Shimura Curves Research Articles

A p-adic Waldspurger formula

In this article, we study p-adic torus periods for certain p-adic-valued functions on Shimura curves of classical origin. We prove a p-adic Waldspurger formula for these periods as a generalization of recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic p-adic L-function of Rankin–Selberg type. At a character of positive weight, the p-adic L-function interpolates the central critical value of the complex Rankin–Selberg L-function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding p-adic torus period.

The Fermat septic and the Klein quartic as moduli spaces of hypergeometric Jacobians

We study the Schwarz triangle function with the monodromy group $\Delta(7,7,7)$, and we construct its inverse by theta constants. As consequences, we give uniformizations of the Klein quartic curve and the Fermat septic curve as Shimura curves parametrizing Abelian $6$-folds with endomorphisms $\mathbb{Z}[\zeta_7]$.

CM points on Shimura curves and $p$-adic binary quadratic forms

We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion $\mathbb{Q}$-algebra, is in bijection with the set of certain classes of $p$-adic binary quadratic forms, where $p$ is a prime dividing the discriminant of the quaternion algebra. The classes of $p$-adic binary quadratic forms are obtain by the action of a discrete and cocompact subgroup of $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$ arising from the $p$-adic uniformization of the Shimura curve. We finally compute families of $p$-adic binary quadratic forms associated to an infinite family of Shimura curves studied in a previous paper of Amor\'os-Milione. This extends results of Alsina-Bayer to the $p$-adic context.

On multiplicities of Galois representations in cohomology groups of Shimura curves

On multiplicities of Galois representations in cohomology groups of Shimura curves

Overconvergent Eichler–Shimura isomorphisms for quaternionic modular forms over ℚ

In this work, we construct overconvergent Eichler–Shimura isomorphisms over Shimura curves over [Formula: see text]. More precisely, for a prime [Formula: see text] and a wide open disk [Formula: see text] in the weight space, we construct a Hecke–Galois-equivariant morphism from the space of families of overconvergent modular symbols over [Formula: see text] to the space of families of overconvergent modular forms over [Formula: see text]. In addition, for all but finitely many weights [Formula: see text], this morphism provides a description of the finite slope part of the space of overconvergent modular symbols of weight [Formula: see text] in terms of the finite slope part of the space of overconvergent modular forms of weight [Formula: see text]. Moreover, for classical weights these overconvergent isomorphisms are compatible with the classical Eichler–Shimura isomorphism.

An application of a theorem of Emerton to mod p representations of GL 2

Let $p$ be a prime and $L$ be a finite extension of $\mathbb{Q}_p$. We study the ordinary parts of $\mathrm{GL}_2(L)$-representations arised in the mod $p$ cohomology of Shimura curves attached to indefinite division algebras which splits at a finite place above $p$. The main tool of the proof is a theorem of Emerton \cite{Em3}.

The -adic Gross–Zagier formula on Shimura curves

We prove a general formula for the$p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above$p$. The formula is in terms of the cyclotomic derivative of a Rankin–Selberg$p$-adic$L$-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan–Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is$+1$rather than$-1$, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a$p$-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups

Abstract Given a modular form f {{f}} of even weight larger than two and an imaginary quadratic field K {{K}} satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga–Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to f {{f}} enjoying the compatibility properties of an Euler system. Then we use Kolyvagin’s method [21], as adapted by Nekovář [28] to higher weight modular forms, to bound the size of the relevant Selmer group associated to f {{f}} and K {{K}} and prove the finiteness of the (primary part) of the Shafarevich–Tate group, provided that a suitable cohomology class does not vanish.

The Oort conjecture on Shimura curves in the Torelli locus of hyperelliptic curves

Oort has conjectured that there do not exist Shimura varieties of dimension >0 contained generically in the Torelli locus of genus-g curves when g is sufficiently large. In this paper we prove the analogue of this conjecture for Shimura curves with respect to the hyperelliptic Torelli locus of genus g>7.

P‐adic heights of Heegner points and Beilinson–Flach classes

AbstractWe give a new proof of Howard's ‐adic Gross–Zagier formula Compos. Math. 141 (2005) 811–846. MR 2148200 (2006f:11074)], which we extend to the context of indefinite Shimura curves over attached to nonsplit quaternion algebras. This formula relates the cyclotomic derivative of a two‐variable ‐adic ‐function restricted to the anticyclotomic line to the cyclotomic ‐adic heights of Heegner points over the anticyclotomic tower, and our proof, rather than inspired by the influential approaches of Gross–Zagier [Invent. Math. 84 (1986) 225–320. MR 833192 (87j:11057)] and Perrin‐Riou [Invent. Math. 89 (1987) 455–510. MR 903381 (89d:11034)], is via Iwasawa theory, based on the connection between Heegner points, Beilinson–Flach elements, and their explicit reciprocity laws.

