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Christian Wuthrich

Publications and source records attributed to Christian Wuthrich.

35 records · Page 2Linked to original sources
arXiv2015

Vanishing of some Galois cohomology groups for elliptic curves

Let E/Q be an elliptic curve and p be a prime number, and let G be the Galois group of the extension of Q obtained by adjoining the coordinates of the p-torsion points on E. We determine all cases when the Galois cohomology group H^1(G, E[p]) does not vanish, and investigate the analogous question for E[p^i] when i>1. We include an application to the verification of certain cases of the Birch and Swinnerton-Dyer conjecture, and another application to the Grunwald-Wang problem for elliptic curves.

math.NT↗
arXiv2015

On Mordell-Weil groups and congruences between derivatives of twisted Hasse-Weil L-functions

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair ( h^1(A/F)(1), Z[Gal(F/k)] ). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell-Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural p-adic congruences between the values at s=1 of derivatives of the Hasse-Weil L-functions associated to twists of A, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate-Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions.

math.NT↗
arXiv2015

On the Galois structure of Selmer groups

Let A be an abelian variety defined over a number field k and F a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we investigate the explicit Galois structure of the p-primary Selmer group of A over F. We also use the results so obtained to derive new bounds on the growth of the Selmer rank of A over extensions of k.

math.NT↗
arXiv2015

What becomes of a causal set

Unlike the relativity theory it seeks to replace, causal set theory has been interpreted to leave space for a substantive, though perhaps 'localized', form of 'becoming'. The possibility of fundamental becoming is nourished by the fact that the analogue of Stein's theorem from special relativity does not hold in causal set theory. Despite this, we find that in many ways, the debate concerning becoming parallels the well-rehearsed lines it follows in the domain of relativity. We present, however, some new twists and challenges. In particular, we show that a novel and exotic notion of becoming is compatible with causal sets. In contrast to the 'localized' becoming considered compatible with the dynamics of causal set theory by its advocates, our novel kind of becoming, while not answering to the typical A-theoretic demands, is 'global' and objective.

physics.hist-ph↗
arXiv2014

Putnam looks at quantum mechanics (again and again)

Hilary Putnam (1965, 2005) has argued that from a realist perspective, quantum mechanics stands in need of an interpretation. Ironically, this hypothesis may appear vulnerable against arguments drawing on Putnam's own work. Nancy Cartwright (2005) has urged that his 1962 essay on the meaning of theoretical terms suggests that quantum mechanics needs no interpretation and thus stands in tension with his claim of three years later. She furthermore contends that this conflict should be resolved in favour of the earlier work, as quantum mechanics, like all successful theories, does not need an interpretation. The first part of this essay deflates both of these objections. The second part addresses and evaluates Putnam's own assessments of the main interpretative options available in 1965 and 2005. Although we may disagree on some aspects, his pessimistic conclusion will come out largely unscathed, and, in fact, enhanced. I will close by briefly stating the historical relevance of this work.

physics.hist-ph↗
arXiv2014

A quantum-information-theoretic complement to a general-relativistic implementation of a beyond-Turing computer

There exists a growing literature on the so-called physical Church-Turing thesis in a relativistic spacetime setting. The physical Church-Turing thesis is the conjecture that no computing device that is physically realizable (even in principle) can exceed the computational barriers of a Turing machine. By suggesting a concrete implementation of a beyond-Turing computer in a spacetime setting, István Németi and Gyula Dávid (2006) have shown how an appreciation of the physical Church-Turing thesis necessitates the confluence of mathematical, computational, physical, and indeed cosmological ideas. In this essay, I will honour István's seventieth birthday, as well as his longstanding interest in, and his seminal contributions to, this field going back to as early as 1987 by modestly proposing how the concrete implementation in Németi and Dávid (2006) might be complemented by a quantum-information-theoretic communication protocol between the computing device and the logician who sets the beyond-Turing computer a task such as determining the consistency of Zermelo-Fraenkel set theory. This suggests that even the foundations of quantum theory and, ultimately, quantum gravity may play an important role in determining the validity of the physical Church-Turing thesis.

physics.hist-ph↗
arXiv2014

Raiders of the lost spacetime

Spacetime as we know and love it is lost in most approaches to quantum gravity. For many of these approaches, as inchoate and incomplete as they may be, one of the main challenges is to relate what they take to be the fundamental non-spatiotemporal structure of the world back to the classical spacetime of GR. The present essay investigates how spacetime is lost and how it may be regained in one major approach to quantum gravity, loop quantum gravity.

