Melanie Matchett Wood's Publications
Papers

The free group on n generators modulo n+u random relations as n goes to
infinity, with Yuan Liu, arxiv preprint.

We show that, as n goes to infinity, the free group on n generators, modulo
n+u random relations, converges to a random group that we give explicitly. This
random group is a nonabelian version of the random abelian groups that feature
in the CohenLenstra heuristics. For each n, these random groups belong to the
few relator model in the Gromov model of random groups. This is the foundational work on the topic of random groups given by free groups modulo random relations, that we expect to be necessary for eventual models for the Galois group of the maximal unramified extension of a random number field, generalizing the CohenLenstra heuristics for class groups and the BostonBushHajir heuristics for the Galois group of the maximal unramified prop extension of a random imaginary quadratic number field.

Nonabelian CohenLenstra Moments, arxiv preprint.

We give a conjecture for the average number of unramified Gextensions of a quadratic field for any finite group G. This specializes to the prediction fromthe CohenLenstra heuristics whenGis odd and abelian. We prove a theorem towards the function field analog of our conjecture, and give additional motivations for the conjecture including the construction of a lifting invariant for the unramified Gextensions that takes the same number of values as the predicted average and an argument using the MalleBhargava principle. We note that for even G, corrections for the roots of unity in Q are required, which can not be seen whenGis abelian. For odd G, better function field results are proven by Boston and Wood.

CohenLenstra heuristics and local conditions, preprint.

We prove function field theorems supporting the CohenLenstra 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 thedistribution of class groups. Our results lead us to make further conjectures refining the CohenLenstra heuristics, including on the distribution of certain elements in class groups. We prove some instances of these conjectures in the number field case.

Coincidences of homological densities, predicted by arithmetic, with Benson Farb, Jesse Wolfson arxiv preprint.

Motivated by the arithmetic fact that the density of squarefree integers (zeta(2)^{1}) is the same as the density of pairs of relatively prime integers, we introduce a new notion of homological density in topology to capture an analogous phenomenon in topology. This phenomenon is a much deeper symmetry between different kinds of configuration spaces in topology, which holds in great generality, but still with some exceptions. The topological proofs involve the BjornerWachs theory of lexicographic shellability.

A heuristic for boundedness of ranks of elliptic curves, with Jennifer Park, Bjorn Poonen,
and John Voight, arxiv preprint.

Inspired by a heuristic for the distribution of ShafarevichTate groups of elliptic curves of a given rank based on skewsymmetric matrices of a given rank (which agrees with Delaunay's predictions for the distribution of ShafarevichTate groups), we develop a heuristic that the ranks of elliptic curves should be modeled by the (co)ranks of skewsymmetric matrices with integral coefficients. The heuristic then predicts many wellknown conjectures, including the asymptotic count of rank 2 elliptic curves. As a more controversial prediction, our hueristic suggests that there are only finitely many elliptic curves of rank greater than 21. To determine what our heuristic predicts, we prove a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.

Random integral matrices and the Cohen Lenstra Heuristics arxiv preprint.

This paper shows that cokernels of random integral matrices with independent entries
are distributed among finite abelian groups in the same distribution as Cohen and Lenstra predicted for the distribution of class groups of imaginary quadratic fields. Since these class groups are naturally cokernels of square matrices, this gives moral support to the CohenLenstra heuristics. These results are a refinement of the determination of the distribution of ranks of random matrices modulo a prime.

On ltorsion in class groups of number fields, with Jordan Ellenberg and Lillian B. Pierce, to appear in Algebra and Number Theory arxiv preprint.

For each integer l, we prove an unconditional upper bound on the size of the ltorsion subgroup of the class group, which holds for all but a zerodensity set of field extensions of Q of degree d, for any fixed d=2,3,4,5 (with the additional restriction in the case d=4 that the field be nonD4). For sufficiently large l (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, powersaving error terms for the asymptotics of quartic (nonD4) and quintic fields with chosen splitting types at a finite set of primes.

The distribution of sandpile groups of random graphs Journal of the American Mathematical Society 30 (2017), pp. 915958. old arxiv preprint.

This paper proves a conjecture of Payne, et. al. on the distribution of Jacobians (sandpile groups) of random graphs. The distribution is a variant of the CohenLenstra distribution that takes into account the symmetric pairing on the Jacobian.
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 growing to fast to use standard methods to deduce this.

