3rd installment of Front Range Algebraic Geometry (FRAG) day happening at CSU in Weber 201 1:00-5:00 https://sites.google.com/colorado.edu/front-range-ag/home
FRAG Day
Thu, Oct. 30 10am (Math 350)
Number Theory
Wissam Ghantous (University of Central Florida)
X
The main hard problem underlying the field of isogeny-based cryptography is that of finding an isogeny between two given supersingular elliptic curves. An associated (and equivalent) problem is that of finding the endomorphism ring of a given elliptic curve. Indeed, endomorphisms of elliptic curves play a crucial role, and appear in many places in the literature: from the special soundness of the SIDH-based Identification Protocol, to the binding property of Sterner's isogeny graph commitment scheme, to that of finding collisions for the CGL hash function.
From a graph theoretic point of view, these endomorphisms correspond to cycles in . We study these cycles in a very general way, and using two different methods: Brandt matrices and ideal counting. Indeed, this study is not limited to the usual isogeny graph. We instead consider the generalized supersingular -isogeny graph, where is a set of primes and isogenies are allowed to have degree in the set .
Cycles in the generalized supersingular -isogeny graph
Thu, Oct. 30 1:20pm (MATH 3…
Number Theory
Gabrielle Scullard (University of Georgia)
X
The study of supersingular elliptic curves oriented by a quadratic imaginary order O, and the horizontal isogenies between them (induced by the action of an ideal in the class group), has become increasingly important in isogeny-based cryptography; these are the curves and isogenies which are fundamental to CSIDH and SCALLOP, for example. We relate N-isogenies between supersingular elliptic curves oriented by an order of discriminant D, to solutions of equations involving positive definite binary quadratic forms of discriminant D. In the case that -DN < 2p, we characterize when non-horizontal N-isogenies arise. As an application, when -D < 2p, we classify when an oriented supersingular elliptic curve has multiple orientations by the order O.
Isogenies between oriented supersingular elliptic curves
Thu, Oct. 30 2:30pm (MATH 3…
Functional Analysis
Maggie Reardon (CU Boulder)
X
Matui’s HK-conjecture proposes a connection between the homology of a nice enough étale groupoid and the -theory of the associated reduced -algebra. The HK-conjecture is not true in general and there are a number of counterexamples. The related AH-conjecture predicts an exact sequence involving the zeroth and first homology groups together with the abelianization of the topological full group. Unlike the HK-conjecture, no counterexamples are known for the AH-conjecture, though it remains unproven. Both conjectures are verified for a number of natural classes of groupoids, including AF groupoids.
Putnam introduced a new class of groupoids in the paper titled “Some classifiable groupoid -algebras with prescribed -theory”. These new groupoids are related to AF groupoids and this prompts a natural question: does this new class of groupoids satisfy the HK- and AH-conjectures?
The HK- and AH-conjectures for certain groupoids constructed by Putnam
FRAG Day