arXiv subjects

Melanie Matchett Wood

Publications and source records attributed to Melanie Matchett Wood.

At least 37 records · Page 2Linked to original sources

arXiv2018

Cokernels of adjacency matrices of random $r$-regular graphs

We study the distribution of the cokernels of adjacency matrices (the Smith groups) of certain models of random $r$-regular graphs and directed graphs, using recent mixing results of Mészáros. We explain how convergence of such distributions to a limiting probability distribution implies asymptotic nonsingularity of the matrices, giving another perspective on recent results of Huang and Mészáros on asymptotic nonsingularity of adjacency matrices of random regular directed and undirected graphs, respectively. We also remark on the new distributions on finite abelian groups that arise, in particular in the $p$-group aspect when $p\mid r$.

math.PR↗

arXiv2018

Random integral matrices: universality of surjectivity and the cokernel

For a random matrix of entries sampled independently from a fairly general distribution in Z we study the probability that the cokernel is isomorphic to a given finite abelian group, or when it is cyclic. This includes the probability that the linear map between the integer lattices given by the matrix is surjective. We show that these statistics are asymptotically universal (as the size of the matrix goes to infinity), given by precise formulas involving zeta values, and agree with distributions defined by Cohen and Lenstra, even when the distribution of matrix entries is very distorted. Our method is robust and works for Laplacians of random digraphs and sparse matrices with the probability of an entry non-zero only n^{-1+epsilon}.

math.PR↗

arXiv2018

Cohen-Lenstra heuristics and local conditions

We prove function field theorems supporting the Cohen-Lenstra heuristics for real quadratic fields, and natural strengthenings of these analogs from the affine class group to the Picard group of the associated curve. Our function field theorems also support a conjecture of Bhargava on how local conditions on the quadratic field do not affect the distribution of class groups. Our results lead us to make further conjectures refining the Cohen-Lenstra heuristics, including on the distribution of certain elements in class groups. We prove instances of these conjectures in the number field case. Our function field theorems use a homological stability result of Ellenberg, Venkatesh, and Westerland.

math.NT↗

arXiv2017

On $\ell$-torsion in class groups of number fields

For each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of $\mathbb{Q}$ of degree $d$, for any fixed $d \in \{2,3,4,5\}$ (with the additional restriction in the case $d=4$ that the field be non-$D_4$). For sufficiently large $\ell$ (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our argument, we develop a probabilistic "Chebyshev sieve," and give uniform, power-saving error terms for the asymptotics of quartic (non-$D_4$) and quintic fields with chosen splitting types at a finite set of primes.

math.NT↗

arXiv2017

Representations of integers by systems of three quadratic forms

It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers $(n_1,\ldots,n_R)$ by a system of quadratic forms $Q_1,\ldots, Q_R$ in $k$ variables, as long as $k$ is sufficiently large; reducing the required number of variables remains a significant open problem. In this work, we consider the case of 3 forms and improve on the classical result by reducing the number of required variables to $k \geq 10$ for "almost all" tuples, under appropriate nonsingularity assumptions on the forms $Q_1,Q_2,Q_3$. To accomplish this, we develop a three-dimensional analogue of Kloosterman's circle method, in particular capitalizing on geometric properties of appropriate systems of three quadratic forms.

math.NT↗

arXiv2017

Irreducibility of Random Polynomials

We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic polynomials. Our data supports conjectures made by Odlyzko and Poonen and by Konyagin, and we formulate a universality heuristic and new conjectures that connect their work with Hilbert's Irreducibility Theorem and work of van der Waerden. The data indicates that the probability that a random polynomial is reducible divided by the probability that there is a linear factor appears to approach a constant and, in the large-degree limit, this constant appears to approach one. In cases where the model makes it impossible for the random polynomial to have a linear factor, the probability of reducibility appears to be close to the probability of having a non-linear, low-degree factor. We also study characteristic polynomials of random matrices with +1 and -1 entries.

math.PR↗

arXiv2017

Nonabelian Cohen-Lenstra Heuristics over Function Fields

Boston, Bush, and Hajir have developed heuristics, extending the Cohen-Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-p extensions of imaginary quadratic number fields for p an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of F_q(t), the Galois groups of the maximal unramified pro-p extensions, as q goes to infinity, have the moments predicted by the Boston, Bush, and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

math.NT↗

arXiv2016

Gauss Composition for P^1, and the universal Jacobian of the Hurwitz space of double covers

We investigate the universal Jacobian of degree n line bundles over the Hurwitz stack of double covers of P^1 by a curve of genus g. Our main results are: the construction of a smooth, irreducible, universally closed (but not separated) moduli compactification of this universal Jacobian; a description of the global geometry and moduli properties of these stacks; and a computation of the Picard groups of these stacks in the cases when n-g is even. An important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms. This parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms, which in particular gives the group law on integral binary quadratic forms first discovered by Gauss.

math.AG↗

arXiv2015

The distribution of $\mathbb{F}_q$-points on cyclic $\ell$-covers of genus $g$

We study fluctuations in the number of points of $\ell$-cyclic covers of the projective line over the finite field $\mathbb{F}_q$ when $q \equiv 1 \mod \ell$ is fixed and the genus tends to infinity. The distribution is given as a sum of $q+1$ i.i.d. random variables. This was settled for hyperelliptic curves by Kurlberg and Rudnick, while statistics were obtained for certain components of the moduli space of $\ell$-cyclic covers by Bucur, David, Feigon and Lalín. In this paper, we obtain statistics for the distribution of the number of points as the covers vary over the full moduli space of $\ell$-cyclic covers of genus $g$. This is achieved by relating $\ell$-covers to cyclic function field extensions, and counting such extensions with prescribed ramification and splitting conditions at a finite number of primes.