RATIONAL POINTS ON SHIMURA CURVES AND THE MANIN OBSTRUCTION

In a previous article, we proved that Shimura curves have no points rational over number fields under a certain assumption. In this article, we give another criterion of the nonexistence of rational points on Shimura curves and obtain new counterexamples to the Hasse principle for Shimura curves. We also prove that such counterexamples obtained from the above results are accounted for by the Manin obstruction.

Height pairings on orthogonal Shimura varieties

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and complex multiplication points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin–Selberg convolution associated with a cusp form of half-integral weight $n/2+1$, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross–Zagier theorem on heights of Heegner points.

8. Overconvergent Modular Forms and Perfectoid Shimura Curves

We give a new construction of overconvergent modular forms of arbitrary weights, defining them in terms of functions on certain affinoid subsets of Scholze's infinite-level modular curve. These affinoid subsets, and a certain canonical coordinate on them, play a role in our construction which is strongly analogous with the role of the upper half-plane and its coordinate `$z$' in the classical analytic theory of modular forms. As one application of these ideas, we define and study an overconvergent Eichler-Shimura map in the context of compact Shimura curves over $\mb{Q}$, proving stronger analogues of results of Andreatta-Iovita-Stevens.

Equations of hyperelliptic Shimura curves

By constructing suitable Borcherds forms on Shimura curves and using Schofer’s formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves $X_{0}^{D}(N)$. As a byproduct, we also address the problem of whether a modular form on Shimura curves $X_{0}^{D}(N)/W_{D,N}$ with a divisor supported on CM divisors can be realized as a Borcherds form, where $X_{0}^{D}(N)/W_{D,N}$ denotes the quotient of $X_{0}^{D}(N)$ by all of the Atkin–Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.

Icosahedral invariants and Shimura curves

Shimura curves are moduli spaces of abelian surfaces with quaternion multiplication. Models of Shimura curves are very important in number theory. Klein's icosahedral invariants $\mathfrak{A},\mathfrak{B}$ and $\mathfrak{C}$ give the Hilbert modular forms for $\sqrt{5}$ via the period mapping for a family of $K3$ surfaces. Using the period mappings for several families of $K3$ surfaces, we obtain explicit models of Shimura curves with small discriminant in the weighted projective space ${\rm Proj} (\mathbb{C}[\mathfrak{A},\mathfrak{B},\mathfrak{C}])$.

Selmer groups and central values of L-functions for modular forms

In this article, we construct an Euler system using CM cycles on Kuga–Sato varieties over Shimura curves and show a relation with the central values of Rankin–Selberg L-functions for elliptic modular forms and ring class characters of an imaginary quadratic field. As an application, we prove that the non-vanishing of the central values of Rankin–Selberg L-functions implies the finiteness of Selmer groups associated to the corresponding Galois representation of modular forms under some assumptions.

$p$-Adic Interpolation of Automorphic Periods for GL$_2$

We give a new and representation theoretic construction of p -adic interpolation series for central values of self-dual Rankin-Selberg L -functions for GL _2 in dihedral towers of CM fields, using expressions of these central values as automorphic periods. The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local. We give the construction here for a cuspidal automorphic representation of GL _2 over a totally real field corresponding to a \mathfrak{p} -ordinary Hilbert modular forms of parallel weight two and trivial character, although a similar approach can be taken in any setting where the underlying GL _2 -representation can be chosen to take values in a discrete valuation ring. A certain choice of vectors allows us to establish a precise interpolation formula thanks to theorems of Martin-Whitehouse and File-Martin-Pitale. Such interpolation formulae had been conjectured by Bertolini-Darmon in antecedent works. Our construction also gives a conceptual framework for the nonvanishing theorems of Cornut-Vatsal in that it describes the underlying theta elements. To highlight this latter point, we describe how the construction extends in the parallel weight two setting to give a p -adic interpolation series for central derivative values when the root number is generically equal to -1 , in which case the formula of Yuan-Zhang-Zhang can be used to give an interpolation formula in terms of heights of CM points on quaternionic Shimura curves.

P-adic modular forms over unitary Shimura curves and local-global compatibility

p-adic modular forms over unitary Shimura curves and local-global compatibility

Overconvergent Modular Forms and Perfectoid Shimura Curves

We give a new construction of overconvergent modular forms of arbitrary weights, defining them in terms of functions on certain affinoid subsets of Scholze's infinite-level modular curve. These affinoid subsets, and a certain canonical coordinate on them, play a role in our construction which is strongly analogous with the role of the upper half-plane and its coordinate `z' in the classical analytic theory of modular forms. As one application of these ideas, we define and study an overconvergent Eichler-Shimura map in the context of compact Shimura curves over $\mathbb{Q}$, proving stronger analogues of results of Andreatta-Iovita-Stevens.

The discriminant 10 Shimura curve and its associated Heun functions

The Shimura curve of discriminant 10 is uniformized by a subgroup of an arithmetic $(2,2,2,3)$ quadrilateral group. We derive the differential structure of the ring of modular forms for the Shimura ...