arXiv subjects
Publications and source records attributed to Will Sawin.
We prove upper and lower bounds on the number of pairs of commuting $n\times n$ matrices with integer entries in $[-T,T]$, as $T\to \infty$. Our work uses Fourier analysis and leads us to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket.
We determine the largest subset $A\subseteq \{1,\dotsc,n\}$ such that for all $a,b\in A$, the product $ab$ is not squarefree. Specifically, the maximum size is achieved by the complement of the odd squarefree numbers. This resolves a problem of Paul Erdős and András Sárközy from 1992.
We propose a refinement of the random matrix model for a certain family of $L$-functions over $\mathbb F_q[u]$, using techniques that we hope will eventually apply to an arbitrary family of $L$-functions. This consists of a probability distribution on power series in $q^{-s}$ which combines properties of the characteristic polynomials of Haar-random unitary matrices and random Euler products over $\mathbb F_q[u]$. The support of our distribution is contained in the intersection of the supports of the two original distributions. The expectations of low-degree polynomials in the coefficients of our series approximate the expectations of the same polynomials in the coefficients of random Euler products, while the expectations of high-degree polynomials approximate the expectations of the same polynomials in the coefficients of the characteristic polynomials of random matrices. Furthermore, the expectations of absolute powers of our series approximate the Conrey-Farmer-Keating-Rubinstein-Snaith/Andrade-Keating prediction for the moments of our family of $L$-functions.
The moment problem in probability theory asks for criteria for when there exists a unique measure with a given tuple of moments. We study a variant of this problem for random objects in a category, where a moment is given by the average number of epimorphisms to a fixed object. When the moments do not grow too fast, we give a necessary and sufficient condition for existence of a distribution with those moments, show that a unique such measure exists, give formulas for the measure in terms of the moments, and prove that measures with those limiting moments approach that particular measure. Our result applies to categories satisfying some finiteness conditions and a condition that gives an analog of the second isomorphism theorem, including the categories of finite groups, finite modules, finite rings, as well as many variations of these categories. This work is motivated by the non-abelian Cohen-Lenstra-Martinet program in number theory, which aims to calculate the distribution of random profinite groups arising as Galois groups of maximal unramified extensions of random number fields.
For $G$ and $H_1,\dots, H_n$ finite groups, does there exist a $3$-manifold group with $G$ as a quotient but no $H_i$ as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with topological results generalizing the theory of semicharacteristics. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard splittings as the genus goes to infinity. We believe this is the first construction of a new distribution of random groups from its moments.
We give an axiomatic characterization of multiple Dirichlet series over the function field $\mathbb F_q(T)$, generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature.
For partially ordered sets $X$ we consider the square matrices $M^{X}$ with rows and columns indexed by linear extensions of the partial order on $X$. Each entry $\left( M^{X}\right)_{PQ}$ is a formal variable defined by a pedestal of the linear order $Q$ with respect to linear order $P$. We show that all the eigenvalues of any such matrix $M^{X}$ are $\mathbb{Z}$-linear combinations of those variables.
Cohen, Lenstra, and Martinet have given conjectures for the distribution of class groups of extensions of number fields, but Achter and Malle have given theoretical and numerical evidence that these conjectures are wrong regarding the Sylow $p$-subgroups of the class group when the base number field contains $p$th roots of unity. We give complete conjectures of the distribution of Sylow $p$-subgroups of class groups of extensions of a number field when $p$ does not divide the degree of the Galois closure of the extension. These conjectures are based on $q\rightarrow\infty$ theorems on these distributions in the function field analog and use recent work of the authors on explicitly giving a distribution of modules from its moments. Our conjecture matches many, but not all, of the previous conjectures that were made in special cases taking into account roots of unity.
Gilmer has recently shown that in any nonempty union-closed family $\mathcal F$ of subsets of a finite set, there exists an element contained in at least a proportion $.01$ of the sets of $\mathcal F$. We improve the proportion from $.01$ to $\frac{ 3 -\sqrt{5}}{2} \approx .38$ in this result. An improvement to $\frac{1}{2}$ would be the Frankl union-closed set conjecture. We follow Gilmer's method, replacing one key estimate by a sharp estimate. We then suggest a new addition to this method and sketch a proof that it can obtain a constant strictly greater than $\frac{ 3 -\sqrt{5}}{2} $. We also disprove a conjecture of Gilmer that would have implied the union-closed set conjecture.
