arXiv subjects

Sa'ar Zehavi

Publications and source records attributed to Sa'ar Zehavi.

9 recordsLinked to original sources

arXiv2026

Arithmetic Wu Formulas and the Generalized Hecke Theorem

We construct canonical stable Steenrod squares on modified compactly supported étale cohomology of separated finite-type schemes over rings of $S$-integers in number fields with $2$ invertible, extending Feng's absolute Wu classes to arithmetic bases. For a flat projective morphism $f:X\to B$ of pure relative dimension, with $X$ regular and $B$ such a base, we prove $v_X=\operatorname{Sq}^{-1}(w_{\mathrm{et}}(τ_f+\mathcal O_X^{\oplus3}))$ in completed mod-$2$ étale cohomology. Here $τ_f$ is the virtual relative tangent bundle, $w_{\mathrm{et}}$ the total étale Stiefel--Whitney class, and $\operatorname{Sq}^{-1}$ the inverse total Steenrod square. Over finite fields of odd characteristic, the formula holds without the three trivial summands. The proof uses a modified compactly supported relative Wu formula extending Benoist's theorem. Our generalized Hecke theorem gives universal mod-$2$ relations involving Chern classes and the Kummer class of $-1$, governed by an arithmetic deformation of Hirzebruch's $2$-Todd series. These hold modulo an explicit archimedean ideal and become vanishing identities over finite fields, over totally imaginary arithmetic bases, or when $-1$ is a square on $X$. Nonempty real loci force infinitely many nonzero Wu components. Applications include Hecke's theorem on the different away from $2$, a finite-field analog of Atiyah's theorem on theta characteristics, and new higher-dimensional relations. We also revisit Serre's Riemann--Hurwitz theorem for spin bundles, recover the Shusterman--Sawin theorem for smooth branched covers of closed $3$-manifolds and prove a function-field analog of the Lusztig--Milnor--Peterson formula, expressing the difference between mod-$2$ and $2$-adic semicharacteristics using the middle Wu class and its Tate-twisted Bockstein.

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arXiv2026

The Unipotent Chabauty-Kim-Kantor Method for Relative Completions

We develop a new $p$-adic analytic method for studying integral points on hyperbolic curves, building on Kantor's relative-completion approach to unifying the Chabauty-Kim and Lawrence-Venkatesh methods. Its central mechanism converts dimension inequalities between algebraic global and local Galois cohomology stacks into nonzero rigid-analytic functions vanishing on the integral points within a residue disk. Assuming the relevant Bloch-Kato dimension formulas, we verify these inequalities for curves of genus at least $2$, as well as for certain modular curves, and thereby obtain a new conditional proof of the theorems of Faltings and Siegel. Our method produces genuinely new functions, different from those produced by Chabauty-Kim, pointing to an improved effective approach. Our key technical contributions include the following. We prove that Kantor's $p$-adic period map is analytic and algebraically Zariski dense in the relevant period domain. We also bypass the conjectural representability of Kantor's Selmer stacks by decomposing the integral-point locus into finitely many Kummer strata. On each stratum, the relative Kummer map factors through a neutral-fibre Selmer stack, which we prove is an algebraic Artin stack of finite type. We also develop a theory of finite-type motivic quotients of the relative completion: these quotients are cofinal among all finite-type quotients and recover the relative completion as their inverse limit. For Kodaira-Parshin families over curves of genus at least $2$, we construct a canonical adjoint-supported tower with fixed finite-dimensional abelianization and controlled graded pieces. At sufficiently deep levels, the Bloch-Kato formulas imply the required dimension inequality and produce genuinely relative rigid-analytic functions that do not arise from the ordinary unipotent completion.

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arXiv2026

Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultants

Given a finite set $S$ of distinct primes, we propose a method to construct polylogarithmic motivic Chabauty-Kim functions for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$ using resultants. For a prime $p\not\in S$, the vanishing loci of the images of such functions under the $p$-adic period map contain the solutions of the $S$-unit equation. In the case $\vert S\vert=2$, we explicitly construct a non-trivial motivic Chabauty-Kim function in depth 6 of degree 18, and prove that there do not exist any other Chabauty-Kim functions with smaller depth and degree. The method, inspired by work of Dan-Cohen and the first author, enhances the geometric step algorithm developed by Corwin and Dan-Cohen, providing a more efficient approach.

