## Departmental PhD Thesis Exam – William George

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, August 26, 2015
2:10 p.m.
BA6183

PhD Candidate: William George
Supervisor:  Kumar Murty
Thesis title:  Lifting Problems, Cross-fiberedness, and Diffusive Properties on Elliptic Surfaces

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Abstract:

Given an elliptic curve $\tilde{E}$ over $\mathbb{F}_p$ the field of $p$ elements, and given points $\tilde{P}$ and $\tilde{Q}$ in $\tilde{E}(\mathbb{F}_p)$ such that $\tilde{Q}=n\tilde{P}$, the Elliptic Curve Discrete Logarithm Problem (ECDLP) is to find $n$. The difficulty of the ECDLP is the foundation for much of modern cryptographic security. S. Miri and V.K. Murty consider a lifting problem where one asks to find the canonical height of lifts of points $P\in \tilde{E}(\mathbb{F}_p)$ to an appropriately chosen elliptic curve over $\mathbb{Q}$. They saw that the ECDLP would be equivalent to this lifting problem if one could find lifts of $\tilde{E}$ with certain properties. In this thesis we consider a variant of this lifting problem where lifts with the required properties can be found and hence we prove that this problem is equivalent to the ECDLP.

Furthermore, we relate these ideas to a conjecture of B. Mazur which asserts that for any non-constant elliptic fibration $\left\{E_t \right\}_{t\in \mathbb{Q}}$, the set $\left\{t\in \mathbb{Q}: rk(E_t(\mathbb{Q}))>0 \right\}$ is finite or dense in $\mathbb{R}$ with respect to the real topology. Particularly interesting to us are elliptic surfaces considered by R. Munshi on which there is a second fibration by genus one curves such that the two fibrations interact in nice ways. On these surfaces, using the second fibration, non-torsion points are “diffused” from one fiber $E_t$ to others and hence positive rank fibers are “diffused” through $\mathbb{R}$ allowing one to conclude Mazur’s conjecture for these families. We show that this process also “diffuses” certain other properties of fibers. We see that, under certain conditions, the set of fibers having a list of properties motivated by our lifting problems is either finite or dense in $\mathbb{R}$. Moreover, we study the dynamics of this process of diffusion allowing us to bound the proportion of times our process produces fibers with the desired properties. One would expect these properties to be largely unrelated to each other; some of them come from viewing the fiber as an elliptic curve over each of $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{F}_p$. Indeed, we show that the probability that the diffusion process produces a lift with all of the properties is the product of each of the respective probabilities up to an error bound that goes to $0$ as $p\rightarrow \infty$. Hence we see that whether the output fiber of the diffusion process has these properties is “asymptotically independent.”

A copy of the thesis can be obtained by contacting William at william.george@utoronto.ca