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In this paper, we study isogeny graphs of supersingular elliptic curves. Supersingular isogeny graphs were introduced as a hard problem into cryptography by Charles, Goren, and Lauter for the construction of cryptographic hash functions [CGL06]. These are large expander graphs, and the hard problem is to find an efficient algorithm for routing, or path-finding, between two vertices of the graph. We consider four aspects of supersingular isogeny graphs, study each thoroughly and, where appropriate, discuss how they relate to one another. First, we consider two related graphs that help us understand the structure: the `spine $mathcal{S}$, which is the subgraph of $mathcal{G}_ell(overline{mathbb{F}_p})$ given by the $j$-invariants in $mathbb{F}_p$, and the graph $mathcal{G}_ell(mathbb{F}_p)$, in which both curves and isogenies must be defined over $mathbb{F}_p$. We show how to pass from the latter to the former. The graph $mathcal{S}$ is relevant for cryptanalysis because routing between vertices in $mathbb{F}_p$ is easier than in the full isogeny graph. The $mathbb{F}_p$-vertices are typically assumed to be randomly distributed in the graph, which is far from true. We provide an analysis of the distances of connected components of $mathcal{S}$. Next, we study the involution on $mathcal{G}_ell(overline{mathbb{F}_p})$ that is given by the Frobenius of $mathbb{F}_p$ and give heuristics on how often shortest paths between two conjugate $j$-invariants are preserved by this involution (mirror paths). We also study the related question of what proportion of conjugate $j$-invariants are $ell$-isogenous for $ell = 2,3$. We conclude with experimental data on the diameters of supersingular isogeny graphs when $ell = 2$ and compare this with previous results on diameters of LPS graphs and random Ramanujan graphs.
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Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere. W.S. Anglin, the Mathematical Intelligencer, 4 (4), 1982.
We obtain an upper bound for the number of pairs $ (a,b) in {Atimes B} $ such that $ a+b $ is a prime number, where $ A, B subseteq {1,...,N }$ with $|A||B| , gg frac{N^2}{(log {N})^2}$, $, N geq 1$ an integer. This improves on a bound given by Balog, Rivat and Sarkozy.
If $mathfrak{p} subseteq mathbb{Z}[zeta]$ is a prime ideal over $p$ in the $(p^d - 1)$th cyclotomic extension of $mathbb{Z}$, then every element $alpha$ of the completion $mathbb{Z}[zeta]_mathfrak{p}$ has a unique expansion as a power series in $p$ with coefficients in $mu_{p^d -1} cup {0}$ called the Teichmuller expansion of $alpha$ at $mathfrak{p}$. We observe three peculiar and seemingly unrelated patterns that frequently appear in the computation of Teichmuller expansions, then develop a unifying theory to explain these patterns in terms of the dynamics of an affine group action on $mathbb{Z}[zeta]$.
For a natural number $Ngeq 2$ and a real $alpha$ such that $0 < alpha leq sqrt{N}-1$, we define $I_alpha:=[alpha,alpha+1]$ and $I_alpha^-:=[alpha,alpha+1)$ and investigate the continued fraction map $T_alpha:I_alpha to I_alpha^-$, which is defined as $T_alpha(x):= N/x-d(x),$ where $d(x):=left lfloor N/x -alpharight rfloor$. For all natural $N geq 7$, for certain values of $alpha$, open intervals $(a,b) subset I_alpha$ exist such that for almost every $x in I_{alpha}$ there is an natural number $n_0$ for which $T_alpha^n(x) otin (a,b)$ for all $ngeq n_0$. These emph{gaps} $(a,b)$ are investigated in the square $Upsilon_alpha:=I_alpha times I_alpha^-$, where the emph{orbits} $T_alpha^k(x), k=0,1,2,ldots$ of numbers $x in I_alpha$ are represented as cobwebs. The squares $Upsilon_alpha$ are the union of emph{fundamental regions}, which are related to the cylinder sets of the map $T_alpha$, according to the finitely many values of $d$ in $T_alpha$. In this paper some clear conditions are found under which $I_alpha$ is gapless. When $I_alpha$ consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of $I_alpha$ with regard to the fixed points of $I_alpha$ under $T_alpha$.
We study the variance of the random variable that counts the number of lattice points in some shells generated by a special class of finite type domains in $mathbb R^d$. The proof relies on estimates of the Fourier transform of indicator functions of convex domains.
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