We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double covers of abelian varieties and reduce the Shafarevich conjecture for hypersurfaces to the case of hypersurfaces of high dimension. These are special cases of a general setup for integral points on moduli stacks of cyclic covers, and our arithmetic results are achieved via a version of the Chevalley–Weil theorem for stacks.

We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation $A^2-DB^2=1$ , with $A,B,D\in \mathbb {C}[t]$ and certain ramified covers $\mathbb {P}^1\to \mathbb {P}^1$ arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of André, Corvaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomials D that fit in a Pell equation inside the space of polynomials of fixed even degree. Moreover, Riemann existence theorem associates to the abovementioned covers certain permutation representations: We are able to characterize the representations corresponding to ‘primitive’ solutions of the Pell equation or to powers of solutions of lower degree and give a combinatorial description of these representations when D has degree 4. In turn, this characterization gives back some precise information about the rational values of the Betti map.

Let $p$ be a prime number. Kęstutis Česnavičius proved that for an abelian variety $A$ over a global field $K$, the $p$-Selmer group $\mathrm {Sel}_{p}(A/L)$ grows unboundedly when $L$ ranges over the $(\mathbb {Z}/p\mathbb {Z})$-extensions of $K$. Moreover, he raised a further problem: is $\dim _{\mathbb {F}_{p}} \text{III} (A/L) [p]$ also unbounded under the above conditions? In this paper, we give a positive answer to this problem in the case $p \neq \mathrm {char}\,K$. As an application, this result enables us to generalize the work of Clark, Sharif and Creutz on the growth of potential $\text{III}$ in cyclic extensions. We also answer a problem proposed by Lim and Murty concerning the growth of the fine Tate–Shafarevich groups.

We define cohomological complexes of locally compact abelian groups associated with varieties over p-adic fields and prove a duality theorem under some assumption. Our duality takes the form of Pontryagin duality between locally compact motivic cohomology groups.

Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline {K}}$ has infinitely many rational curves or X has infinitely many unirational specialisations.

Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.

For a smooth rigid space X over a perfectoid field extension K of $\mathbb {Q}_p$ , we investigate how the v-Picard group of the associated diamond $X^{\diamondsuit }$ differs from the analytic Picard group of X. To this end, we construct a left-exact ‘Hodge–Tate logarithm’ sequence

$$\begin{align*}0\to \operatorname{Pic}_{\mathrm{an}}(X)\to \operatorname{Pic}_v(X^{\diamondsuit})\to H^0(X,\Omega_X^1)\{-1\}. \end{align*}$$

We deduce some analyticity criteria which have applications to p-adic modular forms. For algebraically closed K, we show that the sequence is also right-exact if X is proper or one-dimensional. In contrast, we show that, for the affine space $\mathbb {A}^n$ , the image of the Hodge–Tate logarithm consists precisely of the closed differentials. It follows that, up to a splitting, v-line bundles may be interpreted as Higgs bundles. For proper X, we use this to construct the p-adic Simpson correspondence of rank one.

Given a singular modulus $j_0$ and a set of rational primes S, we study the problem of effectively determining the set of singular moduli j such that $j-j_0$ is an S-unit. For every $j_0 \neq 0$ , we provide an effective way of finding this set for infinitely many choices of S. The same is true if $j_0=0$ and we assume the Generalised Riemann Hypothesis. Certain numerical experiments will also lead to the formulation of a “uniformity conjecture” for singular S-units.

Let E be an elliptic curve with positive rank over a number field K and let p be an odd prime number. Let $K_{\operatorname {cyc}}$ be the cyclotomic $\mathbb {Z}_p$ -extension of K and $K_n$ its nth layer. The Mordell–Weil rank of E is said to be constant in the cyclotomic tower of K if for all n, the rank of $E(K_n)$ is equal to the rank of $E(K)$ . We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in this sense. We then indicate the potential applications to Hilbert’s tenth problem for number rings.

We discuss the $\ell $ -adic case of Mazur’s ‘Program B’ over $\mathbb {Q}$ : the problem of classifying the possible images of $\ell $ -adic Galois representations attached to elliptic curves E over $\mathbb {Q}$ , equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell =2$ and $\ell \ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell = 3, 5, 7, 11$ . For each of these $\ell $ , we compute the directed graph of arithmetically maximal $\ell $ -power level modular curves $X_H$ , compute explicit equations for all but three of them and classify the rational points on all of them except $X_{\mathrm {ns}}^{+}(N)$ , for $N = 27, 25, 49, 121$ and two-level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$ .

Aside from the $\ell $ -adic images that are known to arise for infinitely many ${\overline {\mathbb {Q}}}$ -isomorphism classes of elliptic curves $E/\mathbb {Q}$ , we find only 22 exceptional images that arise for any prime $\ell $ and any $E/\mathbb {Q}$ without complex multiplication; these exceptional images are realised by 20 non-CM rational j-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\mathrm {ns}}^+(\ell )$ with $\ell \ge 19$ , or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell $ -adic images of Galois for any elliptic curve over $\mathbb {Q}$ .

In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on $\Gamma _1(N)$ are of $\operatorname {GL}_2$ -type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$ .

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety X defined over an algebraically closed field K of characteristic $0$ , endowed with a birational self-map $\phi $ of dynamical degree $1$ , we expect that either there exists a nonconstant rational function $f:X\dashrightarrow \mathbb {P} ^1$ such that $f\circ \phi =f$ , or there exists a proper subvariety $Y\subset X$ with the property that, for any invariant proper subvariety $Z\subset X$ , we have that $Z\subseteq Y$ . We prove our conjecture for automorphisms $\phi $ of dynamical degree $1$ of semiabelian varieties X. Moreover, we prove a related result for regular dominant self-maps $\phi $ of semiabelian varieties X: assuming that $\phi $ does not preserve a nonconstant rational function, we have that the dynamical degree of $\phi $ is larger than $1$ if and only if the union of all $\phi $ -invariant proper subvarieties of X is Zariski dense. We give applications of our results to representation-theoretic questions about twisted homogeneous coordinate rings associated with abelian varieties.

Let $p$ be a rational prime, let $F$ denote a finite, unramified extension of ${{\mathbb {Q}}}_p$, let $K$ be the maximal unramified extension of ${{\mathbb {Q}}}_p$, ${{\overline {K}}}$ some fixed algebraic closure of $K$, and ${{\mathbb {C}}}_p$ be the completion of ${{\overline {K}}}$. Let $G_F$ be the absolute Galois group of $F$. Let $A$ be an abelian variety defined over $F$, with good reduction. Classically, the Fontaine integral was seen as a Hodge–Tate comparison morphism, i.e. as a map $\varphi _{A} \otimes 1_{{{\mathbb {C}}}_p}\colon T_p(A)\otimes _{{{\mathbb {Z}}}_p}{{\mathbb {C}}}_p\to \operatorname {Lie}(A)(F)\otimes _F{{\mathbb {C}}}_p(1)$, and as such it is surjective and has a large kernel. This paper starts with the observation that if we do not tensor $T_p(A)$ with ${{\mathbb {C}}}_p$, then the Fontaine integral is often injective. In particular, it is proved that if $T_p(A)^{G_K} = 0$, then $\varphi _A$ is injective. As an application, we extend the Fontaine integral to a perfectoid like universal cover of $A$ and show that if $T_p(A)^{G_K} = 0$, then $A(\overline {K})$ has a type of $p$-adic uniformization, which resembles the classical complex uniformization.

We use the method of Bruinier–Raum to show that symmetric formal Fourier–Jacobi series, in the cases of norm-Euclidean imaginary quadratic fields, are Hermitian modular forms. Consequently, combining a theorem of Yifeng Liu, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension for unitary Shimura varieties defined in these cases.

Let K be a number field. For which primes p does there exist an elliptic curve $E / K$ admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.

Iwasawa theory of elliptic curves over noncommutative $GL(2)$ extension has been a fruitful area of research. Over such a noncommutative p-adic Lie extension, there exists a structure theorem providing the structure of the dual Selmer groups for elliptic curves in terms of reflexive ideals in the Iwasawa algebra. The central object of this article is to study Iwasawa theory over the $PGL(2)$ extension and connect it with Iwasawa theory over the $GL(2)$ extension, deriving consequences to the structure theorem when the reflexive ideal is the augmentation ideal of the center. We also show how the dual Selmer group over the $GL(2)$ extension being torsion is related with that of the $PGL(2)$ extension.

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.

Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently computable formula for the tropical moment of a tropical Jacobian in terms of potential theory on the underlying metric graph. We show that there exists a universal linear relation between the tropical moment, a certain capacity called the tau invariant, and the total length of a metric graph. To put our formula in a broader context, we relate our work to the computation of heights attached to principally polarized abelian varieties.

Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau and Martínez to obtain uniform bounds on the number of maximal torsion cosets in the Manin–Mumford problem across a given isogeny class. For the proofs, we adapt the strategy, due to Lang, Serre, Tate, and Hindry, of using Galois automorphisms that act on the torsion as homotheties to the family setting.

We develop tools for constructing rigid analytic trivializations for Drinfeld modules as infinite products of Frobenius twists of matrices, from which we recover the rigid analytic trivialization given by Pellarin in terms of Anderson generating functions. One advantage is that these infinite products can be obtained from only a finite amount of initial calculation, and consequently we obtain new formulas for periods and quasi-periods, similar to the product expansion of the Carlitz period. We further link to results of Gekeler and Maurischat on the $\infty $ -adic field generated by the period lattice.

Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of $n$ points of an algebraic surface is algebraic at a CM point $\tau$ and rational numbers $z_1$ and $z_2$. Our result gives a refinement of the algebraicity on Betti numbers.

We consider Shimura varieties for orthogonal or spin groups acting on hermitian symmetric domains of type IV. We give regular $p$-adic integral models for these varieties over odd primes $p$ at which the level subgroup is the connected stabilizer of a vertex lattice in the orthogonal space. Our construction is obtained by combining results of Kisin and the first author with an explicit presentation and resolution of a corresponding local model.