Lower bounds for ranks of Mumford-Tate groups

Let A be a complex abelian variety and G its Mumford--Tate group. Supposing that the simple abelian subvarieties of A are pairwise non-isogenous, we find a lower bound for the rank of G, which is a little less than log_2 dim A. If we suppose that End A is commutative, then we show that rk G >= log_2 dim A + 2, and this latter bound is sharp. We also obtain the same results for the rank of the l-adic monodromy group of an abelian variety defined over a number field. ----- Soit A une vari\'et\'e ab\'elienne complexe et G son groupe de Mumford--Tate. En supposant que les sous vari\'et\'es ab\'eliennes simples de A sont deux \`a deux non-isog\`enes, on trouve une minoration du rang rk G de G, l\'eg\`erement inf\'erieure \`a log_2 dim A. Si on suppose que End A est commutatif, alors on montre que rk G >= log_2 dim A + 2, et cette borne-ci est optimale. On obtient les m\^emes resultats pour le rang du groupe de monodromie l-adique d'une vari\'et\'e ab\'elienne d\'efinie sur un corps de nombres

Height bounds and the Siegel property

Let $G$ be a reductive group defined over $\mathbb{Q}$ and let $\mathfrak{S}$ be a Siegel set in $G(\mathbb{R})$. The Siegel property tells us that there are only finitely many $\gamma \in G(\mathbb{Q})$ of bounded determinant and denominator for which the translate $\gamma.\mathfrak{S}$ intersects $\mathfrak{S}$. We prove a bound for the height of these $\gamma$ which is polynomial with respect to the determinant and denominator. The bound generalises a result of Habegger and Pila dealing with the case of $GL_2$, and has applications to the Zilber-Pink conjecture on unlikely intersections in Shimura varieties. In addition we prove that if $H$ is a subset of $G$, then every Siegel set for $H$ is contained in a finite union of $G(\mathbb{Q})$-translates of a Siegel set for $G$.Comment: 24 pages, minor revision

Families of abelian varieties with many isogenous fibres

Let Z be a subvariety of the moduli space of principally polarised abelian varieties of dimension g over the complex numbers. Suppose that Z contains a Zariski dense set of points which correspond to abelian varieties from a single isogeny class. A generalisation of a conjecture of Andr\'e and Pink predicts that Z is a weakly special subvariety. We prove this when dim Z = 1 using the Pila--Zannier method and the Masser--W\"ustholz isogeny theorem. This generalises results of Edixhoven and Yafaev when the Hecke orbit consists of CM points and of Pink when it consists of Galois generic points.Comment: Gap in Lemma 3.3 found and corrected by Gabriel Dil

On compatibility between isogenies and polarisations of abelian varieties

We discuss the notion of polarised isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarisations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarised isogenies can be reduced to questions about unpolarised isogenies or vice versa. Our main theorem concerns abelian varieties B which are isogenous to a fixed abelian variety A. It establishes the existence of a polarised isogeny A to B whose degree is polynomially bounded in n, if there exist both an unpolarised isogeny A to B of degree n and a polarised isogeny A to B of unknown degree. As a further result, we prove that given any two principally polarised abelian varieties related by an unpolarised isogeny, there exists a polarised isogeny between their fourth powers. The proofs of both theorems involve calculations in the endomorphism algebras of the abelian varieties, using the Albert classification of these endomorphism algebras and the classification of Hermitian forms over division algebras

Models for online, open, flexible and technology enhanced higher education across the globe – a comparative analysis

Digital technology has become near ubiquitous in many countries today or is on a path to reach this state in the near future. Across the globe the share of internet users, for instance, has jumped in the last ten years. In Europe most countries have a share of internet users near to or above 90% in 2016 (last year available for international comparisons), in China the current share is 53%, but this has grown from just 16% in 2007, even in Ethiopia the share has grown from 0.4% to 15.4% in the same period (data from ITU). At the same time expectations of widespread adoption of digital solutions in higher education have been rising. In 2017 the New Media Consortium’s Horizon Report predicted that adaptive learning would take less than a year to be widely adopted (Adams Becker et al., 2017). And projects such as ‘Virtually Inspired’ are showcasing creative examples of how new technologies are already being harnessed to improve the quality of teaching and learning. Furthermore, discussion of the United Nations’ Sustainable Development Goals emphasise the key potentials that digital technology holds for achieving the goals for education in 2030 (UNESCO, 2017). These developments lead university and college leadership to the question of how they should position their institution. What type of digitalisation initiatives can be found practice beyond best practices and future potentials? This is the question that this study attempts to answer. It sets out to analyse how higher education providers from across the world are harnessing digitalisation to improve teaching and learning and learner support and to identify emerging types of practice. For this, it focuses on the dimensions of flexibility of provision (in terms of time, place and pace) and openness of provision (in terms of who has access to learning and support and who is involved in the design of learning provision), as both of these dimensions can significantly benefit from integration of digital solutions. The method of information collation used by the study was a global survey of higher education institutions (HEIs) covering all world continents, more than thirty countries and 69 cases. The survey found that nearly three-quarters of all HEIs have at least one strategic focus and typologies were developed based on this analysis to group HEIs with similar strategic focuses. Overall, the findings suggest that most higher education providers are just at the beginning of developing comprehensive strategies for harnessing digitalisation. For this reason, the authors of this study believe that providers can benefit from the outcomes of this study’s research, as it can be used by university and college leadership for benchmarking similarities and differences and for cooperative peer learning between institutions. The database of cases and the guidelines for reviewing current strategies, which accompany this study, aim to facilitate this learning and evaluation process

Clearing Up Some Conceptual Confusions About Conspiracy Theory Theorising

A reply to Gérald Bronner, Véronique Campion-Vincent, Sylvain Delouvée, Sebastian Dieguez, Nicolas Gauvrit, Anthony Lantian, and Pascal Wagner-Egger's piece, '“They” Respond: Comments on Basham et al.’s “Social Science’s Conspiracy-Theory Panic: Now They Want to Cure Everyone”

Galois conjugates of special points and special subvarieties in Shimura varieties

Let be a Shimura variety with reflex field . We prove that the action of on maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih’s construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne