We prove a conjecture of Pappas and Rapoport for all Shimura varieties of abelian type with parahoric level structure when $p>2$ by showing that the Kisin–Pappas–Zhou integral models of Shimura varieties of abelian type are canonical. In particular, this shows that these models are independent of the choices made during their construction, and that they satisfy functoriality with respect to morphisms of Shimura data.

We prove new fundamental lemma and arithmetic fundamental lemma identities for general linear groups over quaternion division algebras. In particular, we verify the transfer conjecture and the arithmetic transfer conjecture from Li and Mihatsch (2023, Preprint, arXiv:2307.11716) in cases of Hasse invariant $1/2$.

Let X be a smooth projective variety over a complete discretely valued field of mixed characteristic. We solve non-Archimedean Monge–Ampère equations on X assuming resolution and embedded resolution of singularities. We follow the variational approach of Boucksom, Favre, and Jonsson proving the continuity of the plurisubharmonic envelope of a continuous metric on an ample line bundle on X. We replace the use of multiplier ideals in equicharacteristic zero by the use of perturbation friendly test ideals introduced by Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron, and Witaszek building upon previous constructions by Hacon, Lamarche, and Schwede.

We consider integral models of Hilbert modular varieties with Iwahori level structure at primes over p, first proving a Kodaira–Spencer isomorphism that gives a concise description of their dualizing sheaves. We then analyze fibres of the degeneracy maps to Hilbert modular varieties of level prime to p and deduce the vanishing of higher direct images of structure and dualizing sheaves, generalizing prior work with Kassaei and Sasaki (for p unramified in the totally real field F). We apply the vanishing results to prove flatness of the finite morphisms in the resulting Stein factorizations, and combine them with the Kodaira–Spencer isomorphism to simplify and generalize the construction of Hecke operators at primes over p on Hilbert modular forms (integrally and mod p).

We examine the maximum dimension of a linear system of plane cubic curves whose $\mathbb {F}_q$-members are all geometrically irreducible. Computational evidence suggests that such a system has a maximum (projective) dimension of $3$. As a step towards the conjecture, we prove that there exists a three-dimensional linear system $\mathcal {L}$ with at most one geometrically reducible $\mathbb {F}_q$-member.

A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb {Z}[t]$ and which number fields K the Hasse principle holds for the affine equation $f(t) = \mathbf {N}_{K/\mathbb {Q}}(\mathbf {x}) \neq 0$. Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving [27], which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.

We explore generalizations of the p-adic Simpson correspondence on smooth proper rigid spaces to principal bundles under rigid group varieties G. For commutative G, we prove that such a correspondence exists if and only if the Lie group logarithm is surjective. Second, we treat the case of general G on ordinary abelian varieties, in which case we prove a generalization of Faltings’ “small” correspondence to general rigid groups. On abeloid varieties, we also prove an analog of the classical Corlette–Simpson correspondence for principal bundles under linear algebraic groups.

We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the Generalized Riemann Hypothesis, we show that the average analytic rank of isomorphism classes of elliptic curves over K is bounded above by $(9\deg (K)+1)/2$, when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.

Building upon the classification by Lacini, we determine the isomorphism classes of log del Pezzo surfaces of rank one over an algebraically closed field of characteristic five either which are not log liftable over the ring of Witt vectors or whose singularities are not feasible in characteristic zero. We also show that such a surface is always constructed from the Du Val del Pezzo surface of Dynkin type $2[2^4]$. Furthermore, We show that the Kawamata–Viehweg vanishing theorem for ample $\mathbb {Z}$-Weil divisors holds for log del Pezzo surfaces of rank one in characteristic five if those singularities are feasible in characteristic zero.

We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over $\mathbb{F}_q$. Among other results, this allows us to prove that the $\mathbb{Q}$-vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.

We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$. We apply our methods to all 28 Atkin–Lehner quotients of $X_0(N)$ of genus $2$, all 97 genus $2$ curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least $2$. We also give an example where we verify that the order of the Tate–Shafarevich group is $7^2$ and agrees with the order predicted by the BSD Conjecture.

We investigate and compare applications of the Zilber–Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way, we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu’s immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion.

Let X be a smooth projective variety defined over a number field K. We give an upper bound for the generalised greatest common divisor of a point $x\in X$ with respect to an irreducible subvariety $Y\subseteq X$ also defined over K. To prove the result, we establish a rather uniform Riemann–Roch-type inequality.

In this paper, we compare the $\mathbb J$-stratification (or the semi-module stratification) and the Ekedahl–Oort stratification of affine Deligne–Lusztig varieties in the superbasic case. In particular, we classify the cases where the $\mathbb J$-stratification gives a refinement of the Ekedahl–Oort stratification, which include many interesting cases such that the affine Deligne-Lusztig variety admits a simple geometric structure.

Let G be a semiabelian variety defined over a finite subfield of an algebraically closed field K of prime characteristic. We describe the intersection of a subvariety X of G with a finitely generated subgroup of $G(K)$.

Given a self-morphism $\phi$ on a projective variety defined over a number field k, we prove two results which bound the largest iterate of $\phi$ whose evaluation at P is quasi-integral with respect to a divisor D, uniformly across P defined over a field of bounded degree over k. The first result applies when the pullback of D by some iterate of $\phi$ breaks up into enough irreducible components which are numerical multiples of each other. The proof uses Le’s algebraic-point version of a result of Ji–Yan–Yu, which is based on Schmidt subspace theorem. The second result applies more generally but relies on a deep conjecture by Vojta for algebraic points. The second result is an extension of a recent result of Matsuzawa, based on the theory of asymptotic multiplicity. Both results are generalisations of Hsia–Silverman, which treated the case of morphisms on ${\mathbb{P}}^1$.

We give a precise classification, in terms of Shimura data, of all $1$-dimensional Shimura subvarieties of a moduli space of polarized abelian varieties.

The blow-up of the anticanonical base point on a del Pezzo surface S of degree 1 gives rise to a rational elliptic surface $\mathscr {E}$ with only irreducible fibers. The sections of minimal height of $\mathscr {E}$ are in correspondence with the $240$ exceptional curves on S. A natural question arises when studying the configuration of these curves: if a point on S is contained in “many” exceptional curves, is it torsion on its fiber on $\mathscr {E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$, where there are 56 exceptional curves, that if “many” equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if ‘many’ equals $9$ or more. Moreover, we give counterexamples where a non-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.

This paper contains a method to prove the existence of smooth curves in positive characteristic whose Jacobians have unusual Newton polygons. Using this method, I give a new proof that there exist supersingular curves of genus $4$ in every prime characteristic. More generally, the main result of the paper is that, for every $g \geq 4$ and prime p, every Newton polygon whose p-rank is at least $g-4$ occurs for a smooth curve of genus g in characteristic p. In addition, this method resolves some cases of Oort’s conjecture about Newton polygons of curves.