higher composition laws i a new view on gauss composition and quadratic generalizations two centuries ago in his celebrated work disquisitiones arithmeticae of 1801 gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow even today two centuries later this law of composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders it is hence only natural to ask whether higher analogues of this composition law exist that could shed light on the structure of other algebraic number rings and fields this article forms the first of a series of four articles in which our aim is precisely to develop such higher composition laws in fact we show that gauss s law of composition is only one of at least fourteen composition laws of its kind which yield information on number rings and their class groups in this paper we begin by deriving a general law of composition on 2 2 2 cubes of integers from which we are able to obtain gauss s composition law on binary quadratic forms as a simple special case in a manner reminiscent of the group law on plane elliptic curves we also obtain from this composition law on 2 2 2 cubes four further new laws of composition these laws of composition are defined on 1 binary cubic forms 2 pairs of binary quadratic forms 3 pairs of quaternary alternating 2 forms and 4 senary six variable alternating 3 forms more precisely gauss s theorem states that the set of sl2 z equivalence classes of primitive binary quadratic forms of a given discriminant d has an inherent group structure the five other spaces of forms mentioned above including the space of 2 2 2 cubes also possess natural actions by special linear groups over z and certain products thereof we prove that just like gauss s space of binary quadratic forms each of these group actions has the following remarkable properties first each of these six spaces possesses only a single polynomial invariant for the corresponding group action which we call the discriminant this discriminant invariant is found to take only values that

on rings with linear resolutions let r denote a positively graded noetherian ring over a field k r eori with r k and m the irrelevant maximal ideal generated by forms of degree 1 or let r be the m adic completion of such a ring if we consider x as a homomorphic image of a polynomial ring resp of a power series ring over k with a homogeneous defining ideal one may ask whether r has a minimal linear resolution as an s module cohen macaulay rings r of maximal embedding dimension e g have a linear resolution as sally showed in 9 she also proved formulas for the hilbert series hr and the poincari series pi pz schenzel lo later characterized cm rings r with a linear resolution and called these rings extremal he also gave formulas for hr and pi s goto see 3 1 below proved besides other facts the linearity of the resolution for buchsbaum rings of maximal embedding dimension i e edim r dim r e r i r 1 i r denotes the invariant of the buchsbaum ring r and a formula for hh in section 1 we generalize these results and characterize rings with a linear resolution which have a parameter system that forms a d sequence in the buchsbaum case we then show that the h pi and pf of such rings are determined by the embedding dimension the krull dimension the dimensions of the local cohomology modules as k vector spaces and what we call the type of r in section 2 we determine all buchsbaum rings r k r of maximal embedding dimension and multiplicity 2 sally proved in 8 that for a local cm ring r m of maximal embedding dimension the associated graded ring is also cm of maximal embedding dimension goto showed a similar result for buchsbaum rings of

on the number of quintic fields we show that the number of quintic number fields whose discriminant does not exceed x in absolute value is bounded by a constant times x1 e for any e 0 this may be compared with the conjecture that the number of such fields is asymptotic to a constant times x as x tends to infinity

prehomogeneous vector spaces and field extensions letk be an infinite field of characteristic not equal to 2 3 5 in this paper we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois homomorphisms

the concomitants of a prehomogeneous vector space abstract the classical method of radial separation of variables is applied to gain some information about the concomitants also known as relatively equivariant polynomial maps or covariants of a class of prehomogeneous vector spaces particular attention is payed to concomitants that are themselves prehomogeneous the results are applied to classify all prehomogeneous concomitants of several examples of interest in arithmetic

secondary terms in counting functions for cubic fields we prove the existence of secondary terms of order x5 6 in the davenport heilbronn theorems on cubic fields and 3 torsion in class groups of quadratic fields for cubic fields this confirms a conjecture of datskovsky wright and roberts we also prove a variety of generalizations including to arithmetic progressions where we discover a curious bias in the secondary term roberts s conjecture has also been proved independently by bhargava shankar and tsimerman in contrast to their work our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms developed by shintani and datskovsky wright

levels of distribution for sieve problems in prehomogeneous vector spaces in our companion paper 28 we developed an efficient algebraic method for computing the fourier transforms of certain functions defined on prehomogeneous vector spaces over finite fields and we carried out these computations in a variety of cases here we develop a method based on fourier analysis and algebraic geometry which exploits these fourier transform formulas to yield level of distribution results in the sense of analytic number theory such results are of the shape typically required for a variety of sieve methods as an example of such an application we prove that there are gg frac x log x quartic fields whose discriminant is squarefree bounded above by x and has at most eight prime factors

