Let $\mathscr{X} \rightarrow C$ be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve $C$ in characteristic $p \geq 5$. We prove that the geometric Picard rank jumps at infinitely many closed points of $C$. More generally, suppose that we are given the canonical model of a Shimura variety $\mathcal{S}$ of orthogonal type, associated to a lattice of signature $(b,2)$ that is self-dual at $p$. We prove that any generically ordinary proper curve $C$ in $\mathcal{S}_{\overline{\mathbb{F}}_p}$ intersects special divisors of $\mathcal{S}_{\overline{\mathbb{F}}_p}$ at infinitely many points. As an application, we prove the ordinary Hecke orbit conjecture of Chai--Oort in this setting; that is, we show that ordinary points in $\mathcal{S}_{\overline{\mathbb{F}}_p}$ have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.
We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula. Along the way we prove new results on characteristic polynomials of Frobenius …
We investigate the geometry of correspondences between curves, and prove that correspondences over a non-Archimedean valued field have potentially stable reduction, generalising and strengthening results of Coleman and Liu. This yields a concrete description of the operator on the cohomology of the generic fibres arising from linearisation of the correspondence, via the weight-monodromy filtration and Picard-Lefschetz theory. We explicitly determine stable models of Hecke operators on various quaternionic Shimura curves, and prove a generalisation of the geometric theory of canonical subgroups…
We determine all complex K3 surfaces with Picard rank 20 over Q. Here the Neron-Severi group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We then apply our methods to general singular K3 surfaces, i.e. with Neron-Severi group of rank 20, but not necessarily generated by divisors over Q.
We compute the cohomology with compact supports of a Picard modular surface as a virtual module over the product of the appropriate Galois group and the appropriate Hecke algebra. We use the method developed by Ihara, Langlands, and Kottwitz: comparison of the Grothendieck-Lefschetz formula and the Arthur-Selberg trace formula. Our implementation of this method takes as its starting point the works of Laumon and Morel.
We use methods for computing Picard numbers of reductions of K3 surfaces in order to study the decomposability of Jacobians over number fields and the variance of Mordell-Weil ranks of families of Jacobians over different ground fields. For example, we give examples of surfaces whose Picard numbers jump in rank at all primes of good reduction using Mordell-Weil groups of Jacobians and show that the genus of curves over number fields whose Jacobians are isomorphic to a product of elliptic curves satisfying certain reduction conditions is bounded. The isomorphism result addresses the number fiel…
We provably compute the full set of rational points on 1403 Picard curves defined over $\mathbb{Q}$ with Jacobians of Mordell-Weil rank $1$ using the Chabauty-Coleman method. To carry out this computation, we extend Magma code of Balakrishnan and Tuitman for Coleman integration. The new code computes $p$-adic (Coleman) integrals on curves to points defined over number fields where the prime $p$ splits completely and implements effective Chabauty for curves whose Jacobians have infinite order points that are not the image of a rational point under the Abel-Jacobi map. We discuss several interes…
We give a new general technique for constructing and counting number fields with an ideal class group of nontrivial m-rank. Our results can be viewed as providing a way of specializing the Picard group of a variety V over $\mathbb{Q}$ to obtain class groups for number fields $\mathbb{Q}(P)$, $P\in V(\Qbar)$, for certain families of points P. In particular, we show how the problem of constructing quadratic number fields with a large-rank ideal class group can be reduced to the problem of finding a hyperelliptic curve with a rational Weierstrass point and a large rational torsion subgroup in its…
We calculate the effect of simple Hecke operators on u-expansions of higher rank Drinfeld modular forms, the eigenvalue for the Drinfeld discriminant function $Δ_t$ and show that a certain natural class of Hecke operators is completely multiplicative.
In this article we construct a $p$-adic three dimensional Eigenvariety for the group $U(2,1)(E)$, where $E$ is a quadratic imaginary field and $p$ is inert in $E$. The Eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta-Iovita-Pilloni by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch-Kato conjecture for some Galois characters of $E$, exte…
We test the methods for computing the Picard group of a $K3$ surface in a situation of high rank. The examples chosen are resolutions of quartics in $\bP^3$ having 14 singularities of type $A_1$. Our computations show that the method of R. van Luijk works well when sufficiently large primes are used.
