We investigate the fate of determinism in general relativity (GR), comparing the philosopher’s account with the physicist’s well-posed initial value formulations. The fate of determinism is interwoven with the question of what it is for a spacetime to be ‘physically reasonable’. A central concern is the status of global hyperbolicity, a putatively necessary condition for determinism in GR. While global hyperbolicity may fail to be true of all physically reasonable models, we analyze whether global hyperbolicity should be (i) imposed by fiat; (ii) established from weaker assumptions, as in cosmic censorship theorems; or (iii) justified by beyond-GR physics.

This article examines two cosmological models of quantum gravity (from string theory and loop quantum gravity) to investigate the foundational and conceptual issues arising from quantum treatments of the big bang. While the classical singularity is erased, the quantum evolution that replaces it may not correspond to classical spacetime: it may instead be a nonspatiotemporal region that somehow transitions to a spatiotemporal state. The different kinds of transition involved are partially characterized, the concept of a physical transition without time is investigated, and the problem of empirical incoherence for regions without spacetime is discussed.

Modular curves like X0(N) and X1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here by Xnsp(p) and Xnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures of p-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc. (3) 77(1) (1998), 1–38; J. Algebra231(1) (2000), 414–448).

We prove the p-parity conjecture for elliptic curves over global fields of characteristic p>3. We also present partial results on the ℓ-parity conjecture for primes ℓ≠p.

Structural realist interpretations of generally relativistic spacetimes have recently come to enjoy a remarkable degree of popularity among philosophers. I present a challenge to these structuralist interpretations that arises from considering cosmological models in general relativity. As a consequence of their high degree of spacetime symmetry, these models resist a structuralist interpretation. I then evaluate the various strategies available to the structuralist to react to this challenge.

Within the Tate–Shafarevich group of an elliptic curve E defined over a number field K, there is a canonical subgroup defined by imposing stronger conditions at the places above a given prime p. This group appears naturally in the Iwasawa theory for E. We propose a study of what one can say about the relation to the full Tate–Shafarevich group. Some numerical examples are included, as well as a few conjectures.

Does the need to find a quantum theory of gravity imply that the gravitational field must be quantized? Physicists working in quantum gravity routinely assume an affirmative answer, often without being aware of the metaphysical commitments that tend to underlie this assumption. The ambition of this article is to probe these commitments and to analyze some recently adduced physical—as opposed to metaphysical—arguments pertinent to the issue of quantization. While there exist good reasons to quantize gravity, as this analysis will show, alternative approaches to gravity challenge the received wisdom. These renegade approaches do not regard gravity as a fundamental force, but rather as effective, i.e., as merely supervening on fundamental physics. I will urge that these alternative accounts at least prove the tenability of an opposition to quantization.

The non-degeneracy of the canonical $p$-adic height pairing defined by Perrin-Riou and Schneider on an elliptic curve over a number field with good, ordinary reduction is still unknown.

Following the work done for the real-valued pairing, the behaviour of the $p$-adic height is analysed as a point varies on a section of a family of elliptic curves, and so new information is obtained about this pairing. In particular, the variation is $p$-adically continuous and the non-degeneracy of a set of sections can be checked simultaneously for almost all elements of the family. The paper contains some conjectures about the valuation of the $p$-adic regulator and its consequences for the growth of the Mordell–Weil group in cyclotomic $\mathbb{Z}_p$-extensions.