Nonabelian CohenLenstra Heuristics over Function Fields, with Nigel Boston, Compositio Mathematica
153 (2017), no. 7, pp. 13721390.
old arxiv preprint.

We find the moments of the BostonBushHajir distribution for the nonabelian CohenLenstra problem and prove the moments determine the distribution. We prove in the function field case that these moments, as q gets large, are as predicted by Boston, Bush, and Hajir. Our function field result suggests new conjectures for the distribution of Galois groups of maximal unramified extensions.

Mass formulas for local Galois representations and quotient singularities II: dualities and resolution of singularities, with Takehiko Yasuda, Algebra and Number Theory 114 (2017), 817840. DOI 10.2140/ant.2017.11.817 old arxiv preprint.

In the preceding paper, the authors observed that in a particular example, two total masses coming from two different weightings of counting extensions of local fields are dual to each other, one coming from Bhargava's local mass formula and one coming from the Hilbert scheme of points in the plane. We discuss how generally such a duality holds and relate it to the existence of simultaneous resolution of singularities, using the wild McKay correspondence and the Poincar\'e duality for stringy invariants.

Gauss Composition for P^1, and the universal Jacobian of the Hurwitz space of double covers, with Daniel Erman, Journal of Algebra 470 (2017) 320352. old arxiv preprint.

This paper uses the correspondence developed in "Gauss composition over an arbitrary base" over the base P^1 to
describe the moduli space of hyperelliptic curves with line bundles. 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 ng is even.

Irreducibility of Random Polynomials, with Christian Borst, Evan Boyd, Claire Brekken, Samantha Solberg, and Philip Matchett Wood, Experimental Mathematics (2017) DOI:10.1080/10586458.2017.1325790. old arxiv preprint.

In joint work with undergraduates, we study the probability that a random polynomial with integer coefficients, from a number of different models, is reducible. Our computer generated data support 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 indicate 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 largedegree limit, this constant appears to approach 1.

Representations of integers by systems of three quadratic forms, with Lillian B. Pierce and Damaris Schindler, Proceedings of the London Mathematical Society(3), 113 (2016), no. 3, 289344.

The circle method produces an asymptotic for the number of representations of a tuple of integers (n_1,...,n_R) by a system of quadratic forms Q_1,...,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>=10 for "almost all" tuples, under appropriate nonsingularity assumptions on the forms Q_1,Q_2,Q_3. To accomplish this, we develop a threedimensional analogue of Kloosterman's circle method, in particular capitalizing on geometric properties of appropriate systems of three quadratic forms.

Asymptotics for number fields and class groups,
in Research
Directions in Number Theory Springer (2016) 291339. old preprint.

This article was developed from lecture notes for a series of five lectures at the 2014 Arizona Winter School on arithmetic statistics.
It is an exposition of some of the basic questions of arithmetic
statistics (counting number fields and distribution of class groups) aimed
at readers new to the area. It treats the simplest cases in detailed way, with an
emphasis on connections and perspectives that are well known to experts
but absent from the literature.

The distribution of Fq points on cyclic lcovers of genus g, with Alina Bucur, Chantal David, Brooke Feigon, Nathan Kaplan, Matilde Lalin, and Ekin Ozman,
International Mathematics Research Notices, (2016) no. 14, 42974340. old arxiv preprint.

We find the distribution of Fq points on cyclic lcovers of the projective line of genus g,
with q fixed and g going to infinity. We adapt the method of the below paper ``On the probabilities of local behaviors in abelian field extensions'' to the function field setting to count cyclic function fields with local conditions corresponding to the number of rational points over each rational point of the line. We order curves by their genus, in contrast to the below paper ``The distribution of points on superelliptic curves over finite fields,'' and the paper ``Biased statistics for
traces of cyclic pfold covers over finite fields,'' by Bucur, David, Feigon, and Lalin, which both consider the distribution of points on the same curves, but with different orderings of the curves.

Discriminants in the Grothendieck Ring, with Ravi Vakil, Duke Mathematical Journal, 164 (2015), no. 6, 11391185, old arxiv preprint.

