Noam D. Elkies
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View article: Equations for a K3 Lehmer map
Equations for a K3 Lehmer map Open
C. T. McMullen proved the existence of a K3 surface with an automorphism of entropy given by the logarithm of Lehmer’s number, which is the minimum possible among automorphisms of complex surfaces. We reconstruct equations for the surface …
View article: New rank records for elliptic curves having rational torsion
New rank records for elliptic curves having rational torsion Open
We present rank-record breaking elliptic curves having torsion subgroups Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.
View article: Periodic continued fractions over $S$-integers in number fields and Skolem’s $p$-adic method
Periodic continued fractions over $S$-integers in number fields and Skolem’s $p$-adic method Open
We generalize the classical theory of periodic continued fractions (PCFs) over ${\mathbf Z}$ to rings ${\mathcal O}$ of $S$-integers in a number field. Let ${\mathcal B}=\{β, {β^*}\}$ be the multi-set of roots of a quadratic polynomial in …
View article: Periodic continued fractions over $S$-integers in number fields and\n Skolem's $p$-adic method
Periodic continued fractions over $S$-integers in number fields and\n Skolem's $p$-adic method Open
We generalize the classical theory of periodic continued fractions (PCFs)\nover ${\\mathbf Z}$ to rings ${\\mathcal O}$ of $S$-integers in a number field.\nLet ${\\mathcal B}=\\{\\beta, {\\beta^*}\\}$ be the multi-set of roots of a\nquadra…
View article: The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k
The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k Open
The elliptic curve $E_k \\colon y^2 = x^3 + k$ admits a natural 3-isogeny\n$\\phi_k \\colon E_k \\to E_{-27k}$. We compute the average size of the\n$\\phi_k$-Selmer group as $k$ varies over the integers. Unlike previous results\nof Bhargav…
View article: Permutations that Destroy Arithmetic Progressions in Elementary $p$-Groups
Permutations that Destroy Arithmetic Progressions in Elementary $p$-Groups Open
Given an abelian group $G$, it is natural to ask whether there exists a permutation $\pi$ of $G$ that "destroys" all nontrivial 3-term arithmetic progressions (APs), in the sense that $\pi(b) - \pi(a) \neq \pi(c) - \pi(b)$ for every ordere…
View article: Permutations that Destroy Arithmetic Progressions in Elementary\n $p$-Groups
Permutations that Destroy Arithmetic Progressions in Elementary\n $p$-Groups Open
Given an abelian group $G$, it is natural to ask whether there exists a\npermutation $\\pi$ of $G$ that "destroys" all nontrivial 3-term arithmetic\nprogressions (APs), in the sense that $\\pi(b) - \\pi(a) \\neq \\pi(c) - \\pi(b)$\nfor eve…
View article: Configurations of Extremal Type II Codes
Configurations of Extremal Type II Codes Open
We prove configuration results for extremal Type II codes, analogous to the configuration results of Ozeki and of the second author for extremal Type II lattices. Specifically, we show that for $n \in \{8, 24, 32, 48, 56, 72, 96\}$ every e…
View article: Genus $1$ fibrations on the supersingular $\mathrm{K}3$ surface in characteristic $2$ with Artin invariant $1$
Genus $1$ fibrations on the supersingular $\mathrm{K}3$ surface in characteristic $2$ with Artin invariant $1$ Open
The supersingular K3 surface X in characteristic 2 with Artin invariant 1 admits several genus 1 fibrations (elliptic and quasi-elliptic).We use a bijection between fibrations and definite even lattices of rank 20 and discriminant 4 to cla…