We study p-adic hyper-Kloosterman sums, a generalization of the Kloosterman sum with a parameter k that recovers the classical Kloosterman sum when k=2, over general p-adic rings and even equal characteristic local rings. These can be evaluated by a simple stationary phase estimate when k is not divisible by p, giving an essentially sharp bound for their size. We give a more complicated stationary phase estimate to evaluate them in the case when k is divisible by p. This gives both an upper bound and a lower bound showing the upper bound is essentially sharp. This generalizes previously known bounds of Cochrane, Liu, and Zhen in the case of $\mathbb Z_p$. The lower bounds in the equal characteristic case have two applications to function field number theory, showing that certain short interval sums and certain moments of Dirichlet L-functions do not, as one might hope, admit square-root cancellation.
Work on generalizations of the Cohen-Lenstra and Cohen-Martinet heuristics has drawn attention to probability measures on the space of isomorphism classes of profinite groups. As is common in probability theory, it would be desirable to know that these measures are determined by their moments, which in this context are the expected number of surjections to a fixed finite group. We show a wide class of measures, including those appearing in a recent paper of Liu, Wood, and Zurieck-Brown, have this property. The method is to work "locally" with groups that are extensions of a fixed group by a product of finite simple groups. This eventually reduces the problem to the case of powers of a fixed finite simple group, which can be handled by a simple explicit calculation. We can also prove a similar theorem for random modules over an algebra.
We establish the exact structure of the cohomology associated to a certain matrix exponential sum investigated in prior work (arXiv:2109.00762) of the first and last author.
We prove estimates for the level of distribution of the Möbius function, von Mangoldt function, and divisor functions in squarefree progressions in the ring of polynomials over a finite field. Each level of distribution converges to $1$ as $q$ goes to $\infty$, and the power savings converges to square-root cancellation as $q$ goes to $\infty$. These results in fact apply to a more general class of functions, the factorization functions, that includes these three. The divisor estimates have applications to the moments of $L$-functions, and the von Mangoldt estimate to one-level densities.
For a connected reductive group $G$ over a non-archime\-dean local field $F$ of positive characteristic, Genestier and Lafforgue have attached a semisimple parameter $\CL^{ss}(π)$ to each irreducible representation $π$. Our first result shows that the Genestier-Lafforgue parameter of a tempered $π$ can be uniquely refined to a tempered L-parameter $\CL(π)$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of $\CL^{ss}(π)$ for unramfied $G$ and supercuspidal $π$ constructed by induction from an open compact (modulo center) subgroup. If $L^{ss}(π)$ is pure in an appropriate sense, we show that $\CL^{ss}(π)$ is ramified (unless $G$ is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal{L}^{ss}(π)$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is $\PP^1$ and a simple application of Deligne's Weil II.
Hecke has shown that the different of an extension of number fields is a square in the class group. We prove an analog for branched covers of closed $3$-manifolds saying that the branch divisor is a square in the first homology group.
We use a function field version of the Hardy-Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre's reformulation of the Batyrev-Manin conjecture in terms of slopes with respect to the tangent bundle.
We prove an asymptotic formula for the number of representations of squares by nonsingular cubic forms in six or more variables. The main ingredients of the proof are Heath-Brown's form of the Circle Method and various exponential sum results. The depth of the exponential sum results is comparable to Hooley's work on cubic forms in nine variables, in particular we prove an analogue of Katz' bound.
We prove that the coefficients of a $\mathrm{GL}_3\times \mathrm{GL}_2$ Rankin--Selberg $L$-function do not correlate with a wide class of trace functions of small conductor modulo primes, generalizing the corresponding result \cite{FKM1} for~$\mathrm{GL}_2$ and \cite{KLMS} for $\mathrm{GL}_3$. This result is inspired by a recent work of P. Sharma who discussed the case of a Dirichlet character of prime modulus.