This paper determines the limiting motive (class in the Grothendieck ring of varieties) of several different sequences of moduli spaces of "nice" objects, or equivalently of their complements the "discriminant" variety of "not so nice" objects. For example, we determine the limit of the motive of smooth divisors (or with s singularities) in increasing multiples of a linear system. We also determine the limit of the motive of configuration spaces of distinct points (or points that are allowed to come together to a limited extent) as the number of points increases. All of these limit motives are given by explicit formulas in terms of motivic zeta values. Our results motivate a large number of conjectures in topology and arithmetic.

Semiample Bertini theorems over finite fields, with Daniel Erman,
Duke Mathematical Journal 164 (2015), no. 1, 138.
old arxiv preprint.

This paper gives the probability that a section of nA+dE is smooth, where A and E are a very ample and globally generated (respectively) divisor on a fixed variety over a finite field, as d goes to infinity.
This gives a semiample generalization of Poonen's Bertini Theorem over a finite field, which is the case A=E of our result.
The probability of smoothness is computed as a product of local probabilities taken over the fibers of the morphism determined by E.
Unlike in Poonen's theorem where smoothness is independent at all points and ampleness of a certain divisor is a key
ingredient in the proof, in our situation there is now dependence among certain points and we develop new tools to replace the use of ampleness.
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.

The distribution of points on superelliptic curves over finite fields, with GilYoung Cheong and Azeem Zaman; Proceedings of the American Mathematical Society, 143 no. 4 (2015), pp. 13651375.
old arxiv preprint.

We give the distribution of points on smooth superelliptic curves and smooth mfold cyclic covers of the line over a fixed finite field, as their degree goes to infinity. In a departure from previous work of distribution of points on curves over a fixed finite field, our cyclic covers are not given by explicit equations but have to be accessed through singular models with a different number of points.

A heuristic for the distribution of point counts for random curves over a finite field, with Jeff Achter, Daniel Erman, Kiran S. Kedlaya, David ZureickBrown, Philosophical Transactions of the Royal Society A 373, no. 2040: 20140310. old arxiv preprint.

We propose a heuristic for the limiting distribution of the number of points on a genus g curve over a fixed finite field, as the genus goes to infinity. The heuristic, roughly, is that only the tautological (equivalently, only the stable) cohomology classes should
have nonnegligible contribution in the limit to the GrothendieckLefschetz trace formula counts for the moments of this distribution. The result is a prediction that the number of points is asymptotically Poisson with mean q + 1 + 1/(q1).

Mass formulas for local Galois representations and quotient singularities I: A comparison of counting functions, with Takehiko Yasuda, International Mathematics Research Notices (2015) no. 23, 1259012619.
old arxiv preprint.

We study a relation between the Artin conductor and the weight coming from the motivic integration over wild DelgineMumford stacks. As an application, we prove some version of the McKay correspondence, which relates Bhargava's mass formula for extensions of a local field and the Hilbert scheme of points.

On a CohenLenstra Heuristic for Jacobians of Random Graphs, with
Julien Clancy, Nathan Kaplan, Timothy Leake, and Sam Payne, Journal of Algebraic Combinatorics 42 (2015), no. 3, 701723,
old arxiv preprint.

We investigate a CohenLenstra type heuristic due to Payne, et. al. 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. A significant piece of this heuristic has been proven in the paper ``The distribution of sandpile groups of random graphs'' above, but that paper uses the fact (proven here) that random symmetric matrices over the padic integers, distributed according to Haar measure, have cokernels distributed according to the heuristic.
Our investigation also leads us to conjecture that the Jacobian of a random graph is cyclic with probability a little over .7935 (the above paper proves the upper bound we expect).

Parametrization of ideal classes in rings associated to binary forms, in Journal für die reine und angewandte Mathematik (Crelle), 689 (2014), 169199.
old arxiv preprint.
 This paper proves that ideal classes of rings associated to binary forms (see "Rings and ideals parametrized by binary nic forms" below) are parametrized by classes of 2xnxn tensors. This generalizes the results of Bhargava's Higher Composition Laws I and II (which include these parametrizations for n=2,3 respectively). Also, these parametrizations are proven with an arbitrary base scheme (or ring) replacing the integers.

Counting polynomials over finite fields with given root multiplicities, with Ayah Almousa; Journal of Number Theory,
136C (2014), pp. 394402.
old arxiv preprint.

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). 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.

The distribution of the number of points on trigonal curves over F_q.
International Mathematics Research Notices (2012)
no. 23, 54445456.
old arxiv preprint.

