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

Clinical classification of variants in the valosin-containing protein gene associated with multisystem proteinopathy

BACKGROUND AND OBJECTIVES: Pathogenic variants in the valosin-containing protein ( METHODS: A 6-item clinical score was developed to evaluate the phenotypic level of evidence to support the pathogenicity of the novel variants. Each item is allocated a value, a score ranging from 0.5 to 5.5 points. A receiver-operating characteristic curve was used to identify a cutoff value of 3 to consider a variant as high likelihood disease associated. The scoring system results were confronted with results of in vitro ATPase activity assays and with in silico analysis. RESULTS: All variants were missense, except for one small deletion-insertion, 18 led to amino acid changes within the N and D1 domains, and 13 increased the enzymatic activity. The clinical score coincided with the functional studies in 17 of 19 variants and with the in silico analysis in 12 of 19. For 12 variants, the 3 predictive tools agreed, and for 7 variants, the predictive tools disagreed. The pooled data supported the pathogenicity of 13 of 19 novel VCP variants identified in the study. DISCUSSION: This study provides data to support pathogenicity of 14 of 19 nove

Independent neural computation of value from other people's confidence

Expectation of reward can be shaped by the observation of actions and expressions of other people in one's environment. A person's apparent confidence in the likely reward of an action, for instance, makes qualities of their evidence, not observed directly, socially accessible. This strategy is computationally distinguished from associative learning methods that rely on direct observation, by its use of inference from indirect evidence. In twenty-three healthy human subjects, we isolated effects of first-hand experience, other people's choices, and the mediating effect of their confidence, on decision-making and neural correlates of value within ventromedial prefrontal cortex (vmPFC). Value derived from first hand experience and other people's choices (regardless of confidence) were indiscriminately represented across vmPFC. However, value computed from agent choices weighted by their associated confidence was represented with specificity for ventromedial area 10. This pattern corresponds to shifts of connectivity and overlapping cognitive processes along a posterior-anterior vmPFC axis. Task behavior and self-reported self-reliance for decision-making in other social contexts correlated. The tendency to conform in other social contexts corresponded to increased activation in cortical regions previously shown to respond to social conflict in proportion to subsequent conformity (Campbell-Meiklejohn et al., 2010). The tendency to self-monitor predicted a selectively enhanced response to accordance with others in the right temporoparietal junction (rTPJ). The findings anatomically decompose vmPFC value representations according to computational requirements and provide biological insight into the social transmission of preference and reassurance gained from the confidence of others. Significance Statement: Decades of research have provided evidence that the ventromedial prefrontal cortex (vmPFC) signals the satisfaction we expect from imminent actions. However, we have a surprisingly modest understanding of the organization of value across this substantial and varied region. This study finds that using cues of the reliability of other peoples'; knowledge to enhance expectation of personal success generates value correlates that are anatomically distinct from those concurrently computed from direct, personal experience. This suggests that representation of decision values in vmPFC is suborganized according to the underlying computation, consistent with what we know about the anatomical heterogeneity of the region. These results also provide insight into the observational learning process by which someone else's confidence can sway and reassure our choices

Asynchronous Transmitter Release from Cholecystokinin-Containing Inhibitory Interneurons Is Widespread and Target-Cell Independent

Neurotransmitter release at most central synapses is synchronized to the timing of presynaptic action potentials. Here we show that 3 classes of DSI-expressing, CCK-containing, hippocampal interneurons show highly asynchronous release in response to trains of action potentials. This asynchrony is correlated to the class of presynaptic interneuron but is unrelated to their postsynaptic cell target. Asynchronous- and synchronous release from CCK-containing interneurons show a slightly different calcium dependence such that the proportion of asynchronous release increases with external calcium concentration possibly suggesting that the modes of release are mediated by different calcium sensors. Asynchronous IPSCs include very large (up to 500 pA/7nS) amplitude events, which persist in low extracellular calcium and strontium, showing that they result from quantal transmitter release at single release sites. Finally we show that asynchronous release is prominent in response to trains of presynaptic spikes which mimic natural activity of CCK-containing interneurons. That asynchronous release from CCK-containing interneurons is a widespread phenomenon indicates a fundamental role for these cells within the hippocampal network that is distinct from the phasic inhibition provided by PV-containing interneurons