d4 modular forms in this paper we study modular forms on two simply connected groups of type d 4 over q one group g s is a globally split group of type d 4 viewed as the group of isotopies of the split rational octonions the other g c is the isotopy group of the rational nonsplit octonions we study automorphic forms on g s in analogy to the work of gross gan and savin on g 2 namely we study automorphic forms whose component at infinity corresponds to a quaternionic discrete series representation we study automorphic forms on g c using gross s formalism of algebraic modular forms finally we follow work of gan savin gross rallis and others to study an exceptional theta correspondence connecting modular forms on g c and g s this can be thought of as an octonionic generalization of the jacquet langlands correspondence

the length of a shortest geodesic loop at a point in this paper we prove that given a point p m n where m n is a closed riemannian manifold of dimension n the length of a shortest geodesic loop lp m n at this point is bounded above by 2nd where d is the diameter of m n moreover we show that on a closed simply connected riemannian manifold m n with a nontrivial second homotopy group there either exist at least three geodesic loops of length less than or equal to 2d at each point of m n or the length of a shortest closed geodesic on m n is bounded from above by 4d

a modular compactification of m_ 1 n from a_ infty structures we show that a certain moduli space of minimal a structures coincides with the modular compactification m1 n n 1 of m1 n constructed by smyth in 22 in addition we describe these moduli spaces and the universal curves over them by explicit equations prove that they are normal and gorenstein show that their picard groups have no torsion and that they have rational singularities if and only if n 11

orbit closure diagram for the space of quadruples of quinary alternating forms abstract we construct the hasse diagram for the closure ordering of the orbits of the irreducible reduced prehomogeneous vector space of quadruples of quinary alternating forms in five variables

higher composition laws ii on cubic analogues of gauss composition in our first article 2 we developed a new view of gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms moreover we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number fields while the spaces underlying those composition laws were closely related to certain exceptional lie groups in this paper our aim is to develop analogous laws of composition on certain spaces of forms so that the resulting groups yield information on the class groups of orders in cubic fields that is we wish to obtain genuine cubic analogues of gauss composition the fundamental object in our treatment of quadratic composition 2 was the space of 2 x 2 x 2 cubes of integers in particular gauss composition arose from the three different ways of slicing a cube a into two 2 x 2 matrices mi ni i 1 2 3 each such pair mi ni gives rise to a binary quadratic form qa x y qi x y defined by qi x y det mix niy the cube law of 2 declares that as a ranges over all cubes the sum of qi q2 q3 is zero it was shown in 2 that the cube law gives a law of addition on binary quadratic forms that is equivalent to gauss composition various other invariant theoretic constructions using the space of 2 x 2 x 2 cubes led to several new composition laws on other spaces of forms furthermore we showed that each of these composition laws gave rise to groups that are closely related to the class groups of orders in quadratic fields based on the quadratic case described above our first inclination for the cubic case might be to examine 3 x 3 x 3 cubes of integers a 3 x 3 x 3 cube c can be sliced in three different ways into three 3 x 3 matrices li mi ni i 1 2 3 we may therefore obtain from c three ternary cubic forms fl x y z f2 x y z f3 x y z defined by

how many rational points does a random curve have a large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties for an elliptic curve over q the set of rational points forms a finitely generated abelian group the ranks of these groups when ranging over all elliptic curves are conjectured to be evenly distributed between rank 0 and rank 1 with higher ranks being negligible we will describe these conjectures and discuss some results on bounds for average rank highlighting the recent work of bhargava and shankar

quotients of some prehomogeneous vector spaces abstract we give quotients v g spec c v g for some prehomogeneous vector spaces g h v

higher composition laws iv the parametrization of quintic rings rst three parts of this series we considered quadratic cubic and quartic rings i e rings free of ranks 2 3 and 4 over z respectively and found that various algebraic structures involving these rings could be completely parametrized by the integer orbits of an appropriate group representation on a vector space these orbit results are summarized in table 1 in particular the theories behind the parametrizations of quadratic cubic and quartic rings noted in items 2 9 and 13 of table 1 were seen to closely parallel the classical developments of the solutions to the quadratic cubic and quartic equations respectively despite the quintic having been shown to be unsolvable nearly two centuries ago by abel it turns out there still remains much to be said regarding the integral theory of the quintic although a solution naturally still is not possible we show in this article that it is nevertheless possible to completely parametrize quintic rings indeed a theory just as complete as in the quadratic cubic and quartic cases exists also in the case of the quintic in fact we present here a unied theory of ring parametrizations which includes the cases n 2