Melanie Matchett Wood's Publications

Papers

• A predicted distribution for Galois groups of maximal unramified extensions, with Yuan Liu and David Zureick-Brown, arxiv preprint.

  • We study the distribution of the Galois groups Gal(K^un/K) of maximal unramified extensions as K ranges over Gamma-extensions of \Q or \F_q(t). We prove two properties of Gal(K^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 goes to infinity, agreement of Gal(K^un/K) as K varies over totally real Gamma-extensions of \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 large q limit, the prime-to-q(q-1)|Gamma|-moments of the distribution of class groups of totally real Gamma-extensions of \F_q(t) agree with the prediction of the Cohen--Lenstra--Martinet heuristics.

• Moments and interpretations of the Cohen-Lenstra-Martinet heuristics, with Weitong Wang arxiv preprint.

  • The goal of this paper is to further the understanding of the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields. 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.

• An algebraic lifting invariant of Ellenberg, Venkatesh, and Westerland preprint.

  • In this note, we define and prove basic properties of a lifting invariant of curves over an algebraically closed field k with a map to the projective line P^1_k that was introduced by Ellenberg, Venkatesh, and Westerland.

• On a conjecture for l-torsion in class groups of number fields: from the perspective of moments, with Lillian B. Pierce and Caroline L. Turnage-Butterbaugh, arxiv preprint.

  • In this paper we make explicit the quantitative relationships between the l-torsion conjecture (that the l-torsion in the class group of a number field is bounded by an epsilon power of the discriminant) 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. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the "method of moments."

• Cokernels of adjacency matrices of random r-regular graphs, with Hoi H. Nguyen, arxiv preprint.

  • We give the distribution of the cokernels of adjacency matrices (the Smith groups) of certain models of random r-regular graphs and directed graphs, using 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 divides r.

• Random integral matrices: universality of surjectivity and the cokernel, with Hoi H. Nguyen, arxiv preprint.

  • For a random matrix of entries sampled independently from a fairly general distribution in the integers 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}. My paper below ``Random integral matrices and the Cohen Lenstra Heuristics'' understands the Sylow p-subgroups of the cokernels of (not too sparse) random integral matrices for at most finitely many fixed primes, and this paper determines the distribution of the entire cokernel and can handle sparser matrices, both of which require substantial new ideas.

• An effective Chebotarev density theorem for families of number fields, with an application to l-torsion in class groups, with Lillian B. Pierce and Caroline L. Turnage-Butterbaugh, to appear in Inventiones Mathematicae,, arxiv preprint.

  • In this work we prove a new effective Chebotarev density theorem, independent of GRH, that improves the previously known unconditional error term and allows primes to be taken quite small (certainly 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. Such a family has fixed degree, fixed Galois group of the Galois closure, and in certain cases a ramification restriction on all tamely ramified primes in each field; examples include totally ramified cyclic fields, degree n Sn-fields with square-free discriminant, and degree n An-fields. In all cases, our work is independent of GRH; in some cases we assume the strong Artin conjecture or hypotheses on counting number fields. The technical innovation leading to our main theorem is a new idea to extend a result of Kowalski and Michel, a priori for the average density of zeroes of a family of cuspidal L-functions, to a family of non-cuspidal L-functions. The new effective Chebotarev theorem is expected to have many applications, of which we demonstrate two: on nontrivial bounds for l-torsion in the class groups of "almost all" fields in the families of fields we consider, and, in answer to a question of Ruppert, we prove that within each family, "almost all" fields have a small generator.

• Coincidences of homological densities, predicted by arithmetic, with Benson Farb, Jesse Wolfson, to appear in Advances, arxiv preprint.

  • Motivated by the arithmetic fact that the density of square-free 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 Bjorner--Wachs theory of lexicographic shellability.

• Nonabelian Cohen-Lenstra Moments (including an appendix with Philip Matchett Wood), to appear Duke Mathematical Journal, arxiv preprint.

  • We give a conjecture for the average number of unramified G-extensions of a quadratic field for any finite group G. This specializes to the prediction fromthe Cohen-Lenstra heuristics when G is 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 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 Q are required, which can not be seen when G is abelian. For odd G, better function field results are proven by Boston and Wood. The appendix gives numerical computations of unramified A_4 extensions of quadratic fields in support of the conjecture.

• Random integral matrices and the Cohen Lenstra Heuristics, to appear American Journal of Mathematics, 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 Cohen-Lenstra heuristics. These results are a refinement of the determination of the distribution of ranks of random matrices modulo a prime.

