Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders

Orientations of supersingular elliptic curves encode the information of an endomorphism of the curve. Computing the full endomorphism ring is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is $\mathfrak{O}$-orientable for a fixed imaginary quadratic order $\mathfrak{O}$ provides non-trivial information towards computing an endomorphism corresponding to the $\mathfrak{O}$-orientation. We provide explicit algorithms and in-depth complexity analysis. We also consider the question in terms of quaternion algebras. We provide algorithms which compute an embedding of a fixed imaginary quadratic order into a maximal order of the quaternion algebra ramified at $p$ and $\infty$. We provide code implementations in Sagemath which is efficient for finding embeddings of imaginary quadratic orders of discriminants up to $O(p)$, even for cryptographically sized $p$

Sato-Tate groups of abelian threefolds

Given an abelian variety over a number field, its Sato-Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L-function of the abelian variety. It was previously shown by Fit\'e, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato-Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato-Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato-Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura's theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations in Gap, SageMath, and Magma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated L-functions.Comment: Simplified a calculation in Section 6.4; 87 page

Mobile sound: media art in hybrid spaces

The thesis explores the relationships between sound and mobility through an examination of sound art. The research engages with the intersection of sound, mobility and art through original empirical work and theoretically through a critical engagement with sound studies. In dialogue with the work of De Certeau, Lefebvre, Huhtamo and Habermas in terms of the poetics of walking, rhythms, media archeology and questions of publicness, I understand sound art as an experimental mobile and public space. The thesis establishes and situates the emerging field of mobile sound art by mapping three key traditions of mobile sound art - locative art, sound art and public art - and creates a taxonomy of mobile sound art by defining four categories: 'placing sounds', 'sound platforms', 'sonifying mobility' and 'musical instruments' (each represented by one case study). In doing so it develops a methodology that is attentive to the specifics of the sonic and mobile of media experience. I demonstrate how sonic interactions and embodied mobility are designed and experienced in specific ways in each of the four case studies - 'Aura' by Symons (UK), 'Pophorns' by Torstensson and Sandelin (Sweden), 'SmSage' by Redfern and Borland (US) and 'Core Sample' by Rueb (US) (all 2007). In tracing the topos of the musical telephone, discussing the making and breaking of relevant micro publics, accounting for the polyphonies of footsteps and unwrapping bundles of rhythms, this thesis contributes to understanding complex media experiences in hybrid spaces. In doing so it critically sheds light on the quality of sonic artistic experiences, the audience engagement with urban, public and networked spaces and the relationship between sound art and everyday media experience. My thesis provides valuable insight into auditory ways of mobilising and making public spaces, non-verbal and embodied media practices, and rhythms and scales of mobile media experiences