Travis Morrison
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View article: Zeta functions of abstract isogeny graphs and modular curves
Zeta functions of abstract isogeny graphs and modular curves Open
We introduce a ``non-orientable'' variation of Serre's definition of a graph, which we call an abstract isogeny graph. These objects capture the combinatorics of the graphs $G(p,\ell,H)$, the $\ell$-isogeny graphs of supersingular elliptic…
View article: The SEA algorithm for endomorphisms of supersingular elliptic curves
The SEA algorithm for endomorphisms of supersingular elliptic curves Open
For a prime $p{\,>\,}3$ and a supersingular elliptic curve $E$ defined over $\mathbb{F}_{p^2}$ with ${j(E)\notin\{0,1728\}}$, consider an endomorphism $α$ of $E$ represented as a composition of $L$ isogenies of degree at most $d$. We prove…
View article: Masking Countermeasures Against Side-Channel Attacks on Quantum Computers
Masking Countermeasures Against Side-Channel Attacks on Quantum Computers Open
We propose a modification to the transpiler of a quantum computer to safeguard against side-channel attacks aimed at learning information about a quantum circuit. We demonstrate that if it is feasible to shield a specific subset of gates f…
View article: Computing Isogenies at Singular Points of the Modular Polynomial
Computing Isogenies at Singular Points of the Modular Polynomial Open
In this paper we present a method which, given a singular point $(j_1, j_2)$ on $Y_0(\ell)$ with $j_1, j_2 \neq 0, 1728$ and an elliptic curve $E$ with $j$-invariant ${j_1}$, returns an elliptic curve $\widetilde{E}$ with $j$-invariant ${j…
View article: Towards a classification of isolated $j$-invariants
Towards a classification of isolated $j$-invariants Open
We develop an algorithm to test whether a non-CM elliptic curve $E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing …
View article: Curve-lifted codes for local recovery using lines
Curve-lifted codes for local recovery using lines Open
In this paper, we introduce curve-lifted codes over fields of arbitrary characteristic, inspired by Hermitian-lifted codes over $\mathbb{F}_{2^r}$. These codes are designed for locality and availability, and their particular parameters dep…
View article: Computing supersingular endomorphism rings using inseparable endomorphisms
Computing supersingular endomorphism rings using inseparable endomorphisms Open
We give an algorithm for computing an inseparable endomorphism of a supersingular elliptic curve $E$ defined over $\mathbb F_{p^2}$, which, conditional on GRH, runs in expected $O(p^{1/2}(\log p)^2(\log\log p)^3)$ bit operations and requir…
View article: Computing endomorphism rings of supersingular elliptic curves and connections to path-finding in isogeny graphs
Computing endomorphism rings of supersingular elliptic curves and connections to path-finding in isogeny graphs Open
Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems.We give a n…
View article: Computing endomorphism rings of supersingular elliptic curves and connections to pathfinding in isogeny graphs
Computing endomorphism rings of supersingular elliptic curves and connections to pathfinding in isogeny graphs Open
Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems. In this pa…
View article: Chabauty-Coleman computations on rank 1 Picard curves
Chabauty-Coleman computations on rank 1 Picard curves Open
We provably compute the full set of rational points on 1403 Picard curves defined over $\mathbb{Q}$ with Jacobians of Mordell-Weil rank $1$ using the Chabauty-Coleman method. To carry out this computation, we extend Magma code of Balakrish…
View article: Diophantine definability of nonnorms of cyclic extensions of global fields
Diophantine definability of nonnorms of cyclic extensions of global fields Open
We show that, for any square-free natural number and any global field with containing a primitive th root of unity, the pairs such that is not a relative norm of form a diophantine set over . We use the Hasse norm theorem, Kummer the…
View article: Cycles in the supersingular $\ell$-isogeny graph and corresponding endomorphisms
Cycles in the supersingular $\ell$-isogeny graph and corresponding endomorphisms Open
We study the problem of generating the endomorphism ring of a supersingular elliptic curve by two cycles in $\ell$-isogeny graphs. We prove a necessary and sufficient condition for the two endomorphisms corresponding to two cycles to be li…
View article: Universally and existentially definable subsets of global fields
Universally and existentially definable subsets of global fields Open
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 $\mathbb{Z}$ in $\mathbb{Q}$ and the ri…