Simultaneous Kummer congruences and E∞\mathbb{E}_\infty-orientations of KO and tmf

Building on results of M. Ando, M.J. Hopkins and C. Rezk, we show the existence of uncountably many $\mathbb{E}_\infty$-String orientations of real K-theory KO and of topological modular forms tmf, generalizing the $\hat{A}$- (resp. the Witten) genus. Furthermore, the obstruction to lifting an $\mathbb{E}_\infty$-String orientations from KO to tmf is identified with a classical Iwasawa-theoretic condition. The common key to all these results is a precise understanding of the classical Kummer congruences, imposed for all primes simultaneously. This result is of independent arithmetic interest.Comment: final versio

Five squares in arithmetic progression over quadratic fields

We give several criteria to show over which quadratic number fields Q(sqrt{D}) there should exists a non-constant arithmetic progressions of five squares. This is done by translating the problem to determining when some genus five curves C_D defined over Q have rational points, and then using a Mordell-Weil sieve argument among others. Using a elliptic Chabauty-like method, we prove that the only non-constant arithmetic progressions of five squares over Q(sqrt{409}), up to equivalence, is 7^2, 13^2, 17^2, 409, 23^2. Furthermore, we give an algorithm that allow to construct all the non-constant arithmetic progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem.Comment: To appear in Revista Matem\'atica Iberoamerican

Primes represented by incomplete norm forms

Let $K=\mathbb{Q}(\omega)$ with $\omega$ the root of a degree $n$ monic irreducible polynomial $f\in\mathbb{Z}[X]$. We show the degree $n$ polynomial $N(\sum_{i=1}^{n-k}x_i\omega^{i-1})$ in $n-k$ variables formed by setting the final $k$ coefficients to 0 takes the expected asymptotic number of prime values if $n\ge 4k$. In the special case $K=\mathbb{Q}(\sqrt[n]{\theta})$, we show $N(\sum_{i=1}^{n-k}x_i\sqrt[n]{\theta^{i-1}})$ takes infinitely many prime values provided $n\ge 22k/7$. Our proof relies on using suitable `Type I' and `Type II' estimates in Harman's sieve, which are established in a similar overall manner to the previous work of Friedlander and Iwaniec on prime values of $X^2+Y^4$ and of Heath-Brown on $X^3+2Y^3$. Our proof ultimately relies on employing explicit elementary estimates from the geometry of numbers and algebraic geometry to control the number of highly skewed lattices appearing in our final estimates.Comment: 103 pages; v2 is significant rewrite of v1, main results unchange

Computing in Algebraic Closures of Finite Fields

We present algorithms to construct and perform computations in algebraic closures of finite fields. Inspired by algorithms for constructing irreducible polynomials, our approach for constructing closures consists of two phases; First, extension towers of prime power degree are built, and then they are glued together using composita techniques. To be able to move elements around in the closure we give efficient algorithms for computing isomorphisms and embeddings. In most cases, our algorithms which are based on polynomial arithmetic, rather than linear algebra, have quasi-linear complexity

Globally nilpotent differential operators and the square Ising model

We recall various multiple integrals related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their lambda-extensions. These integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations. These differential operators are very selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, and even a remarkable weight-1 modular form emerging in the three-particle contribution $\chi^{(3)}$ of the magnetic susceptibility of the square Ising model. In the case where we do not have G-functions, but Hamburger functions (one irregular singularity at 0 or $\infty$) that correspond to the confluence of singularities in the scaling limit, the p-curvature is also found to verify new structures associated with simple deformations of the nilpotent property.Comment: 55 page

The Explicit Sato-Tate Conjecture and Densities Pertaining to Lehmer-Type Questions

Let $f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (\Gamma_0(N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$, define $\theta_p\in[0,\pi]$ to be the angle for which $a(p)=2p^{( k -1)/2}\cos \theta_p$. Let $I\subset[0,\pi]$ be a closed subinterval, and let $d\mu_{ST}=\frac{2}{\pi}\sin^2\theta d\theta$ be the Sato-Tate measure of $I$. Assuming that the symmetric power $L$-functions of $f$ satisfy certain analytic properties (all of which follow from Langlands functoriality and the Generalized Riemann Hypothesis), we prove that if $x$ is sufficiently large, then $$\left|\#\{p\leq x:\theta_p\in I\} -\mu_{ST}(I)\int_2^x\frac{dt}{\log t}\right|\ll\frac{x^{3/4}\log(N k x)}{\log x}$$ with an implied constant of $3.34$. By letting $I$ be a short interval centered at $\frac{\pi}{2}$ and counting the primes using a smooth cutoff, we compute a lower bound for the density of positive integers $n$ for which $a(n)\neq0$. In particular, if $\tau$ is the Ramanujan tau function, then under the aforementioned hypotheses, we prove that $$\lim_{x\to\infty}\frac{\#\{n\leq x:\tau(n)\neq0\}}{x}>1-1.54\times10^{-13}.$$ We also discuss the connection between the density of positive integers $n$ for which $a(n)\neq0$ and the number of representations of $n$ by certain positive-definite, integer-valued quadratic forms.Comment: 29 pages. Significant revisions, including improvements in Theorems 1.2, 1.3, and 1.5 and a more detailed account of the contour integration, are included. Acknowledgements are update

Lower Bounds for Straight Line Factoring

Straight line factoring algorithms include a variant Lenstra\u27s elliptic curve method. This note proves lower bounds on the length of straight line factoring algorithms

An Invitation to Formal Power Series

This is an account on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able to prove Newton’s binomial theorem, Jacobi’s triple product, the Rogers–Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan’s partition congruences, generating functions of Stirling numbers and Jacobi’s four-square theorem. We further discuss formal Laurent series and multivariate power series and end with a proof of MacMahon’s master theorem

Algorithmic Views of Vectorized Polynomial Multipliers – NTRU Prime

In this paper, we explore the cost of vectorization for polynomial multiplication with coefficients in $\mathbb{Z}_q$ for an odd prime $q$. If there is a large power of two dividing $q−1$, we can apply radix-2 Cooley–Tukey fast Fourier transforms to multiply polynomials in $\mathbb{Z}_q[x]$. The radix-2 nature admits efficient vectorization. Conversely, if 2 is the only power of two dividing $q−1$, we can apply Schönhage’s and Nussbaumer’s FFTs to craft radix-2 roots of unity, but these double the number of coefficients. We show how to avoid this doubling while maintaining vectorization friendliness with Good–Thomas, Rader’s, and Bruun’s FFTs. In particular, we exploit the existing Fermat-prime factor of $q − 1$ for Rader’s FFT and the power-of-two factor of $q + 1$ for Bruun’s FFT. We implement these ideas for the NTRU Prime instances ntrulpr761/sntrup761, operating over the coefficient ring $\mathbb{Z}_{4591}$ on a Cortex-A72. sntrup761 is currently used in OpenSSH 9.0 by default. Our polynomial multiplication outperforms the state-of-the-art vector-optimized implementation by 6.1×. For ntrulpr761, our keygen, encap, and decap are 2.98×, 2.79×, and 3.07× faster than the state-of-the-art vector-optimized implementation. For sntrup761, we outperform the reference implementation significantly