rational and contained in the stable space.

We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck’s trace map of the surface as a sum of residues. Points at infinity are then incorporated into the theory and the reciprocity law is extended to all curves on the surface. Applications to adelic duality for the arithmetic surface are discussed.

Let $p>2$ be prime. We complete the proof of the weight part of Serre’s conjecture for rank-two unitary groups for mod $p$ representations in the totally ramified case by proving that any Serre weight which occurs is a predicted weight. This completes the analysis begun by Barnet-Lamb, Gee, and Geraghty, who proved that all predicted Serre weights occur. Our methods are a mixture of local and global techniques, and in the course of the proof we use global techniques (as well as local arguments) to establish some purely local results on crystalline extension classes. We also apply these local results to prove similar theorems for the weight part of Serre’s conjecture for Hilbert modular forms in the totally ramified case.

Our goal in this note is to show that cohomology classes with coefficients in finite flat group schemes can be killed by finite covers of the base scheme, and similarly for abelian schemes with “finite covers” replaced by “proper covers.” We apply this result to commutative algebra to give by a conceptual proof of Hochster–Huneke’s theorem on the existence of big Cohen–Macaulay algebras in positive characteristic; all previous proofs of this result were equational or cocycle-theoretic in nature.