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arXiv2025

Modular Chabauty: Effective S-Integral Point Computation On Curves with Elliptic Fibrations

We present a practical, unconditional algorithm for determining the $S$-integral points on any elliptic moduli problem $\mathcal{Y}/\mathbb{Z}[1/S]$ -- that is, on any geometrically connected curve carrying a non-isotrivial elliptic fibration $\mathcal{E} \to \mathcal{Y}$. The associated map $Φ_M\colon \mathcal{Y} \to \mathcal{M}_{1,1}$ (the modular period map) plays the role ordinarily filled by a $p$-adic period map in Chabauty-type methods. Our Modular Chabauty method studies the image and fibres of $Φ_M$, and proceeds in two steps: an Effective Shafarevich step, in which we combine the modularity theorem with Cremona's enumeration of elliptic curves by conductor and list all rational elliptic curves with good reduction outside $S$; and a Fibre Computation step, in which we compute the $S$-integral points in the corresponding fibre of $Φ_M$. A Python/Sage implementation computes $\mathcal{Y}(\mathbb{Z}[1/S])$ for $\mathcal{Y}=\mathbb{P}^1\setminus\{0,1,\infty\}$ and for every modular curve $Y_1(N)$ with $4\le N\le 10$ or $N=12$, for all sets $S$ with $\prod_{p\in S} p^{2}\le 5\cdot 10^{5}$, within $3.5$ seconds on a standard computer.

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arXiv2022

Pointwise bounds for Eisenstein series on $Γ_0(q)\setminus SL_2(\mathbb{R})$

We construct pointwise bounds in the weight aspect for Eisenstein series on $X_0(q) = Γ_0(q)\setminus SL_2(\mathbb{R})$, with squarefree level $q$, using a Sobolev technique. More specifically, we show that for an Eisenstein series $E$ on $X_0(q)$ of weight parameter $n$ and type $t$, one has for all $x\in X_0(q)$: $|E(x,1/2 + it)| \ll_ε q^ε(1 + |n|^{1/2 + ε} + |t|^{1/2 + ε})\sqrt{y(x) + y(x)^{-1}}$, where $y(x)$ is the Iwasawa $y$-coordinate of the point $x$.

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arXiv2022

Sectorial equidistribution of the roots of $x^2 + 1$ modulo primes

The equation $x^2 + 1 = 0\mod p$ has solutions whenever $p = 2$ or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec from 1995. We show that a subsequence of the roots of the equation remains equidistributed even when one adds a restriction on the primes which has to do with the angle in the plane formed by their corresponding representation as a sum of squares. Similar to Duke, Friedlander and Iwaniec, we reduce the problem to the study of certain Poincare series, however, while their Poincare series were functions on an arithmetic quotient of the upper half plane, our Poincare series are functions on arithmetic quotients of $SL_2(\mathbb{R})$, as they have a nontrivial dependence on their Iwasawa $θ$-coordinate. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, which posed a few new challenges. A couple of notable ones were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula.

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arXiv2020

On the Joint Distribution of the Roots of Pairs of Polynomial Congruences

Let f(x) be a primitive irreducible polynomial with integer coefficients of degree greater than one. In 1964, Hooley showed that the sequence of normalized roots u/n, where f(u) = 0(n), ordered in the obvious way, is uniformly distributed modulo one. It is the goal of this paper to show that if f(x) and g(x) are a pair of primitive irreducible polynomials of degree greater than one, not necessarily distinct, then the sequence (u/n,v/n), with f(u) = 0(n) and g(v) = 0(n), ordered in the obvious way, is uniformly distributed modulo one in the unit torus.

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arXiv2019

On Cilleruelo's conjecture for the least common multiple of polynomial sequences

A conjecture due to Cilleruelo states that for an irreducible polynomial $f$ with integer coefficients of degree $d\geq 2$, the least common multiple $L_f(N)$ of the sequence $f(1), f(2), \dots, f(N)$ has asymptotic growth $\log L_f(N)\sim (d-1)N\log N$ as $N\to \infty$. We establish a version of this conjecture for almost all shifts of a fixed polynomial, the range of $N$ depending on the range of shifts.

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arXiv2017

Not Conway's 99-Graph Problem

Conway's 99-graph problem is the second problem amongst the five 1000\$ 2017 open problems set. Four out of the five remain unsolved to this day, including the 99-graph problem. In this paper we quote Conway's definition of the problem and give an alternative interpretation of it, which we humorously name "not Conway's 99-graph problem". We solve the alternative interpretation completely.

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