arXiv subjects
Publications and source records attributed to Josep M. Miret.
We generalize the notion of twisting endomorphisms, first defined by Castryck-Panny-Vercauteran, to the setting of $\mathcal{O}$-oriented supersingular elliptic curves. We give an algorithm to find supersingular elliptic curves over $\mathbb{F}_p$ with a twisting endomorphism of prime degree $\ell$, and we use it to compute a basis of their full endomorphism rings.
A digraph $G$ is \emph{$k$-geodetic} if for any pair $u,v \in V(G)$ there is at most one $u,v$-walk of length not exceeding $k$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the directed Moore bound $M(d,k) = 1 + d + d^2+ \cdots +d^k$. It is known that the Moore bound cannot be achieved for $d,k \geq 2$. A $k$-geodetic digraph with minimum degree $d$ and order one greater than the Moore bound has \emph{excess one}. In this paper we prove a conjecture that no excess one digraphs exist for $d,k \geq 2$, thus complementing the result of Bannai and Ito on the non-existence of undirected graphs with excess one.
We give a characterization of the codomain $[\ell]E(k)$ of the multiplication-by-$\ell$ map $[\ell]$ in the case of elliptic curves over a field $k$ of characteristic $\ne 2,3$ with $\ell$-torsion $E[\ell]=\langle W_1,W_2 \rangle$ fully defined over $k$, for primes $\ell$ different from the characteristic. We show that a point $Q\in E(k)$ lies in $[\ell]E(k)$ if and only if $h_{W_1}(-Q)$ and $h_{W_2}(-Q)$ are $\ell$-powers of $k$, where $h_{W_1}$ and $h_{W_2}$ are functions on $E$ with divisor ${\rm div}(h_{W_i})=\ell W_i- \ell P_{\infty}$. Our characterization leads to an effective procedure to find pre-images of $[\ell]$ by solving an order $\ell$ system of linear equations and computing a polynomial gcd.