We show that there are no non-trivial stratified bundles over a smooth simply connected quasi-projective variety over an algebraic closure of a finite field if the variety admits a normal projective compactification with boundary locus of codimension greater than or equal to $2$.

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$ -sentence having a unique model of cardinality $\aleph _{1}$ is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$ -sentence has a unique model in every infinite cardinality. In the process, we prove a new characterisation of the special points on any Shimura variety.

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

We solve the Diophantine equation $Y^{2}=X^{3}+k$ for all nonzero integers $k$ with $|k|\leqslant 10^{7}$ . Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

Consider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p ≥ 5 and with complex multiplication by the full ring of integers of K. In this paper, we construct p-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

We prove an analog of the Yomdin–Gromov lemma for $p$-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C}(\!(t)\!)$, in analogy to results by Pila and Wilkie, and by Bombieri and Pila, respectively. Along the way we prove, for definable functions in a general context of non-Archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.

We prove that formal Fourier Jacobi expansions of degree two are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension two, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree two. Combining both results enables us to determine relations of special cycles in the second Chow group.

Van Wamelen [Math. Comp. 68 (1999) no. 225, 307–320] lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over  $\mathbf{Q}$ , and the generic case of a non-Galois quartic CM-field did not feature in this list. The reason is that the field of definition in that case always contains the real quadratic subfield of the reflex field.

We extend Van Wamelen’s list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest ‘generic’ examples of CM curves of genus two.

We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.

We study elliptic curves over quadratic fields with isogenies of certain degrees. Let $n$ be a positive integer such that the modular curve $X_{0}(n)$ is hyperelliptic of genus ${\geqslant}2$ and such that its Jacobian has rank  $0$ over  $\mathbb{Q}$ . We determine all points of $X_{0}(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to $\overline{\mathbb{Q}}$ -isomorphism, every elliptic curve over a quadratic field  $K$ admitting an $n$ -isogeny is $d$ -isogenous, for some $d\mid n$ , to the twist of its Galois conjugate by a quadratic extension $L$ of  $K$ . We determine $d$ and  $L$ explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to $\overline{\mathbb{Q}}$ -isomorphism, all elliptic curves with $n$ -isogenies over quadratic fields are in fact $\mathbb{Q}$ -curves.

We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.

Suppose E is an elliptic curve over $\Bbb Q$, and p>3 is a split multiplicative prime for E. Let q ≠ p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields $K\subset \Bbb Q\big(\mu_{{q^{\infty}}},\;\!^{q^{\infty}\!\!\!\!}\sqrt{m}\big)$ such that p remains inert in $K\cap\Bbb Q(\mu_{{q^{\infty}}})^+$. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.

We study the growth of $\unicode[STIX]{x0428}$ and $p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive ${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are $p$-adic and $l$-adic Lie extensions for $l\neq p$, in particular cyclotomic and other $\mathbb{Z}_{l}$-extensions.

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

In this paper we examine solutions in the Gaussian integers to the Diophantine equation $ax^{4}+by^{4}=cz^{2}$ for different choices of $a,b$ and $c$. Elliptic curve methods are used to show that these equations have a finite number of solutions or have no solution.

In a recent important paper, Hoffstein and Hulse [Multiple Dirichlet series and shifted convolutions, arXiv:1110.4868v2] generalized the notion of Rankin–Selberg convolution $L$-functions by defining shifted convolution$L$-functions. We investigate symmetrized versions of their functions, and we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and “mixed mock modular” forms.

We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. Furthermore, we find a necessary and sufficient condition for a meromorphic Siegel modular function of degree g to have neither a zero nor a pole on a certain subset of the Siegel upper half-space .

We discuss heuristic asymptotic formulae for the number of isogeny classes of pairing-friendly abelian varieties of fixed dimension $g\geqslant 2$ over prime finite fields. In each formula, the embedding degree $k\geqslant 2$ is fixed and the rho-value is bounded above by a fixed real ${\it\rho}_{0}>1$ . The first formula involves families of ordinary abelian varieties whose endomorphism ring contains an order in a fixed CM-field $K$ of degree $g$ and generalizes previous work of the first author when $g=1$ . It suggests that, when ${\it\rho}_{0}<g$ , there are only finitely many such isogeny classes. On the other hand, there should be infinitely many such isogeny classes when ${\it\rho}_{0}>g$ . The second formula involves families whose endomorphism ring contains an order in a fixed totally real field $K_{0}^{+}$ of degree $g$ . It suggests that, when ${\it\rho}_{0}>2g/(g+2)$ (and in particular when ${\it\rho}_{0}>1$ if $g=2$ ), there are infinitely many isogeny classes of $g$ -dimensional abelian varieties over prime fields whose endomorphism ring contains an order of $K_{0}^{+}$ . We also discuss the impact that polynomial families of pairing-friendly abelian varieties has on our heuristics, and review the known cases where they are expected to provide more isogeny classes than predicted by our heuristic formulae.

In this article we explain how the results in our previous article on ‘algebraic Hecke characters and compatible systems of mod $p$ Galois representations over global fields’ allow one to attach a Hecke character to every cuspidal Drinfeld modular eigenform from its associated crystal that was constructed in earlier work of the author. On the technical side, we prove along the way a number of results on endomorphism rings of ${\it\tau}$-sheaves and crystals. These are needed to exhibit the close relation between Hecke operators as endomorphisms of crystals on the one side and Frobenius automorphisms acting on étale sheaves associated to crystals on the other. We also present some partial results on the ramification of Hecke characters associated to Drinfeld modular eigenforms. An important phenomenon absent from the case of classical modular forms is that ramification can also result from places of modular curves of good but non-ordinary reduction. In an appendix, jointly with Centeleghe we prove some basic results on $p$-adic Galois representations attached to $\text{GL}_{2}$-type cuspidal automorphic forms over global fields of characteristic $p$.

Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf{Q}$ . The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let $p$ be an odd prime split in $K/\mathbf{Q}$ . We prove the non-triviality of the $p$ -adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf{Z}_{p}$ -anticyclotomic extension of  $K$ . The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the $\mathbf{Z}_{p}$ -anticyclotomic extension of  $K$ .