• A heuristic for boundedness of ranks of elliptic curves, with Jennifer Park, Bjorn Poonen, and John Voight, Journal of the European Mathematical Society, Published online: 2019-05-21, DOI: 10.4171/JEMS/893. old arxiv preprint.

  • Inspired by a heuristic for the distribution of Shafarevich-Tate groups of elliptic curves of a given rank based on skew-symmetric matrices of a given rank (which agrees with Delaunay's predictions for the distribution of Shafarevich-Tate groups), we develop a heuristic that the ranks of elliptic curves should be modeled by the (co)ranks of skew-symmetric matrices with integral coefficients. The heuristic then predicts many well-known 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.

• The free group on n generators modulo n+u random relations as n goes to infinity, with Yuan Liu, Journal für die reine und angewandte Mathematik (Crelle), , Published online: 2018-10-30, DOI: 10.1515/crelle-2018-0025. old 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 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. 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 Cohen-Lenstra heuristics for class groups and the Boston-Bush-Hajir heuristics for the Galois group of the maximal unramified pro-p extension of a random imaginary quadratic number field.

• Cohen-Lenstra heuristics and local conditions, Research in Number Theory, (2018) 4:41. old arxiv preprint.

  • 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 thedistribution 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 some instances of these conjectures in the number field case.

• On l-torsion in class groups of number fields, with Jordan Ellenberg and Lillian B. Pierce, Algebra and Number Theory 11-8 (2017), 1739--1778. old arxiv preprint.

  • For each integer l, we prove an unconditional upper bound on the size of the l-torsion subgroup of the class group, which holds for all but a zero-density 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 non-D4). 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, power-saving error terms for the asymptotics of quartic (non-D4) 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. 915-958. 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 Cohen-Lenstra 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 Cohen-Lenstra Heuristics over Function Fields, with Nigel Boston, Compositio Mathematica 153 (2017), no. 7, pp. 1372-1390. old arxiv preprint.

  • We find the moments of the Boston-Bush-Hajir distribution for the non-abelian Cohen-Lenstra 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 11-4 (2017), 817--840. DOI 10.2140/ant.2017.11.817 ERRATA Addenda 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) 320-352. 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 n-g 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 large-degree 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, 289-344. old arxiv preprint.

  • 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 three-dimensional 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) 291-339. 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 l-covers of genus g, with Alina Bucur, Chantal David, Brooke Feigon, Nathan Kaplan, Matilde Lalin, and Ekin Ozman, International Mathematics Research Notices, (2016) no. 14, 4297-4340. old arxiv preprint.

  • We find the distribution of Fq points on cyclic l-covers 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 p-fold 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, 1139-1185. ERRATA 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, 1-38. 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. 1365-1375. old arxiv preprint.

  • We give the distribution of points on smooth superelliptic curves and smooth m-fold 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 Zureick-Brown, 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 non-negligible contribution in the limit to the Grothendieck-Lefschetz 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/(q-1).

• 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, 12590-12619. ERRATA old arxiv preprint.

  • We study a relation between the Artin conductor and the weight coming from the motivic integration over wild Delgine-Mumford 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 Cohen-Lenstra Heuristic for Jacobians of Random Graphs, with Julien Clancy, Nathan Kaplan, Timothy Leake, and Sam Payne, Journal of Algebraic Combinatorics 42 (2015), no. 3, 701-723, old arxiv preprint.

  • We investigate a Cohen-Lenstra 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 p-adic 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), 169-199. old arxiv preprint.

  • This paper proves that ideal classes of rings associated to binary forms (see "Rings and ideals parametrized by binary n-ic 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. 394-402. 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, 5444-5456. 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 n-gonal 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, 1300-1320. 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 n-ic 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, 1756-1771. 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 n-ic forms," "Parametrizing quartic algebras over an arbitrary base," and "Parametrization of ideal classes.."

• Rings and ideals parametrized by binary n-ic forms Journal of the London Mathematical Society (2) 83 (2011) 208--231. 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 5-8 (2011), 1069--1094. 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) 433--44. 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, 102--128. ERRATA old arxiv preprint.

  • This paper determines the probabilities of various splitting types of a fixed prime in a random G-extension 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, 391-405. 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, 1581--1587.

  • This paper counts Galois sextic number fields (with Galois group S_3) asymptotically by discriminant.

• Belyi-extending maps and the Galois action on dessins d'enfants, Publications of the Research Institute for Mathematical Sciences 42 (2006), no. 3, 721--738.

• P-orderings: a metric viewpoint and the non-existence of simultaneous orderings. Journal of Number Theory 99 (2003), no. 1, 36--56.

Thesis

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