We consider diagonal cubic surfaces defined by an equation of the form ax^3+by^3+cz^3+dt^3 = 0. Numerically, one can find all rational points of height < B for B in the range of up to 100 000, thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the authors. Changing the coefficients one can obtain cubic surfaces with rank of the Picard group varying between 1 and 4. We check that numerical data are compatible with the above conjectures. In a previo…
In the early 1980s, Rohrlich began a study of canonical Hecke characters, which are closely related to the simplest examples of CM elliptic curves. He and Montgomery showed the non-vanishing of the central value when the L-function has an even functional equation, and we now show the non-vanishing of the central derivative when the functional equation is odd. Using the results of Gross-Zagier and Kolyvagin-Logachev, we can apply the non-vanishing to the ranks and Shafarevitch-Tate groups of the Q-curves "A(p)" studied by Gross in his thesis. In particular, their rank is determined by a congrue…
Let $N$ and $p$ be primes such that $p$ divides the numerator of $\frac{N-1}{12}$. In this paper, we study the rank $g_p$ of the completion of the Hecke algebra acting on cuspidal modular forms of weight $2$ and level $Γ_0(N)$ at the $p$-maximal Eisenstein ideal. We give in particular an explicit criterion to know if $g_p \geq 3$, thus answering partially a question of Mazur. In order to study $g_p$, we develop the theory of \textit{higher Eisenstein elements}, and compute the first few such elements in four different Hecke modules. This has applications such as generalizations of the Eichler …
In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Neron-Severi group over an algebraic closure of the base field, is high enough, more structure is known and more can be said. However, until recently not a single K3 surface was known to have geometric Picard number one. We give explicit examples of such surfaces over the rational numbers. This solves an old problem that has been attributed to Mumford. The examples we give also contain infinitely many rational points, thereby answering a question of Swinnerton-Dyer and …
Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight $k$ and level $Γ_0(\ell)$ at the maximal Eisenstein ideal containing $p$. We give a necessary and sufficient condition for the $\mathbb{Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for …
We obtain two-variable Hecke-Rogers identities for three universal mock theta functions. This implies that many of Ramanujan's mock theta functions, including all the third order functions, have a Hecke-Rogers-type double sum representation. We find new generating function identities for the Dyson rank function, the overpartition rank function, the M2-rank function and related spt-crank functions. Results are proved using the theory of basic hypergeometric functions.
We construct a Taylor-Wiles system using a family of $p$-adically completed modular Jacobians over suitable Hecke algebras and prove that certain $p$-adically completed Mordell-Weil groups of these Jacobians is free of finite rank over a Hecke algebra.
We use pseudodeformation theory to study Mazur's Eisenstein ideal. Given prime numbers $N$ and $p>3$, we study the Eisenstein part of the $p$-adic Hecke algebra for $Γ_0(N)$. We compute the rank of this Hecke algebra (and, more generally, its Newton polygon) in terms of Massey products in Galois cohomology, answering a question of Mazur and generalizing a result of Calegari-Emerton. We also also give new proofs of Merel's result on this rank and of Mazur's results on the structure of the Hecke algebra.
We address the problem of identifying a Hecke-Maass cusp form $f$ of full level from the central values of the Rankin-Selberg $L$-functions $L(1/2,f\otimes h)$ where $h$ runs through the set of Hecke-Maass eigenforms of full level. We prove a quantitative result in this direction.
We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N-cusped surface. We also discuss a higher-rank analogue.
We test R. van Luijk's method for computing the Picard group of a $K3$ surface. The examples considered are the resolutions of Kummer quartics in $\bP^3$. Using the theory of abelian varieties, in this case, the Picard group may be computed directly. Our experiments show that the upper bounds provided by R. van Luijk's method are sharp when sufficiently many primes are used. In fact, there are a lot of primes that yield a value close to the exact one. However, for many but not all Kummer surfaces $V$ of Picard rank 18, we have $\smash{\rk\Pic(V_{\bar\bbF_{p}}) \geq 20}$ for a set of primes of …
Given a curve X of the form y^p = h(x) over a number field, one can use descents to obtain explicit bounds on the Mordell-Weil rank of the Jacobian or to prove that the curve has no rational points. We show how, having performed such a descent, one can easily obtain additional information which may rule out the existence of rational divisors on X of degree prime to p. This can yield sharper bounds on the Mordell-Weil rank by demonstrating the existence of nontrivial elements in the Shafarevich-Tate group. As an example we compute the Mordell-Weil rank of the Jacobian of a genus 4 curve over Q …
It is shown that extremal $2$-modular lattices of ranks $32$ and $48$ are generated by their vectors of minimal norm. In the proof, we use certain properties of the difference of normalized Hecke eigenforms. We refer to them as the pseudo-normalized Hecke eigenform, the concept of which is introduced in this paper.
We study the behavior of geometric Picard ranks of K3 surfaces over the rationals under reduction modulo primes. We compute these ranks for reductions of smooth quartic surfaces modulo all primes $p<2^{16}$ in several representative examples and investigate the resulting statistics.
Let $π$ be an irreducible cuspidal automorphic generic representation of $\mathrm{Sp}_{2n}(\mathbb{A})$ and let $χ:F^\times\backslash \mathbb{A}^\times\to \mathbb{C}^\times$ be a unitary idele class character. In this note, we present a Rankin-Selberg integral of Hecke type for the twisted standard partial $L$-function $L^S(s,π\timesχ)$ of degree $2n+1$.
Over a p-adic field of odd residual characteristic, Gan and Savin proved a correspondence between the Bernstein components of the even and odd Weil representations of the metaplectic group and the components of the trivial representation of the equal rank odd orthogonal groups. In this paper, we extend their result to the case of even residual characteristic.
Let $K$ be a non-discrete non-Archimedean local field with residue characteristic $p$. Let $G$ be the group of $K$ rational points of a algebraic connected reductive group defined over $K$. In this article we compute the extensions between characters of affine pro-$p$ Iwahori--Hecke algebra $\mathcal{H}^{\text{aff}}$ over an algebraically closed field $R$ of characteristic $p$. In rank one case we deduce the relation between the blocks and $L$-packets.
