Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.05515

The main conjecture for a Vinogradov subsystem

Trevor D. Wooley
View original source ↗Read source document ↗Export metadata ↓

Abstract

When $k\ge 2$ and $s>0$, we establish that for each $\varepsilon>0$, one has \[ \int_{[0,1)^{k-1}}\biggl| \sum_{1\le x\le X}e(α_2x^2+\ldots +α_k x^k)\biggr|^{2s}\,{\rm d}\boldsymbol α\ll X^{s+\varepsilon}+X^{2s-(k^2+k-2)/2}, \] confirming the main conjecture for a new family of exponential sums beyond those of Vinogradov (translation-dilation invariant) type. Our methods are based on very recent work of the author employing the nested efficient congruencing method. Consequently, analogues of our results hold also in the setting of number fields and function fields.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Trevor D. Wooley. 2026-10-04. The main conjecture for a Vinogradov subsystem. https://arxiv.org/abs/2610.05515

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the average $2$-torsion in class groups and narrow class groups of cubic orders with prescribed shape

We study the distribution of $2$-torsion in class groups and narrow class groups of cubic fields and cubic orders subject to prescribed shape conditions. The \emph{shape} of a cubic order in a number field is a natural geometric invariant taking values in the modular surface $\mathbb{H}/\operatorname{GL}_2(\mathbb{Z})$. Fix a subset $W$ of the modular surface with positive hyperbolic measure and boundary of measure zero. Refining the methods of Bhargava and Varma, we prove that among cubic fields with shape in $W$, the average size of the $2$-torsion subgroup of the class group is $5/4$ for totally real fields and $3/2$ for complex fields, while the average size of the $2$-torsion subgroup of the narrow class group for totally real cubic fields is $2$. We also obtain analogous results for cubic orders satisfying prescribed local conditions at all primes.

math.NT↗
arXiv2026

Kobayashi length bounds on bordered surfaces and generalized integral points on abelian varieties

Let $B$ be a compact Riemann surface and $B_0\subseteq B$ a hyperbolic surface obtained by removing finitely many disjoint closed disks. Fix a nontrivial loop $α$ in $B_0$. For $s\ge0$, let $L(α,s)$ denote the supremum, over all finite subsets $S\subset B_0$ with $\#S\le s$, of the minimal Kobayashi length of a loop in $B_0\smallsetminus S$ that is freely homotopic to $α$ in $B_0$. Phung in [14] proved that $L(α,s)$ grows at most linearly and at least as $\sqrt{s}/\log s$. We sharpen the upper bound to $O(\sqrt{s+1})$, which determines $\lim_{s\to\infty}\log L(α,s)/\log s=1/2$, answering [14, Question 1.4]. As an application, we improve Phung's counting bound for generalized integral points on abelian varieties over complex function fields. For an abelian variety of dimension $n$ over $\mathbb C(B)$ satisfying his geometric hypotheses, Phung proved a bound of order $(s+1)^{2nk}$ modulo the constant trace, where $k$ is the minimal number of generators of $π_1(B_0)$. We explain his method in detail and insert our stronger length estimate to obtain $(s+1)^{nk}$, halving the exponent. Counting in the Mordell--Weil lattice of rank $r$ gives the more precise bound $O((s+1)^{r/2})$ and a linear bound in $s+1$ for the canonical heights, with the family and $B_0$ fixed. For normal-crossings $D$ with Cartier model closure, this proves the fixed-model geometric Lang-Vojta conjecture in the stated abelian setting, without an exceptional set.

math.NT↗
arXiv2026

The Multidimensional Bateman Horn Conjecture: Effective Estimates for the Tail of the Singular Series and an Algorithm for Computation with Guaranteed Accuracy

We propose a method for computing the leading term of the asymptotics of the multidimensional Bateman Horn conjecture for systems of integer polynomials in several variables with controlled relative accuracy. The Bateman Horn conjecture is used as an explicit assumption about the growth rate of the number of prime values, the computational difficulty is concentrated in the singular series: an infinite product over all primes. For systems of general type, we obtain a universal estimate for the tail of the singular series in terms of a geometric constant depending on the Betti numbers of projective closures of hypersurfaces. For systems whose projective closures are smooth hypersurfaces (including inhomogeneous diagonal ones), using the exact Griffiths formula for Betti numbers, we obtain an improved constant in the tail estimate. The estimates are valid for all dimensions and degrees and do not require additional restrictions on the number of variables. We establish a connection between the truncation error of the singular series and the error in computing the number of prime values: under the Bateman Horn condition, the relative error in determining the singular constant coincides with the relative error in determining the number of prime values in the asymptotic regime. Based on the obtained estimates, we develop an algorithm for computing the leading term of the asymptotics with prescribed accuracy, classifying the system by its structure and producing a guaranteed interval for the number of prime values. Numerical experiments confirm the theory.

math.NT↗