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Noam D. Elkies

Publications and source records attributed to Noam D. Elkies.

At least 19 recordsLinked to original sources
arXiv2026

An elliptic K3 surface X/Q(t) with Mordell-Weil rank 17, I: Formulas for X and base changes of rank 18 and 19

In [Elkies 2006, Elkies 2007] we announced a K3 surface X/Q with Néron--Severi group NS(X) = NS_Q(X) of rank 19 and an elliptic fibration of Mordell--Weil rank 17. This is the largest possible Mordell--Weil rank over Q(t) for an elliptic K3 surface. One of the fibers of this surface is the elliptic curve of rank at least 28 that was the elliptic curve of highest rank known during the years 2006--2024. We further announced that there are quadratic base changes of this fibration to elliptic surfaces of Mordell--Weil rank 18 over Q(t), and that there are pairs of such quadratic base changes whose compositum is an elliptic surface of Mordell--Weil rank 19 over an elliptic curve E_0/Q with #(E_0(Q)) = \infty, whence there are infinitely many elliptic curves of rank at least 19 over Q. We exhibit one such pair. A subsequent paper will show (using a technique also announced in [Elkies 2007] how we computed X and its fibration.

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arXiv2026

The constructive inverse Galois problem via Hilbert modular forms: realizing the transitive group 17T7

We show how Hilbert modular forms can be used in the constructive inverse Galois problem over the rationals. In particular, we prove that the transitive permutation group 17T7, isomorphic to a split extension of C_2 by PSL_2(FF_16), is a Galois group over the rationals and exhibit an explicit degree 17 polynomial with this Galois group. The group arises from the field of definition of the 2-torsion on an abelian fourfold with real multiplication defined over a real quadratic field; we find such a fourfold attached to a Hilbert modular form. Building upon work of Dembele, we describe a method for reconstructing a period matrix attached to a Hilbert modular form, and we use it to construct the 2-isogeny polynomial. We also rigorously identify the relevant fourfold as the Jacobian of a genus 4 Shimura curve and compute explicit equations for this curve.

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arXiv2025

Fast evaluation of Riemann theta functions in any dimension

We describe an algorithm to numerically evaluate Riemann theta functions in any dimension in quasi-linear time in terms of the required precision, uniformly on reduced input. This algorithm is implemented in the FLINT number theory library and vastly outperforms existing software. As an application, we evaluate the theta constants attached to certain special abelian varieties of dimension 6 to construct explicit polynomials of degree 65 over $\mathbb{Q}$ with conjectural Galois group $\mathrm{SL}_2(\mathbb{F}_{64})$.

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arXiv2022

Equations for a K3 Lehmer map

McMullen proved that there exists an automorphism of minimal topological entropy on a projective K3 surface. We derive equations for the surface and its automorphism. We reconstruct the surface and its automorphism from the Hodge theoretic model provided by McMullen. The approach is computer aided and relies on finite non-symplectic automorphisms, $p$-adic lifting, elliptic fibrations and the Kneser neighbor method for integer lattices.

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arXiv2019

Periodic continued fractions over $S$-integers in number fields and Skolem's $p$-adic method

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 ${\mathcal O}[x]$. We show that PCFs $P=[b_1,\ldots,b_N,\bar{a_1\ldots ,a_k}]$ of type $(N,k)$ potentially converging to a limit in ${\mathcal B}$ are given by ${\mathcal O}$-points on an affine variety $V:=V({\mathcal B})_{N,k}$ generically of dimension $N+k-2$. We give the equations of $V$ in terms of the continuant polynomials of Wallis and Euler. The integral points $V({\mathcal O})$ are related to writing matrices in $\textrm{SL}_2({\mathcal O})$ as products of elementary matrices. We give an algorithm to determine if a PCF converges and, if so, to compute its limit. Our standard example generalizes the PCF $\sqrt{2}=[1,\bar{2}]$ to the ${\mathbf Z}_2$-extension of ${\mathbf Q}$: $F_n={\mathbf Q}(α_n)$, $α_{n}:=2\cos(2π/2^{n+2})$, with integers ${\mathcal O}_n={\mathbf Z}[α_n]$. We want to find the PCFs of $α_{n+1}$ over ${\mathcal O}_{n}$ of type $(N,k)$ by finding the ${\mathcal O}_{n}$-points on $V({\mathcal B}_{n+1})_{N,k}$ for ${\mathcal B}_{n+1}:=\{α_{n+1}, -α_{n+1}\}$. There are three types $(N,k)=(0,3), (1,2), (2,1)$ such that the associated PCF variety $V({\mathcal B})_{N,k}$ is a curve; we analyze these curves. For generic ${\mathcal B}$, Siegel's theorem implies that each of these three $V({\mathcal B})_{N,k}({\mathcal O})$ is finite. We find all the ${\mathcal O}_n$-points on these PCF curves $V({\mathcal B}_{n+1})_{N,k}$ for $n=0,1$. When $n=1$ we make extensive use of Skolem's $p$-adic method for $p=2$, including its application to Ljunggren's equation $x^2 + 1 =2y^4$.

