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We consider the probability theory, and in particular the moment problem and universality theorems, for random groups of the sort of that arise or are conjectured to arise in number theory, and in related situations in topology and combinatorics. The distributions of random groups that are discussed include those conjectured in the Cohen-Lenstra-Martinet heuristics to be the distributions of class groups of random number fields, as well as distributions of non-abelian generalizations, and those conjectured to be the distributions of Selmer groups of random elliptic curves. For these sorts of distributions on finite and profinite groups, we survey what is known about the moment problem and universality, give a few new results including new applications, and suggest open problems., Comment: ICM 2022 paper from my talk, references/status of problems have not been updated since November 2021

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., Comment: 35 pages

In this paper we study the cokernels of various random integral matrix models, including random symmetric, random skew-symmetric, and random Laplacian matrices. We provide a systematic method to establish universality under very general randomness assumption. Our highlights include both local and global universality of the cokernel statistics of all these models. In particular, we find the probability that a sandpile group of an Erdos-Renyi random graph is cyclic, answering a question of Lorenzini from 2008., Comment: 68 pages

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., Comment: 71 pages

For $2 \leq d \leq 5$, we show that the class of the Hurwitz space of smooth degree $d$, genus $g$ covers of $\mathbb P^1$ stabilizes in the Grothendieck ring of stacks as $g \to \infty$, and we give a formula for the limit. We also verify this stabilization when one imposes ramification conditions on the covers, and obtain a particularly simple answer for this limit when one restricts to simply branched covers., Comment: Many changes were made in response to careful readings by the referees. This version was accepted to Compositio Mathematica

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., Comment: 75 pages

We determine the average size of the 3-torsion in class groups of $G$-extensions of a number field when $G$ is any transitive $2$-group containing a transposition, for example $D_4$. It follows from the Cohen--Lenstra--Martinet heuristics that the average size of the $p$-torsion in class groups of $G$-extensions of a number field is conjecturally finite for any $G$ and most $p$ (including $p\nmid|G|$). Previously this conjecture had only been proven in the cases of $G=S_2$ with $p=3$ and $G=S_3$ with $p=2$. We also show that the average $3$-torsion in a certain relative class group for these $G$-extensions is as predicted by Cohen and Martinet, proving new cases of the Cohen--Lenstra--Martinet heuristics. Our new method also works for many other permutation groups $G$ that are not $2$-groups.

The goal of this paper is to prove theorems that elucidate the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional the to number of automorphisms of structures slightly larger than the class groups. We find the moments of the Cohen-Lenstra-Martinet distributions and prove that the distributions are determined by their moments. In order to apply these conjectures to class groups of non-Galois fields, we prove a new theorem on the capitulation kernel (of ideal classes that become trivial in a larger field) to relate the class groups of non-Galois fields to the class groups of Galois fields. We then construct an integral model of the Hecke algebra of a finite group, show that it acts naturally on class groups of non-Galois fields, and prove that the Cohen-Lenstra-Martinet conjectures predict a distribution for class groups of non-Galois fields that involves the inverse of the number of automorphisms of the class group as a Hecke-module., Comment: updated exposition

We consider the distribution of the Galois groups $\operatorname{Gal}(K^{\operatorname{un}}/K)$ of maximal unramified extensions as $K$ ranges over $\Gamma$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We prove two properties of $\operatorname{Gal}(K^{\operatorname{un}}/K)$ coming from number theory, which we use as motivation to build a probability distribution on profinite groups with these properties. In Part I, we build such a distribution as a limit of distributions on $n$-generated profinite groups. In Part II, we prove as $q\rightarrow\infty$, agreement of $\operatorname{Gal}(K^{\operatorname{un}}/K)$ as $K$ varies over totally real $\Gamma$-extensions of $\mathbb{F}_q(t)$ with our distribution from Part I, in the moments that are relatively prime to $q(q-1)|\Gamma|$. In particular, we prove for every finite group $\Gamma$, in the $q\rightarrow\infty$ limit, the prime-to-$q(q-1)|\Gamma|$-moments of the distribution of class groups of totally real $\Gamma$-extensions of $\mathbb{F}_q(t)$ agree with the prediction of the Cohen--Lenstra--Martinet heuristics., Comment: contains minor corrections and updates from the previous version

It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the $\ell$-torsion conjecture has crucially relied on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the $\ell$-torsion conjecture and other well-known conjectures: the Cohen-Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), and counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the $\ell$-torsion conjecture is true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the "method of moments.", Comment: The paper arXiv:1709.09637v1 by the same authors was submitted for publication in a significantly shorter version; some of the excluded material appears in this new paper, which also proves further results. Version 2 updates some references, with corresponding improvements in Section 6. Version 3 has a few small changes to streamline exposition after referee process

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 M\'esz\'aros. 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 M\'esz\'aros 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 $p\mid r$.

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}., Comment: 44 pages

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.

