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 consider the distribution of $p$ -power group schemes among the torsion of abelian varieties over finite fields of characteristic $p$ , as follows. Fix natural numbers $g$ and $n$ , and let ${\it\xi}$ be a non-supersingular principally quasipolarized Barsotti–Tate group of level $n$ . We classify the $\mathbb{F}_{q}$ -rational forms ${\it\xi}^{{\it\alpha}}$ of ${\it\xi}$ . Among all principally polarized abelian varieties $X/\mathbb{F}_{q}$ of dimension $g$ with $X[p^{n}]_{\bar{\mathbb{F}}_{q}}\cong {\it\xi}_{\bar{\mathbb{F}}_{q}}$ , we compute the frequency with which $X[p^{n}]\cong {\it\xi}^{{\it\alpha}}$ . The error in our estimate is bounded by $D/\sqrt{q}$ , where $D$ depends on $g$ , $n$ , and $p$ , but not on ${\it\xi}$ .

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.

For an infinite cardinal ${\it\kappa}$ , let $\text{ded}\,{\it\kappa}$ denote the supremum of the number of Dedekind cuts in linear orders of size ${\it\kappa}$ . It is known that ${\it\kappa}<\text{ded}\,{\it\kappa}\leqslant 2^{{\it\kappa}}$ for all ${\it\kappa}$ and that $\text{ded}\,{\it\kappa}<2^{{\it\kappa}}$ is consistent for any ${\it\kappa}$ of uncountable cofinality. We prove however that $2^{{\it\kappa}}\leqslant \text{ded}(\text{ded}(\text{ded}(\text{ded}\,{\it\kappa})))$ always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as large as in the class of all countable theories.

We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set $K$ is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects $K$ at exactly one point.

We define a theory of Goodwillie calculus for enriched functors from finite pointed simplicial $G$ -sets to symmetric $G$ -spectra, where $G$ is a finite group. We extend a notion of $G$ -linearity suggested by Blumberg to define stably excisive and ${\it\rho}$ -analytic homotopy functors, as well as a $G$ -differential, in this equivariant context. A main result of the paper is that analytic functors with trivial derivatives send highly connected $G$ -maps to $G$ -equivalences. It is analogous to the classical result of Goodwillie that ‘functors with zero derivative are locally constant’. As the main example, we show that Hesselholt and Madsen’s Real algebraic $K$ -theory of a split square zero extension of Wall antistructures defines an analytic functor in the $\mathbb{Z}/2$ -equivariant setting. We further show that the equivariant derivative of this Real $K$ -theory functor is $\mathbb{Z}/2$ -equivalent to Real MacLane homology.