We calculate the orbifold Euler characteristics of all the degree d fine universal compactified Jacobians over the moduli space of stable curves of genus g with n marked points, as defined by Pagani and Tommasi. We show that this orbifold Euler characteristic agrees with the Euler characteristic of $\overline{\mathcal{M}}_{0, 2g+n}$ up to a combinatorial factor, and in particular, is independent of the degree d and the choice of degree d fine compactified universal Jacobian.

We study the rationality properties of the moduli space ${\mathcal{A}}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular, we show that any principally polarised abelian 3-fold over ${\mathbb{F}}_p$ may be lifted to an abelian variety over $\mathbb{Q}$. This is a phenomenon of low dimension: assuming the Bombieri–Lang conjecture, we also show that this is not the case for abelian varieties of dimension at least 7. Concerning moduli spaces, we show that ${\mathcal{A}}_g$ is unirational over $\mathbb{Q}$ for $g\le 5$ and stably rational for $g=3$. This also allows us to make unconditional one of the results of Masser and Zannier about the existence of abelian varieties over $\mathbb{Q}$ that are not isogenous to Jacobians.

We introduce the abstract notion of a smoothable fine compactified Jacobian of a nodal curve, and of a family of nodal curves whose general element is smooth. Then we introduce the combinatorial notion of a stability assignment for line bundles and their degenerations.

We prove that smoothable fine compactified Jacobians are in bijection with these stability assignments.

We then turn our attention to fine compactified universal Jacobians – that is, fine compactified Jacobians for the moduli space $\overline {\mathcal {M}}_g$ of stable curves (without marked points). We prove that every fine compactified universal Jacobian is isomorphic to the one first constructed by Caporaso, Pandharipande and Simpson in the nineties. In particular, without marked points, there exists no fine compactified universal Jacobian unless $\gcd (d+1-g, 2g-2)=1$.

We consider cyclic unramified coverings of degree d of irreducible complex smooth genus $2$ curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order d. The rich geometry of the associated Prym map has been studied in several papers, and the cases $d=2, 3, 5, 7$ are quite well understood. Nevertheless, very little is known for higher values of d. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for $d\ge 11$ prime such that $\frac {d-1}2$ is also prime. We use results of arithmetic nature on $GL_2$-type abelian varieties combined with theta-duality techniques.

We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups.

We study the p-rank stratification of the moduli space of cyclic degree $\ell $ covers of the projective line in characteristic p for distinct primes p and $\ell $. The main result is about the intersection of the p-rank $0$ stratum with the boundary of the moduli space of curves. When $\ell =3$ and $p \equiv 2 \bmod 3$ is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type $(r,s)$, and p-rank f satisfying the clear necessary conditions.

We apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti–Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl–Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

Modular curves like X0(N) and X1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here by Xnsp(p) and Xnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures of p-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc. (3) 77(1) (1998), 1–38; J. Algebra 231(1) (2000), 414–448).

Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$ , there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level- $m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes.

We construct one-dimensional families of Abelian surfaces with quaternionic multiplication, which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space that parametrizes Abelian surfaces with real multiplication by $\mathbf{Z}\left[ \sqrt{2} \right]$ .

We consider the distribution of $p$ -power group schemes among the torsion of abelian varieties over finite fields of characteristic $p$ , as follows. Fix natural numbers $g$ and $n$ , and let ${\it\xi}$ be a non-supersingular principally quasipolarized Barsotti–Tate group of level $n$ . We classify the $\mathbb{F}_{q}$ -rational forms ${\it\xi}^{{\it\alpha}}$ of ${\it\xi}$ . Among all principally polarized abelian varieties $X/\mathbb{F}_{q}$ of dimension $g$ with $X[p^{n}]_{\bar{\mathbb{F}}_{q}}\cong {\it\xi}_{\bar{\mathbb{F}}_{q}}$ , we compute the frequency with which $X[p^{n}]\cong {\it\xi}^{{\it\alpha}}$ . The error in our estimate is bounded by $D/\sqrt{q}$ , where $D$ depends on $g$ , $n$ , and $p$ , but not on ${\it\xi}$ .

We prove the congruence relation for the $\bmod -p$ reduction of Shimura varieties associated with a unitary similitude group $GU(n\,-\,1,\,1)$ over $\mathbb{Q}$ when $p$ is inert and $n$ odd. The case when $n$ is even was obtained by T. Wedhorn and O. Bültel, as a special case of a result of B. Moonen, when the $\mu$ –ordinary locus of the $p$ –isogeny space is dense. This condition fails in our case. We show that every supersingular irreducible component of the special fiber of $p-I\text{sog}$ is annihilated by a degree one polynomial in the Frobenius element $F$ , which implies the congruence relation.

We illustrate the theory of log abelian varieties and their moduli in the case of log elliptic curves.

In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $\text{GU}\left( 1,\,s \right)$ in the case of an inert prime $p$ . Using Dieudonné theory we define a stratification of the corresponding moduli space of $p$ -divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat–Tits building of a unitary group.

In the case of $\text{GU}\left( 1,\,2 \right)$ , we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

We investigate the special fibres of Siegel modular varieties with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz–Rapoport (KR) stratification; one would like to understand how these stratifications are related to each other. We give a simple description of all KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne–Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties.

We develop the algebraic theory of log abelian varieties. This is Part II of our series of papers on log abelian varieties, and is an algebraic counterpart of the previous Part I ([6]), where we developed the analytic theory of log abelian varieties.

We determine the intersection theory on the Igusa compactification and the second Voronoi compactification of ${\cal A}_4$.

In this paper we study the reduction to characteristic $p$ of the Shimura variety associated to a unitary group which has signature $(n-1,1)$ at its real place. We describe the Newton polygon, the Ekedahl–Oort, and the final stratification. In addition we examine the moduli space of $p$-isogenies using a variant of the local model for Shimura varieties. We apply our results to obtain a proof of the Eichler–Shimura congruence relation for the case that $n$ is even.

in this article we compare different conditions on abelian schemes with real multiplication which occur in the integral models of the hilbert–blumenthal shimura variety considered by rapoport, deligne, pappas and kottwitz. we show that the models studied by deligne/pappas and kottwitz are isomorphic over $\mathrm{spec}\mathbb{z}_{(p)}$. we also examine the associated local models and prove that they are equal.

Some smooth Calabi–Yau threefolds in characteristic two and three that do not lift to characteristic zero are constructed. These threefolds are pencils of supersingular K3-surfaces. The construction depends on Moret-Bailly's pencil of abelian surfaces and Katsura's analysis of generalized Kummer surfaces. The threefold in characteristic two turns out to be nonrigid.