No abstract

Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.

We show that rings of S-integers of a global function field K of odd characteristic are first-order universally definable in K. This extends work of Koenigsmann and Park who showed the same for Z in Q and the ring of integers in a number field, respectively. We also give another proof of a theorem of Poonen and show that the set of non-squares in a global field of characteristic = 2 is diophantine. Finally, we show that the set of pairs (x, y) ∈ K × × K × such that x is not a norm in K( √ y) is diophantine over K for any global field K of characteristic = 2.

We study the problem of generating the endomorphism ring of a supersingular elliptic curve by two cycles in ℓ-isogeny graphs. We prove a necessary and sufficient condition for the two endomorphisms corresponding to two cycles to be linearly independent, expanding on the work by Kohel in his thesis. We also give a criterion under which the ring generated by two cycles is not a maximal order. We give some examples in which we compute cycles which generate the full endomorphism ring. The most difficult part of these computations is the calculation of the trace of these cycles. We show that a generalization of Schoof's algorithm can accomplish this computation efficiently. that led to a significant improvement in the running time analysis of the generalization of Schoof's algorithm, and for outlining the proofs of Proposition 2.4 and Lemma 6.9. Finally, we thank the Women in Numbers 4 conference and BIRS, for enabling us to start this project in a productive environment. K.E. and T.M. were partially supported by National Science Foundation awards DMS-1056703 and CNS-1617802. Isogeny graphs2.1. Definitions and properties. In this section, we recall several definitions and notation that are used throughout. We refer the reader to [Sil09] and [Koh96] for a detailed overview on some of the below. Let k be a field of characteristic p > 3.

Seven axenic wild-type and repair-deficient mutant strains of the cellular slime mold Dicryusrelium discoideum have been treated with the furocoumarin 8-methoxypsoralen (8-MOP) up to 50 pgirne and then exposed to near ultraviolet light (UVA 320-400 nm) up to 21 kJlm2. Fluence-response survival curves exhibit shoulders at lower fluences and an exponential lethal response at higher fluences. Neither the psoralen alone nor the irradiation alone produced any measurable lethal effect. Wild-type strains, which show resistance to 254 nm UV and gamma radiation, also show resistance to psoralen plus UVA. The moderate sensitivity of a rad D repair-deficient mutant strain and the extreme sensitivity of a rad B mutant strain to 8-MOP plus UVA parallel their responses to UV and gamma radiation. However a rad C mutant which is sensitive to UV, exhibits wild-type response to photoactivated psoralen.