On Manin's conjecture for Châtelet surfaces
We resolve Manin's conjecture for all Châtelet surfaces over $\mathbb{Q}$.
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arXiv subjects
Katharine Woo
Publications and source records attributed to Katharine Woo.
Genuine and strongly genuine polynomials: With an application to the persistence of Galois groups under specialization
We develop the theory of strongly $n$-genuine polynomials $F(Y,X_1,\ldots,X_n)$, which have the property that the number of specializations $F(Y,X_1,\mathbf{x}')$ with $\mathbf{x}'=(x_2,\ldots,x_n) \in \mathbb{Z}^{n-1}$ (respectively $\mathbf{x}' \in \mathbb{F}_p^{n-1}$) such that $F(Y,X_1,\mathbf{x}')$ is reducible over $\overline{\mathbb{Q}}$ (respectively over $\overline{\mathbb{F}}_p$) can be well-controlled quantitatively. We also develop the theory of a larger class of $n$-genuine polynomials $F(Y,X_1,\ldots,X_n)$, which have the property that the number of specializations $F(Y,X_1,\mathbf{x}')$ with $\mathbf{x}' \in \mathbb{Z}^{n-1}$ (respectively $\mathbf{x}' \in \mathbb{F}_p^{n-1}$) such that $F(Y,X_1,\mathbf{x}')$ splits completely over $\overline{\mathbb{Q}}$ (respectively over $\overline{\mathbb{F}}_p$) into factors that are linear in $Y$ can be well-controlled quantitatively. For each of these classes, we prove that there are four equivalent characterizations. As an application, we demonstrate that $n$-genuine and strongly $n$-genuine polynomials can be used to prove, for any polynomial $F(Y,X_1,\ldots,X_n)$, an upper bound for the number of specializations $F(Y,\mathbf{x})$ with $\mathbf{x}=(x_1,\ldots,x_n) \in \mathbb{Z}^n$ such that the Galois group of the splitting field of $F(Y,\mathbf{x})$ over $\mathbb{Q}$ is not isomorphic to the Galois group of the splitting field of $F(Y,X_1,\ldots,X_n)$ over $\mathbb{Q}(X_1,\ldots,X_n)$. We simultaneously prove analogous results over any number field.
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Stratification theorems for exponential sums in families
We survey some of the stratification theorems concerning exponential sums over finite fields, especially those due to Katz-Laumon and Fouvry-Katz, as well as some of their applications. Moreover, motivated partly by recent work of Bonolis, Pierce and Woo (arXiv:2505.11226), we prove that these stratification statements admit uniform variants in families, both algebraically and analytically. The paper includes an Appendix by Forey, Fresán and Kowalski (excerpted from arXiv:2109.11961), which provides an elementary intuitive introduction to trace functions in more than one variable over finite fields.
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Counting points on a family of degree one del Pezzo surfaces
We study rational points on the elliptic surface given by the equation: $$y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3,$$ where $A,B\in \mathbb{Z}$ satisfy that $4A^3-27B^2\neq 0$ and $Q(u,v)$ is a positive-definite quadratic form. We prove asymptotics for a special subset of the rational points, specifically those that are integral with respect to the singularity. This method utilizes Mordell's parameterization of integral points on quadratic twists on elliptic curves, which is based on a syzygy for invariants of binary quartic forms. Let $F(A,B)$ denote the set of binary quartic forms with invariants $-4A$ and $-4B$ under the action of $\textrm{SL}_2(\mathbb{Z})$. We reduce the point-counting problem to the question of determining an asymptotic formula for the correlation sums of representation numbers of binary quadratic and binary quartic forms, where the quartic forms range in $F(A,B)$. These sums are then treated using a connection to modular forms.
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Sums of Hecke eigenvalues along polynomial sequences and base change for $\text{GL}(2)$
We study sums of absolute values of Hecke eigenvalues of $\textrm{GL}(2)$ representations that are tempered at all finite places. We show that these sums exhibit logarithmic savings over the trivial bound if and only if the representation is cuspidal. Further, we connect the problem of studying the sums of Hecke eigenvalues along polynomial values to the base change problem for $\textrm{GL}(2).$
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Counting points in thin sets: A survey
In the 1980's Serre asked how many points of bounded height can lie in a thin set. This has motivated significant research ever since, culminating in a series of recent breakthroughs. It is a good time to take stock of the central questions that have been resolved, and also to highlight remaining open questions. First, we survey recent progress on counting points of bounded height in the four types of thin sets, according to the projective/affine and type I/type II designations. Second, we turn to questions of uniformity. Famously, in the setting of type I thin sets, the best-known upper bound for the number of points of bounded height is independent of the maximum size, say $\|F\|$, of the coefficients of the polynomials that define the thin set; such an upper bound is called uniform. A uniform upper bound in the setting of type II thin sets is not known. For type II thin sets, we explore the dependence on $\|F\|$ via several strategies, and construct counterexamples that suggest the question of uniformity is quite subtle in the setting of type II thin sets.