math.NT↗

arXiv2015

On a Cohen-Lenstra Heuristic for Jacobians of Random Graphs

In this paper, we make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are based on a Cohen-Lenstra type heuristic saying that a finite abelian group with duality pairing appears with frequency inversely proportional to the size of the group times the size of the group of automorphisms that preserve the pairing. We conjecture that the Jacobian of a random graph is cyclic with probability a little over .7935. We determine the values of several other statistics on Jacobians of random graphs that would follow from our conjectures. In support of the conjectures, we prove that random symmetric matrices over the p-adic integers, distributed according to Haar measure, have cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.

math.CO↗

arXiv2015

Random integral matrices and the Cohen Lenstra Heuristics

We prove that given any $ε>0$, random integral $n\times n$ matrices with independent entries that lie in any residue class modulo a prime with probability at most $1-ε$ have cokernels asymptotically (as $n\rightarrow\infty$) distributed as in the distribution on finite abelian groups that Cohen and Lenstra conjecture as the distribution for class groups of imaginary quadratic fields. This is a refinement of a result on the distribution of ranks of random matrices with independent entries in $\mathbb{Z}/p\mathbb{Z}$. This is interesting especially in light of the fact that these class groups are naturally cokernels of square matrices. We also prove the analogue for $n\times (n+u)$ matrices.

math.NT↗

arXiv2014

A heuristic for the distribution of point counts for random curves over a finite field

How many rational points are there on a random algebraic curve of large genus $g$ over a given finite field $\mathbb{F}_q$? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean $q+1+1/(q-1)$. We prove a weaker version of this statement in which $g$ and $q$ tend to infinity, with $q$ much larger than $g$.

math.NT↗

arXiv2014

The distribution of sandpile groups of random graphs

We determine the distribution of the sandpile group (a.k.a. Jacobian) of the Erdős-Rényi random graph G(n,q) as n goes to infinity. Since any particular group appears with asymptotic probability 0 (as we show), it is natural ask for the asymptotic distribution of Sylow p-subgroups of sandpile groups. We prove the distributions of Sylow p-subgroups converge to specific distributions conjectured by Clancy, Leake, and Payne. These distributions are related to, but different from, the Cohen-Lenstra distribution. Our proof involves first finding the expected number of surjections from the sandpile group to any finite abelian group (the "moments" of a random variable valued in finite abelian groups). To achieve this, we show a universality result for the moments of cokernels of random symmetric integral matrices that is strong enough to handle dependence in the diagonal entries. We then show these moments determine a unique distribution despite their p^{k^2}-size growth.

math.PR↗

arXiv2013

Discriminants in the Grothendieck Ring

We consider the "limiting behavior" of *discriminants*, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motivated by our results, we conjecture that the symmetric powers of geometrically irreducible varieties stabilize in the Grothendieck ring (in an appropriate sense). Our results extend parallel results in both arithmetic and topology. We give a number of reasons for considering these questions, and propose a number of new conjectures, both arithmetic and topological.

math.AG↗

arXiv2012

The distribution of points on superelliptic curves over finite fields

We give the distribution of points on smooth superelliptic curves over a fixed finite field, as their degree goes to infinity. We also give the distribution of points on smooth m-fold cyclic covers of the line, for any m, as the degree of their superelliptic model goes to infinity. This builds on previous work of Kurlberg, Rudnick, Bucur, David, Feigon, and Lalin for p-fold cyclic covers, but the limits taken differ slightly and the resulting distributions are interestingly different.

math.NT↗

arXiv2012

Counting polynomials over finite fields with given root multiplicities

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an analogous result on configuration spaces in the Grothendieck ring of varieties, suggesting new homological stabilization conjectures for configuration spaces of the plane.

math.NT↗

arXiv2012

Semiample Bertini theorems over finite fields

We prove a semiample generalization of Poonen's Bertini Theorem over a finite field that implies the existence of smooth sections for wide new classes of divisors. The probability of smoothness is computed as a product of local probabilities taken over the fibers of the morphism determined by the relevant divisor. We give several applications including a negative answer to a question of Baker and Poonen by constructing a variety (in fact one of each dimension) which provides a counterexample to Bertini over finite fields in arbitrarily large projective spaces. As another application, we determine the probability of smoothness for curves in Hirzebruch surfaces, and the distribution of points on those smooth curves.

math.AG↗

arXiv2011

The distribution of the number of points on trigonal curves over $\F_q$

We give a short determination of the distribution of the number of $\F_q$-rational points on a random trigonal curve over $\F_q$, in the limit as the genus of the curve goes to infinity. In particular, the expected number of points is $q+2-\frac{1}{q^2+q+1}$, contrasting with recent analogous results for cyclic $p$-fold covers of $\mathbb P^1$ and plane curves which have an expected number of points of $q+1$ (by work of Kurlberg, Rudnick, Bucur, David, Feigon and Lalín) and curves which are complete intersections which have an expected number of points $<q+1$ (by work of Bucur and Kedlaya). We also give a conjecture for the expected number of points on a random $n$-gonal curve with full $S_n$ monodromy based on function field analogs of Bhargava's heuristics for counting number fields.

math.NT↗