physics.hist-ph↗
arXiv2013

No categorial support for radical ontic structural realism

Radical ontic structural realism (ROSR) asserts an ontological commitment to 'free-standing' physical structures understood solely in terms of fundamental relations, without any recourse to relata which stand in these relations. Bain (2011) has recently defended ROSR against the common charge of incoherence by arguing that a reformulation of fundamental physical theories in category-theoretic terms (rather than the usual set-theoretic ones) offers a coherent and precise articulation of the commitments accepted by ROSR. In this essay, we argue that category theory does not offer a more hospitable environment to ROSR than set theory. We also show that the application of category-theoretic tools to topological quantum field theory and to algebraic generalisations of general relativity do not warrant the claim that these theories describe 'object-free' structures. We conclude that category theory offers little if any comfort to ROSR.

physics.hist-ph↗
arXiv2012

Emergent spacetime and empirical (in)coherence

Numerous approaches to a quantum theory of gravity posit fundamental ontologies that exclude spacetime, either partially or wholly. This situation raises deep questions about how such theories could relate to the empirical realm, since arguably only entities localized in spacetime can ever be observed. Are such entities even possible in a theory without fundamental spacetime? How might they be derived, formally speaking? Moreover, since by assumption the fundamental entities can't be smaller than the derived (since relative size is a spatiotemporal notion) and so can't 'compose' them in any ordinary sense, would a formal derivation actually show the physical reality of localized entities? We address these questions via a survey of a range of theories of quantum gravity, and generally sketch how they may be answered positively.

physics.hist-ph↗
arXiv2012

The structure of causal sets

More often than not, recently popular structuralist interpretations of physical theories leave the central concept of a structure insufficiently precisified. The incipient causal sets approach to quantum gravity offers a paradigmatic case of a physical theory predestined to be interpreted in structuralist terms. It is shown how employing structuralism lends itself to a natural interpretation of the physical meaning of causal sets theory. Conversely, the conceptually exceptionally clear case of causal sets is used as a foil to illustrate how a mathematically informed rigorous conceptualization of structure serves to identify structures in physical theories. Furthermore, a number of technical issues infesting structuralist interpretations of physical theories such as difficulties with grounding the identity of the places of highly symmetrical physical structures in their relational profile and what may resolve these difficulties can be vividly illustrated with causal sets.

physics.hist-ph↗
arXiv2012

The fate of presentism in modern physics

There has been a recent spate of essays defending presentism, the view in the metaphysics of time according to which all and only present events or entities exist. What is particularly striking about this resurgence is that it takes place on the background of the significant pressure exerted on the position by the relativity of simultaneity asserted in special relativity, and yet in several cases invokes modern physics for support. I classify the presentist arguments into a two by two matrix depending on whether they take a compatibilist or incompatibilist stance with respect to both special relativity in particular and modern physics in general. I then review and evaluate what I take to be some of the most forceful and intriguing presentist arguments turning on modern physics. Although nothing of what I will say eventuates its categorical demise, I hope to show that whatever presentism remains compatible with empirical facts and our best physics is metaphysically repugnant.

physics.hist-ph↗
arXiv2012

Time in quantum gravity

Quantum gravity--the marriage of quantum physics with general relativity--is bound to contain deep and important lessons for the nature of physical time. Some of these lessons shall be canvassed here, particularly as they arise from quantum general relativity and string theory and related approaches. Of particular interest is the question of which of the intuitive aspects of time will turn out to be fundamental, and which 'emergent' in some sense.

physics.hist-ph↗
arXiv2011

The class group pairing and $p$-descent on elliptic curves

We give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field. Then we relate it to the descent relative to a suitable cyclic isogeny. This allows us to connect the resulting Selmer group with the logarithmic class group of the base. These constructions are explicit and suitable for computer experimentation. From a conceptual point of view, the questions that arise here are analogues of "visibility" questions in the sense of Cremona and Mazur.

math.NT↗
arXiv2008

A twist in the geometry of rotating black holes: seeking the cause of acausality

We investigate Kerr-Newman black holes in which a rotating charged ring-shaped singularity induces a region which contains closed timelike curves (CTCs). Contrary to popular belief, it turns out that the time orientation of the CTC is opposite to the direction in which the singularity or the ergosphere rotates. In this sense, CTCs "counter-rotate" against the rotating black hole. We have similar results for all spacetimes sufficiently familiar to us in which rotation induces CTCs. This motivates our conjecture that perhaps this counter-rotation is not an accidental oddity particular to Kerr-Newman spacetimes, but instead there may be a general and intuitively comprehensible reason for this.

gr-qc↗
arXiv2003

Generic Incomparability of Infinite-Dimensional Entangled States

In support of a recent conjecture by Nielsen (1999), we prove that the phenomena of 'incomparable entanglement'--whereby, neither member of a pair of pure entangled states can be transformed into the other via local operations and classical communication (LOCC)--is a generic feature when the states at issue live in an infinite-dimensional Hilbert space.

quant-ph↗