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The distribution of prime values of random polynomials
The Bateman--Horn Conjecture predicts how often an irreducible polynomial $f(x) \in \mathbb{Z}[x]$ assumes prime values. We demonstrate that with sufficient averaging in the coefficients of $f$ (viz. exponential in the size of the inputs), one can not only prove Bateman--Horn results on average but also pin down precise information about the distribution of prime values. We show that 100\% of polynomials (in an $L^k$ sense for all $k \in \mathbb{N}$) satisfy the Bateman--Horn Conjecture, and that that 100\% of polynomials (in an $L^2$ sense) satisfy an appropriate polynomial analogue of the Hardy--Littlewood Prime Tuples Conjecture. We use the latter to prove that 100\% of polynomials satisfy the appropriate analogue of the Poisson Tail Conjecture, in the sense that the distribution of the gaps between consecutive prime values around the average spacing is Poisson. We also study the frequencies of sign patterns of the Liouville function evaluated at the consecutive outputs of $f$; viewing $f$ as a random variable, we establish the limiting distribution for every sign pattern. The Chowla problem along random polynomials is a special case. A key input behind all of our arguments is Leng's recent quantitative work on the higher-order Fourier uniformity of the von Mangoldt and Möbius functions (in turn relying on Leng, Sah, and Sawhney's quantitative inverse theorem for the Gowers norms).
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Counting integral points in thin sets of type II: singularities, sieves, and stratification
Consider an absolutely irreducible polynomial $F(Y,X_1,\ldots,X_n) \in \mathbb{Z}[Y,X_1,\ldots,X_n]$ that is monic in $Y$ and is a polynomial in $Y^m$ for an integer $m \geq 1$. Let $N(F,B)$ count the number of $\mathbf{x} \in [-B,B]^n \cap \mathbb{Z}^n$ such that $F(y,\mathbf{x})=0$ is solvable for $y \in\mathbb{Z}$. In nomenclature of Serre, bounding $N(F,B)$ corresponds to counting integral points in an affine thin set of type II. Previously, in this generality Serre proved $N(F,B) \ll_F B^{n-1/2}(\log B)^γ$ for some $γ<1$. When $m \geq 2$, this new work proves $N(F,B) \ll_{n,F,ε} B^{n-1+1/(n+1) + ε}$ under a nondegeneracy condition that encapsulates that $F(Y,\mathbf{X})$ is truly a polynomial in $n+1$ variables, even after performing any $\text{GL}_n(\mathbb{Q})$ change of variables on $X_1,\ldots,X_n$. Under GRH, this result also holds when $m=1$. We show that generic polynomials satisfy the relevant nondegeneracy condition. Moreover, for a certain class of polynomials, we prove the stronger bound $N(F,B) \ll_{F} B^{n-1}(\log B)^{e(n)}$, comparable to a conjecture of Serre. A key strength of these results is that they require no nonsingularity property of $F(Y,\mathbf{X})$. The Katz-Laumon stratification for character sums, in a new uniform formulation appearing in a companion paper of Bonolis, Kowalski and Woo, is a key ingredient in the sieve method we develop to prove upper bounds that explicitly control any dependence on the size of the coefficients of $F$.
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Integer-valued o-minimal functions
We study $\mathbb{R}_{\textrm{an},\exp}$-definable functions $f:\mathbb{R}\to \mathbb{R}$ that take integer values at all sufficiently large positive integers. If $|f(x)|= O\big(2^{(1+10^{-5})x}\big)$, then we find polynomials $P_1, P_2$ such that $f(x)=P_1(x)+P_2(x)2^x$ for all sufficiently large $x$. Our result parallels classical theorems of Pólya and Selberg for entire functions and generalizes Wilkie's classification for the case of $|f(x)|= O(C^x)$, for some $C<2$. Let $k\in \mathbb{N}$ and $γ_k=\sum_{j=1}^{k} 1/j$. Extending Wilkie's theorem in a separate direction, we show that if $f$ is $k$-$\textit{concordant}$ and $|f(x)|= O(C^{x})$, for some $C<e^{γ_k}+1$, then $f$ must eventually be given by a polynomial. This is an analog of a result by Pila for entire functions.
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Prime Number Theorems for Polynomials from Homogeneous Dynamics
We establish a new class of examples of the multivariate Bateman-Horn conjecture by using tools from dynamics. These cases include the determinant polynomial on the space of $n\times n$ matrices, the Pfaffian on the space of skew-symmetric $2n\times 2n$ matrices, and the determinant polynomial on the space of symmetric $n\times n$ matrices. In particular, let $(V,F)$ be any pair among the following: $(\textrm{Mat}_n, \det)$, $(\textrm{Skew}_{2n},\textrm{Pff})$, and $(\textrm{Sym}_n, \det).$ We then obtain an asymptotic for $$π_{V,F}(T)= \#\{v\in V: \max(|v_i|)\leq T, F(v) \text{ is prime}\},$$ that matches the Bateman-Horn prediction. The key ingredients of our proof are an asymptotic count for integral points on the level sets of $F$ given by Linnik equidistribution, a geometric approximation of the box by cones, and an upper bound sieve to bound the number of prime values missed by the approximation. In the case of the determinant polynomial on symmetric matrices, we must also use the Siegel mass formula to compute the product of local densities for the main term.
