arXiv subjects
Publications and source records attributed to Christian Wuthrich.
We study the action of the Galois group $G$ of a finite extension $K/k$ of number fields on the points on an elliptic curve $E$. For an odd prime $p$, we aim to determine the structure of the $p$-adic completion of the Mordell-Weil group $E(K)$ as a $\mathbb{Z}_p[G]$-module only using information of $E$ over $k$ and the completions of $K$.
We present computational algorithms to work with points on the modular curve associated to the normaliser of a non-split Cartan group of prime level $p$. Rather than working with explicit equations, we represent these points using the moduli interpretation of necklaces in the $p$-torsion of elliptic curves. We use our methods to investigate for which primes $\ell\neq p$ two rational points with complex multiplication can have equal reduction modulo $\ell$.
Under suitable, fairly weak hypotheses on an elliptic curve $E/\mathbb{Q}$ and a primitive non-trivial Dirichlet character $χ$, we show that the algebraic $L$-value $\mathscr{L}(E,χ)$ at $s=1$ is an algebraic integer. For instance, for semistable curves $\mathscr{L}(E,χ)$ is integral whenever $E$ admits no isogenies defined over $\mathbb{Q}$. Moreover we give examples illustrating that our hypotheses are necessary for integrality to hold.
This is a chapter of the planned monograph "Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and Christian Wüthrich and under contract with Oxford University Press. (More information at https://beyondspacetime.net/.) This chapter introduces the problem of emergence of spacetime in quantum gravity. It introduces the main philosophical challenge to spacetime emergence and sketches our preferred solution to it.
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.
Contemporary research programs in fundamental physics appear to suggest that there could be two (physical) times---or none at all. This essay articulates these possibilities in the context of quantum gravity, and in particular of cosmological models developed in an approach called `loop quantum gravity', and explains how they could nevertheless underwrite our manifestly temporal world. A proper interpretation of these models requires a negotiation of an atemporal and a temporal sense of the emergence of (space)time.
In prior work, we have argued that spacetime functionalism provides tools for clarifying the conceptual difficulties specifically linked to the emergence of spacetime in certain approaches to quantum gravity. We argue in this article that spacetime functionalism in quantum gravity is radically different from other functionalist approaches that have been suggested in quantum mechanics and general relativity: in contrast to these latter cases, it does not compete with purely interpretative alternatives, but is rather intertwined with the physical theorizing itself at the level of quantum gravity. Spacetime functionalism allows one to articulate a coherent realist perspective in the context of quantum gravity, and to relate it to a straightforward realist understanding of general relativity.
Most approaches to quantum gravity suggest that relativistic spacetime is not fundamental, but instead emerges from some non-spatiotemporal structure. This paper investigates the implications of this suggestion for the possibility of time travel in the sense of the existence of closed timelike curves in some relativistic spacetimes. In short, will quantum gravity reverse or strengthen general relativity's verdict that time travel is possible?
Analogue experiments have attracted interest for their potential to shed light on inaccessible domains. For instance, `dumb holes' in fluids and Bose-Einstein condensates, as analogues of black holes, have been promoted as means of confirming the existence of Hawking radiation in real black holes. We compare analogue experiments with other cases of experiment and simulation in physics. We argue---contra recent claims in the philosophical literature---that analogue experiments are not capable of confirming the existence of particular phenomena in inaccessible target systems. As they must assume the physical adequacy of the modelling framework used to describe the inaccessible target system, arguments to the conclusion that analogue experiments can yield confirmation for phenomena in those target systems, such as Hawking radiation in black holes, beg the question.
Although general relativity is a predictively successful theory, it treats matter as classical rather than as quantum. For this reason, it will have to be replaced by a more fundamental quantum theory of gravity. Attempts to formulate a quantum theory of gravity suggest that such a theory may have radical consequences for the nature, and indeed the fate, of spacetime. The present article articulates what this problem of spacetime is and traces it through three approaches to quantum gravity taking general relativity as their vantage point: semi-classical gravity, causal set theory, and loop quantum gravity.
Approaches to quantum gravity often involve the disappearance of space and time at the fundamental level. The metaphysical consequences of this disappearance are profound, as is illustrated with David Lewis's analysis of modality. As Lewis's possible worlds are unified by the spatiotemporal relations among their parts, the non-fundamentality of spacetime---if borne out---suggests a serious problem for his analysis: his pluriverse, for all its ontological abundance, does not contain our world. Although the mere existence---as opposed to the fundamentality---of spacetime must be recovered from the fundamental structure in order to guarantee the empirical coherence of the non-spatiotemporal fundamental theory, it does not suffice to salvage Lewis's theory of modality from the charge of rendering our actual world impossible.
Theories of quantum gravity generically presuppose or predict that the reality underlying relativistic spacetimes they are describing is significantly non-spatiotemporal. On pain of empirical incoherence, approaches to quantum gravity must establish how relativistic spacetime emerges from their non-spatiotemporal structures. We argue that in order to secure this emergence, it is sufficient to establish that only those features of relativistic spacetimes functionally relevant in producing empirical evidence must be recovered. In order to complete this task, an account must be given of how the more fundamental structures instantiate these functional roles. We illustrate the general idea in the context of causal set theory and loop quantum gravity, two prominent approaches to quantum gravity.
Research in quantum gravity strongly suggests that our world in not fundamentally spatiotemporal, but that spacetime may only emerge in some sense from a non-spatiotemporal structure, as this paper illustrates in the case of causal set theory and loop quantum gravity. This would raise philosophical concerns regarding the empirical coherence and general adequacy of theories in quantum gravity. If it can be established, however, that spacetime emerges in the appropriate circumstances and how all its relevant aspects are explained in fundamental non-spatiotemporal terms, then the challenge is fully met. It is argued that a form of spacetime functionalism offers the most promising template for this project.
This paper 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 non-spatiotemporal region, which 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.
Information theory is increasingly invoked by physicists concerned with fundamental physics, including black hole physics. But to what extent is the application of information theory in those contexts legitimate? Using the case of black hole thermodynamics and Bekenstein's celebrated argument for the entropy of black holes, I will argue that information-theoretic notions are problematic in the present case. Bekenstein's original argument, as suggestive as it may appear, thus fails. This example is particularly pertinent to the theme of the present collection because the Bekenstein-Hawking formula for black hole entropy is widely accepted as 'empirical data' in notoriously empirically deprived quantum gravity, even though the laws of black hole thermodynamics have so far evaded empirical confirmation.
Modular curves like X_0(N) and X_1(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 SL_2(Z), they allow for a more arithmetic description as a solution to a moduli problem. This description turns out to be very useful in many applications. We wish to give such a moduli description for two modular curves: those associated to non-split Cartan subgroups and their normaliser in GL_2(F_p). 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. Some classical results about the geometry of those curves can be proven using this moduli description. For instance, we can count the number of elliptic points, describe the cusps and the degeneracy maps. We also give a moduli-theoretic interpretation and a new proof of a result of Chen.
We present a detailed analysis of how to implement the computation of modular symbols for a given elliptic curve by using numerical approximations. This method turns out to be more efficient than current implementation as the conductor of the curve increases.
In this very short note, we show that there is a relation between the leading term at $s=1$ of an $L$-function of an elliptic curve defined over an number field and the term that follows.