This paper finds the distribution of the number of F_q points on a trigonal curve, in the limit as the genus goes to infinity. The method is through the correspondence with cubic extensions of the rational function field and application of the work of Datskovsky and Wright. The surprise is that the expected number of points is near q+2 (perhaps 1 more than expected). We also give conjectures for the distribution of points on a random
ngonal curve with S_n monodromy, based on function field analogs of Bhargava's number field counting heuristics.

Quartic rings associated to binary quartic forms,
International Mathematics Research Notices (2012)
no. 6, 13001320.
old arxiv preprint.
 This paper shows that classes of integral binary quartic forms correspond to quartic rings with monogenic cubic resolvents. This is an different approach to studying rings associated to binary quartic forms than in the paper "Rings and ideals parametrized by binary nic forms," but the technique only works for n=4 (with some parts possible for n=5 but not discussed here).

Gauss composition over an arbitrary base,
Advances in Mathematics 226 (2011) no. 2, 17561771.
old arxiv preprint.
 This paper generalizes the correspondence between binary quadratic forms and ideal classes of quadratic rings that is classical over the integers to any ring, or even scheme. It can serve as an introduction to some of the methods used to parametrize rings and ideals over an arbitrary base that are seen in "Rings and ideals parametrized by binary nic forms," "Parametrizing quartic algebras over an arbitrary base," and "Parametrization of ideal classes.."

Rings and ideals parametrized by binary nic forms
Journal of the London Mathematical Society (2) 83 (2011) 208231.
old arxiv preprint.
 This paper determines what rings and ideals are associated to binary forms, over the integers and also over an arbitrary base. The associated rings and ideals are constructed in several ways, including very concretely and also geometrically.

Parametrizing quartic algebras over an arbitrary base, Algebra & Number Theory 58 (2011), 10691094.
old arxiv preprint.
 This paper shows that quartic algebras, with their cubic resolvent algebras, are parametrized by pair of ternary quadratic forms over any base ring or scheme. This generalizes the main result of Bhargava's Higher Composition Laws III from the integers to any base. Moreover, geometric constructions are given of the quartic algebra from the forms.

Mapping Incidences,
with Van H. Vu, and Philip Matchett Wood, Journal of the London Mathematical Society 2011; (2) 84 (2011) 43344. doi: 10.1112/jlms/jdr017
old arxiv preprint.
 This paper shows that finite systems of complex numbers and their algebraic relations can be mapped to the integers modulo a prime, for some prime, preserving all the relations. The methods are from number theory and algebraic geometry, but the applications are to combinatorial problems.

On the probabilities of local behaviors in abelian field extensions, Compositio Mathematica 146 (2010), no. 1, 102128.
old arxiv preprint.
 This paper determines the probabilities of various splitting types of a fixed prime in a random Gextension of a number field, when G is an abelian group. When the number fields are counted by conductor, the probabilities are as predicted by a heuristic and independent at distinct primes (but things are worse when counting by discriminant!).

Mass formulas for local Galois representations to wreath products and cross products, Algebra and Number Theory,
Vol. 2 (2008), No. 4, 391405.
old arxiv preprint
 This paper proves that there are mass formulas that count Galois representations of local fields to wreath products and cross products of symmetric groups, that are independent of the local field (in an appropriate sense).

The density of discriminants of $S_3$sextic number fields, with Manjul Bhargava,
Proceedings of the American Mathematical Society 136 (2008), no. 5, 15811587.
 This paper counts Galois sextic number fields (with Galois group S_3) asymptotically by discriminant.

Belyiextending maps and the Galois action on dessins d'enfants, Publications of the Research
Institute for Mathematical Sciences 42 (2006), no. 3, 721738.

Porderings:
a metric viewpoint and the nonexistence of simultaneous orderings.
Journal of Number Theory 99 (2003), no. 1, 3656.
Thesis
My PhD Thesis: Moduli Spaces for Rings and Ideals
This thesis is mostly superseded by the published works with the titles agreeing with the chapter titles of the thesis, with the exception of Chapter 7. Chapter 7 gives a parametrization of D_4 quartic rings and some other special quartic rings.
Books
Feng, Zuming; Wood, Melanie Matchett; Rousseau, Cecil.
USA and International Mathematical Olympiads 2005.
Mathematical Association of America, 2006.