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Small scale distribution of linear patterns of primes
Let $Ψ$ be a system of linear forms with finite complexity. In their seminal paper, Green and Tao showed the following prime number theorem for values of the system $Ψ$: $$\sum_{x\in [-N,N]^d} \prod_{i=1}^t \mathbf{1}_{\mathcal{P}}(ψ_i(x)) \sim \frac{(2N)^d}{(\log N)^t} \prod_{p} β_p,$$ where $β_p$ are the corresponding local densities. In this paper, we demonstrate limits to equidistribution of these primes on small scales; we show the analog to Maier's result on primes in short intervals. In particular, we show that for all $λ> 1$, there exist $δ_λ^\pm > 0$ such that for $N$ sufficiently large, there exist boxes $B^\pm\subset [-N, N]^d$ of sidelengths at least $(\log N)^λ$ such that $$\sum_{x\in B^+} \prod_{i=1}^t \mathbf{1}_{\mathcal{P}}(ψ_i(x)) > (1+δ^+) \frac{\mathrm{vol}(B^+)}{(\log N)^t} \prod_{p}β_p,$$ $$\sum_{x\in B^-} \prod_{i=1}^t \mathbf{1}_{\mathcal{P}}(ψ_i(x)) < (1-δ^-) \frac{\mathrm{vol}(B^-)}{(\log N)^t} \prod_{p}β_p.$$
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On the zeros of a class of modular functions
We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose $q$-expansions satisfy \[ f_k(A, τ) \colon = q^{-k}(1+a(1)q+a(2)q^2+...) + O(q),\] where $a(n)$ are numbers satisfying a certain analytic condition. We show that the zeros of such $f_k(τ)$ in the fundamental domain of $SL_2(\mathbb{Z})$ lie on $|τ|=1$ and are transcendental. We recover as a special case earlier work of Witten on extremal "partition" functions $Z_k(τ)$. These functions were originally conceived as possible generalizations of constructions in three-dimensional quantum gravity.
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Formulas for Chebotarev densities of Galois extensions of number fields
We generalize the Chebotarev density formulas of Dawsey (2017) and Alladi (1977) to the setting of arbitrary finite Galois extensions of number fields $L/K$. In particular, if $C \subset G = \textrm{Gal}(L/K)$ is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of $K$: \[ -\lim_{X\rightarrow\infty} \sum_{\substack{2\leq N(I)\leq X \\ I \in S(L/K; C)}} \frac{μ_K(I)}{N(I)} = \frac{|C|}{|G|}, \] where $μ_K(I)$ denotes the generalized Möbius function and $S(L/K;C)$ is the set of ideals $I\subset \mathcal{O}_K$ such that $I$ has a unique prime divisor $\mathfrak{p}$ of minimal norm and the Artin symbol $\left[\frac{L/K}{\mathfrak{p}}\right]$ is $C$. To obtain this formula, we generalize several results from classical analytic number theory, as well as Alladi's concept of duality for minimal and maximal prime divisors, to the setting of ideals in number fields.
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Generating functions for power moments of elliptic curves over $\mathbb{F}_p$
Seminal works by Birch and Ihara gave formulas for the $m$th power moments of the traces of Frobenius endomorphisms of elliptic curves over $\mathbb{F}_{p}$ for primes $p \geq 5$. Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a fixed finite abelian group $A$. We revisit these formulas and determine a simple expression for the zeta function $Z_p(A; t)$, the generating function for these $m$th power moments. In particular, we find that \[ Z_p(A;t) = \frac{\widehat{Z}_p(A; t)}{\displaystyle \prod_{a \in \textrm{Frob}_p(A)}(1 - at)},\] where $\textrm{Frob}_p(A) := \{ a \, \colon -2\sqrt{p} \leq a \leq 2\sqrt{p}\, \text{ and } a \equiv p+1 \pmod{|A|}\}$, and $\widehat{Z}_p(A;t)$ is an easily computed polynomial that is determined by the first $\Big\lceil\frac{2\lfloor 2\sqrt{p}\rfloor}{|A|}\Big\rceil$ power moments. These rational zeta functions have two natural applications. We find rational generating functions in weight aspect for traces of Hecke operators on $S_k(Γ)$ for various congruence subgroups $Γ$. We also prove congruence relations for power moments by making use of known congruences for traces of Hecke operators.
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