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

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 non-abelian version of the random abelian groups that feature in the Cohen-Lenstra heuristics. For each n, these random groups belong to the few relator model in the Gromov model of random groups., Comment: minor corrections added

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., Comment: 14 pages, 9 figures

In this paper we give a conjecture for the average number of unramified $G$-extensions of a quadratic field for any finite group $G$. The Cohen-Lenstra heuristics are the specialization of our conjecture to the case that $G$ is abelian of odd order. 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 $G$-extensions that takes the same number of values as the predicted average and an argument using the Malle-Bhargava principle. We note that for even $|G|$, corrections for the roots of unity in $\mathbb{Q}$ are required, which can not be seen when $G$ is abelian., Comment: main article by Melanie Matchett Wood, appendix by Melanie Matchett Wood and Philip Matchett Wood, to appear in Duke Mathematical Journal

Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil's number field/ function field analogy to predict coincidences for limiting homological densities of various sequences $\mathcal{Z}^{(d_1,\ldots,d_m)}_n(X)$ of spaces of $0$-cycles on manifolds $X$. The main theorem in this paper is that these topological predictions, which seem strange from a purely topological viewpoint, are indeed true. The obstacle to proving such a theorem with current technology is how to deal with the combinatorial complexity of all possible "collisions" of points, this problem does not arise in the simplest (and classical) case $(m,n)=(1,2)$ of configuration spaces. To overcome this obstacle we develop a method that uses the Bj\"orner--Wachs theory of lexicographic shellability from algebraic combinatorics to study such problems. As a consequence we derive new homological stability theorems for broad classes of $0$-cycles on manifolds. Even in the classical case $(m,n)=(1,2)$ this gives a new, simplified proof of classical results, and also of recent theorems of Church and others., Comment: 41 pages. Final Version

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., Comment: 27 pages, updated from v1 with minor edits to exposition

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., Comment: minor corrections made, to appear in Compositio

We present a heuristic that suggests that ranks of elliptic curves over the rationals are bounded. In fact, it suggests that there are only finitely many elliptic curves of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields., Comment: 41 pages, typos fixed in torsion table in section 8

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., Comment: 64 pages, minor edits to exposition to agree with published version

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 Lal\'{i}n. 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.

We prove that given any $\epsilon>0$, random integral $n\times n$ matrices with independent entries that lie in any residue class modulo a prime with probability at most $1-\epsilon$ 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.

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 $g$., Comment: 16 pages; v2: refereed version, Philosophical Transactions of the Royal Society A 2015

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., Comment: 20 pages. v2: Improved exposition and appended code used to generate experimental evidence after the \end{document} line in the source file. To appear in J. Algebraic Combin

We determine the distribution of the sandpile group (a.k.a. Jacobian) of the Erd\H{o}s-R\'enyi 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.

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.

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.

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.

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., Comment: 39 pages, updated with progress by others on various conjectures

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., Comment: mathematical and expositional updates and improvements, 29 pages, 5 figures

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 Lal\'in) and curves which are complete intersections which have an expected number of points $

We give a parametrization of the ideal classes of rings associated to integral binary forms by classes of tensors in $\mathbb Z^2\tensor \mathbb Z^n\tensor \mathbb Z^n$. This generalizes Bhargava's work on Higher Composition Laws, which gives such parametrizations in the cases $n=2,3$. We also obtain parametrizations of 2-torsion ideal classes by symmetric tensors. Further, we give versions of these theorems when $\mathbb Z$ is replaced by an arbitrary base scheme $S$, and geometric constructions of the modules from the tensors in the parametrization.

The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied quadratic rings and ideals associated to binary quadratic forms, and found that ideal classes of quadratic rings are exactly parametrized by equivalence classes of integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic rings are parametrized by equivalence classes of integral binary cubic forms. Birch, Merriman, Nakagawa, Corso, Dvornicich, and Simon have all studied rings associated to binary forms of degree n for any n, but it has not previously been known which rings, and with what additional structure, are associated to binary forms. In this paper, we show exactly what algebraic structures are parametrized by binary n-ic forms, for all n. The algebraic data associated to an integral binary n-ic form includes a ring isomorphic to $\mathbb{Z}^n$ as a $\mathbb{Z}$-module, an ideal class for that ring, and a condition on the ring and ideal class that comes naturally from geometry. In fact, we prove these parametrizations when any base scheme replaces the integers, and show that the correspondences between forms and the algebraic data are functorial in the base scheme. We give geometric constructions of the rings and ideals from the forms that parametrize them and a simple construction of the form from an appropriate ring and ideal., Comment: submitted

We give a bijection between binary quartic forms and quartic rings with a monogenic cubic resolvent ring, relating the rings associated to binary quartic forms with Bhargava's cubic resolvent rings. This gives a parametrization of quartic rings with monogenic cubic resolvents. We also give a geometric interpretation of this parametrization., Comment: to appear in IMRN, section on geometric analogs over an arbitrary base added since version 1

We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree four $S$-schemes), with their cubic resolvents, by pairs of ternary quadratic forms over the base. This generalizes Bhargava's parametrization of quartic rings with their cubic resolvent rings over $\mathbb{Z}$ by pairs of integral ternary quadratic forms, as well as Casnati and Ekedahl's construction of Gorenstein quartic covers by certain rank 2 families of ternary quadratic forms. We give a geometric construction of a quartic algebra from any pair of ternary quadratic forms, and prove this construction commutes with base change and also agrees with Bhargava's explicit construction over $\mathbb{Z}$., Comment: submitted

The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the integers. However, such extensions have always included hypotheses on the rings, and the theorems involve only binary quadratic forms satisfying further hypotheses. We give a complete statement of the relationship between binary quadratic forms and modules for quadratic algebras over any base ring, or in fact base scheme. The result includes all binary quadratic forms, and commutes with base change. We give global geometric as well as local explicit descriptions of the relationship between forms and modules., Comment: submitted