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

For each integer ℓ ≥ 1, 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-D 4 ). 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-D 4 ) and quintic fields with chosen splitting types at a finite set of primes.for every ε > 0. (Throughout, ε > 0 is allowed to be arbitrarily small (possibly taking a different value in different occurrences), and A ≪ B indicates that |A| ≤ cB for an implied constant c, which we allow in any instance to depend on ℓ, d, ε.)It is conjectured thatfor every ε > 0, but improving on the trivial bound (1.1) has proved difficult. (Impetus for this conjecture may be found in Duke [Duk98], Zhang [Zha05, page 10], and Brumer and Silverman [BS96, "Question CL(ℓ, d)"].) For K quadratic, Gauss's genus theory [Gau01] implies (1.2) in the case ℓ = 2. Recently, [BST + 17] obtained nontrivial upper bounds for 2-torsion in fields of degree d for all d ≥ 3, proving |Cl K [2]| ≪ D 0.2784...+ε K for d = 3, 4 and |Cl K [2]| ≪ D 1/2−1/2d+ε K for d ≥ 5. For ℓ = 3, after initial incremental improvement in [HV06], [Pie05], [Pie06] over the trivial bound (1.1) for quadratic fields, Ellenberg and Venkatesh proved [EV07, Prop. 3.4, Cor. 3.7] that (1.3) |Cl K [3]| ≪ D 1/3+ε K 2010 Mathematics Subject Classification. 11R29, 11N36 11R45 .

ABSTRACT. 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.

For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime ℘ of K, we determine the probability that ℘ splits into r primes in a random G-extension of K that is unramified at ℘. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev's density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant.

The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied composition of binary quadratic forms, which we now understand via Dedekind's association of ideal classes of quadratic rings to integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic rings are parameterized by equivalence classes of integral binary cubic forms. Birch, Merriman, Nakagawa, del 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 parameterized by binary n-ic forms, for all n. The algebraic data associated to an integral binary n-ic form includes a ring isomorphic to Z n as a 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 parameterizations 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 parameterize them and a simple construction of the form from an appropriate ring and ideal.