A simplified proof of the Andre-Oort conjecture for products of modular curves

In this paper we give a short proof of the André-Oort conjecture for products of modular curves under the Generalised Riemann Hypothesis using only simple Galois-theoretic and geometric arguments. We believe this method represents a strategy for proving the conjecture for a general Shimura variety under GRH without using ergodic theory. We also demonstrate a short proof of the Manin–Mumford conjecture for Abelian varieties using similar arguments

Canonical models of modular curves and the Galois action on CM-points

We use the theory of canonical models of Shimura varieties to describe the projective limit of the curves Y(N), all N, and its automorphism group. In particular we prove that the Galois group of Q(CM) over Q is an extension of a certain abelian group by a 2-element group, where Q(CM) stands for the the extension of Q by all the CM-points on all the curves Y(N)

Unlikely intersections with ExCM curves in A2

The Zilber--Pink conjecture predicts that an algebraic curve in A2 has only finitely many intersections with the special curves, unless it is contained in a proper special subvariety. Under a large Galois orbits conjecture, we prove the finiteness of the intersection with the special curves parametrising abelian surfaces isogenous to the product of two elliptic curves, at least one of which has complex multiplication. Furthermore, we show that this large Galois orbits conjecture holds for curves satisfying a condition on their intersection with the boundary of the Baily--Borel compactification of A2. More generally, we show that a Hodge generic curve in an arbitrary Shimura variety has only finitely many intersection points with the generic points of a Hecke--facteur family, again under a large Galois orbits conjecture

Conjugation of semisimple subgroups over real number fields of bounded degree

Let G be a linear algebraic group over a field k of characteristic 0. We show that any two connected semisimple k-subgroups of G that are conjugate over an algebraic closure of k are actually conjugate over a finite field extension of k of degree bounded independently of the subgroups. Moreover, if k is a real number field, we show that any two connected semisimple k-subgroups of G that are conjugate over the field of real numbers are actually conjugate over a finite real extension of k of degree bounded independently of the subgroups

Effective estimates for the degrees of maximal special subvarieties

Let Z be an algebraic subvariety of a Shimura variety. We extend results of the first author to prove an effective upper bound for the degree of a non-facteur maximal special subvariety of Z

The Andre-Oort conjecture via o-minimality

The purpose of this article is to explain the Pila-Zannier strategy for proving the Andr\'e-Oort conjecture. First, however, we will provide a brief introduction to the theory of Shimura varieties.Comment: Preprint, to appear in the forthcoming LMS Lecture Note series, O-Minimality and Diophantine Geometry, published by Cambridge Univeristy Press. Originating from a short LMS-EPSRC course held at the Univeristy of Mancheste

How fasting and vegetarianism is perceived to support psi among adepts

Many spiritual traditions include accounts of miracles and supernormal powers (known as ‘psi’, Parapsychological Association, 2015) which are associated with fasting and vegetarianism. Some authors have related these dietary practices to both spirituality and psi. However, there has so far been little empirical research in this area. We present a thematic analysis of interviews with seven ‘psi adepts’ (those who practise psi in a professional capacity) to examine their understanding of how fasting and vegetarianism are perceived to affect their sensitivity to psi. Results demonstrate that participants regard fasting as facilitating psi through purification and its effects on the gut; and that vegetarianism facilitates psi because it enhances mental clarity, supports general health, and because food has purported subtle energies. If supported by subsequent empirical work, these results may help to improve performance in psi tasks; support the health of adepts; and offer insight into proposed theories of psi

How fasting and vegetarianism is perceived to support psi among adepts

Many spiritual traditions include accounts of miracles and supernormal powers (known as ‘psi’, Parapsychological Association, 2015) which are associated with fasting and vegetarianism. Some authors have related these dietary practices to both spirituality and psi. However, there has so far been little empirical research in this area. We present a thematic analysis of interviews with seven ‘psi adepts’ (those who practise psi in a professional capacity) to examine their understanding of how fasting and vegetarianism are perceived to affect their sensitivity to psi. Results demonstrate that participants regard fasting as facilitating psi through purification and its effects on the gut; and that vegetarianism facilitates psi because it enhances mental clarity, supports general health, and because food has purported subtle energies. If supported by subsequent empirical work, these results may help to improve performance in psi tasks; support the health of adepts; and offer insight into proposed theories of psi

Two surveys to explore associations between dietary practices and spontaneous anomalous experiences

Contemporary and historical accounts from spiritual traditions and elsewhere appear to suggest a role for fasting and vegetarianism in spiritual development and access to ‘supernormal powers’. We recently conducted interviews with psi adepts who use fasting and vegetarianism to support their work with psi and reported findings that are consistent with such accounts. However, it was not clear whether these associations obtained only for particular elite practitioners or whether they might apply to more general populations. We therefore proposed to conduct online questionnaire-based surveys of two separate samples recruited through Facebook groups consisting of vegans and vegetarians (N = 804) and those who practise fasting (N = 154) to see if these associations could be confirmed. Working with a vegan/vegetarian group allowed us to compare those who fast with those who do not; similarly, the fasting sample allowed us to compare vegans and vegetarians with those who eat meat. Preliminary findings indicate that significantly higher levels of self-reported anomalous experiences and abilities are reported by those who: fast; engage in longer fasts; have a longer history of fasting; practise vegan or vegetarian over meat-eating diets; practise vegan over vegetarian diets; have increased adherence to a wholefood diet; and habitually abstain from alcohol. The implications of these findings are discussed