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arXiv2017

Permutations that Destroy Arithmetic Progressions in Elementary $p$-Groups

Given an abelian group $G$, it is natural to ask whether there exists a permutation $π$ of $G$ that "destroys" all nontrivial 3-term arithmetic progressions (APs), in the sense that $π(b) - π(a) \neq π(c) - π(b)$ for every ordered triple $(a,b,c) \in G^3$ satisfying $b-a = c-b \neq 0$. This question was resolved for infinite groups $G$ by Hegarty, who showed that there exists an AP-destroying permutation of $G$ if and only if $G/Ω_2(G)$ has the same cardinality as $G$, where $Ω_2(G)$ denotes the subgroup of all elements in $G$ whose order divides $2$. In the case when $G$ is finite, however, only partial results have been obtained thus far. Hegarty has conjectured that an AP-destroying permutation of $G$ exists if $G = \mathbb{Z}/n\mathbb{Z}$ for all $n \neq 2,3,5,7$, and together with Martinsson, he has proven the conjecture for all $n > 1.4 \times 10^{14}$. In this paper, we show that if $p$ is a prime and $k$ is a positive integer, then there is an AP-destroying permutation of the elementary $p$-group $(\mathbb{Z}/p\mathbb{Z})^k$ if and only if $p$ is odd and $(p,k) \not\in \{(3,1),(5,1), (7,1)\}$.

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arXiv2015

Configurations of Extremal Type II Codes

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 extremal Type II code of length $n$ is generated by its codewords of minimal weight. Where Ozeki and Kominers used spherical harmonics and weighted theta functions, we use discrete harmonic polynomials and harmonic weight enumerators. Along we way we introduce "$t\frac12$-designs" as a discrete analog of Venkov's spherical designs of the same name.

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arXiv2014

Genus 1 fibrations on the supersingular K3 surface in characteristic 2 with Artin invariant 1

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 classify the fibrations, and exhibit isomorphisms between the resulting models of X. We also study a configuration of (-2)-curves on X related to the incidence graph of points and lines of IP^2(IF_4).

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arXiv2013

Modular forms and K3 surfaces

For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over QQ associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author.

math.AG↗
arXiv2013

On real part theorem for the higher derivatives of analytic functions in the unit disk

Let $n$ be a positive integer. Let $\mathbf U$ be the unit disk, $p\ge 1$ and let $h^p(\mathbf U)$ be the Hardy space of harmonic functions. Kresin and Maz'ya in a recent paper found the representation for the function $H_{n,p}(z)$ in the inequality $$|f^{(n)} (z)|\leq H_{n,p}(z)|\Re(f-\mathcal P_l)|_{h^p(\mathbf U)}, \Re f\in h^p(\mathbf U), z\in \mathbf U,$$ where $\mathcal P_l$ is a polynomial of degree $l\le n-1$. We find or represent the sharp constant $C_{p,n}$ in the inequality $H_{n,p}(z)\le \frac{C_{p,n}}{(1-|z|^2)^{1/p+n}}$. This extends a recent result of the second author and Marković, where it was considered the case $n=1$ only. As a corollary, an inequality for the modulus of the $n-{th}$ derivative of an analytic function defined in a complex domain with the bounded real part is obtained. This result improves some recent result of Kresin and Maz'ya.

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arXiv2012

Point configurations that are asymmetric yet balanced

A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.

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arXiv2011

Weighted Generating Functions for Type II Lattices and Codes

We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of Euclidean lattices. Namely, we use the finite-dimensional representation theory of sl_2 to derive a decomposition theorem for the spaces of discrete homogeneous polynomials in terms of the spaces of discrete harmonic polynomials, and prove a generalized MacWilliams identity for harmonic weight enumerators. We then present several applications of harmonic weight enumerators, corresponding to some uses of weighted theta functions: an equivalent characterization of t-designs, the Assmus-Mattson Theorem in the case of extremal Type II codes, and configuration results for extremal Type II codes of lengths 8, 24, 32, 48, 56, 72, and 96.

math.NT↗
arXiv2011

Minimal S-universality criteria may vary in size

In this note, we give simple examples of sets S of quadratic forms that have minimal S-universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh in the negative.

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arXiv2010

Every even number greater than 454 is the sum of seven cubes

It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is congruent to 2 mod 4. This result, together with a recent proof for 4|N, shows that the conjecture is true for all even N.

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