Shimura Curves Research Articles

Finite polynomial cohomology with coefficients

We introduce a theory of finite polynomial cohomology with coefficients in this paper. We prove several basic properties and introduce an Abel–Jacobi map with coefficients. As applications, we use this cohomology theory to study the arithmetic of compact Shimura curves over \mathbb{Q} and to simplify the proofs of the works of Darmon–Rotger and Bertolini–Darmon–Prasanna.

Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves

AbstractSarnak's conjecture in quantum computing concerns when the groups and over cyclotomic rings with , , are generated by the Clifford‐cyclotomic gate set. We previously settled this using Euler–Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group has only if is not generated by torsion elements. In this paper, we study the corank of these cyclotomic unitary groups in the families and , , by letting them act on Bruhat–Tits trees. The quotients by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for the families and , the corank grows doubly exponentially in as ; it is 0 precisely when , and indeed, the cyclotomic unitary groups are generated by torsion elements (in fact by Clifford‐cyclotomic gates) for these . We give explicit lower bounds for the corank in two different ways. The first is to bound the isotropy subgroups in the action on the tree by explicit cyclotomy. The second is to relate our graphs to Shimura curves over via interchanging local invariants and applying a result of Selberg and Zograf. We show that the cyclotomy arguments give the stronger bounds. In a final section, we execute a program of Sarnak to show that our results for the and families are sufficient to give a second proof of Sarnak's conjecture.

Arithmetic Gan–Gross–Prasad conjecture for RSZ unitary Shimura curves

Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan–Gross–Prasad conjecture via the Gross–Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a variant of the conjecture, using modified PEL type Shimura varieties, which we call RSZ Shimura varieties. We reinterpret the result of Xue in terms of the modified Shimura curves. We then use the reinterpretation to prove a case of the variant of the conjecture. Our result combined with the work of Xue establishes a connection between the variant and the Gross–Zagier formula.

WILES DEFECT OF HECKE ALGEBRAS VIA LOCAL-GLOBAL ARGUMENTS

Abstract In his work on modularity of elliptic curves and Fermat’s last theorem, A. Wiles introduced two measures of congruences between Galois representations and between modular forms. One measure is related to the order of a Selmer group associated to a newform $f \in S_2(\Gamma _0(N))$ (and closely linked to deformations of the Galois representation $\rho _f$ associated to f), whilst the other measure is related to the congruence module associated to f (and is closely linked to Hecke rings and congruences between f and other newforms in $S_2(\Gamma _0(N))$ ). The equality of these two measures led to isomorphisms $R={\mathbf T}$ between deformation rings and Hecke rings (via a numerical criterion for isomorphisms that Wiles proved) and showed these rings to be complete intersections. We continue our study begun in [BKM21] of the Wiles defect of deformation rings and Hecke rings (at a newform f) acting on the cohomology of Shimura curves over ${\mathbf Q}$ : It is defined to be the difference between these two measures of congruences. The Wiles defect thus arises from the failure of the Wiles numerical criterion at an augmentation $\lambda _f:{\mathbf T} \to {\mathcal O}$ . In situations we study here, the Taylor–Wiles–Kisin patching method gives an isomorphism $ R={\mathbf T}$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of the results in [BKM21] that compute the Wiles defect at $\lambda _f: R={\mathbf T} \to {\mathcal O}$ , and explain in an a priori manner why the answer in [BKM21] is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet–Takahashi that computes change of degrees of optimal parametrizations of elliptic curves over ${\mathbf Q}$ by Shimura curves as we vary the Shimura curve. The results we prove are not attainable using only the methods of Ribet–Takahashi.

On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting

Abstract We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form f and an imaginary quadratic field satisfying a “relaxed” Heegner hypothesis. Let Λ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo–Vigni, we construct the Λ-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa–Greenberg main conjecture for the p-adic L-function defined by Magrone.

Points on Shimura curves rational over imaginary quadratic fields in the non-split case

Points on Shimura curves rational over imaginary quadratic fields in the non-split case

Towards a mod-p Lubin–Tate theory for GL2 over totally real fields

In this paper, we show that the conjectural mod [Formula: see text] local Langlands correspondence can be realized in the mod [Formula: see text] cohomology of the Lubin–Tate towers. The proof utilizes a well-known conjecture of Buzzard–Diamond–Jarvis [8, Conjecture 4.9], a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field [Formula: see text] and of mod [Formula: see text] local Langlands correspondence as given by Emerton–Helm [20]. In the case of modular curves, a similar theorem was obtained by Chojecki [13].

Shimura curves and the abc conjecture

In this work we develop a framework that enables the use of Shimura curve parametrizations of elliptic curves to approach the abc conjecture, leading to a number of new unconditional applications over Q and, more generally, totally real number fields.Several results of independent interest are obtained along the way, such as bounds for the Manin constant, a study of the congruence number, extensions of the Ribet-Takahashi formula, and lower bounds for the L2-norm of integral quaternionic modular forms.The methods require a number of tools from Arakelov geometry, analytic number theory, Galois representations, complex-analytic estimates on Shimura curves, automorphic forms, known cases of the Colmez conjecture, and results on generalized Fermat equations.

Gelfand–Kirillov dimension and the p-adic Jacquet–Langlands correspondence

Abstract We bound the Gelfand–Kirillov dimension of unitary Banach space representations of p-adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and p-adic Banach space representations of the group of units of a quaternion algebra over ℚ p {\mathbb{Q}_{p}} appearing in the p-adic Jacquet–Langlands correspondence, deducing finiteness results in favorable cases.

Anticyclotomic main conjecture and the non-triviality of Rankin–Selberg L-values in Hida families

We prove the two-variable anticyclotomic Iwasawa main conjecture for Hida families and discuss its arithmetic application to a definite version of the horizontal non-vanishing conjecture, which is formulated in [LV11]. Our approach is based on the two-variable anticyclotomic control theorem for Selmer groups and the relation between the two-variable anticyclotomic L-function for Hida families built out of p-adic families of Gross points on definite Shimura curves studied in [CL16] and [CKL17] and the self-dual twist of the specialisation to the anticyclotomic line of the three-variable p-adic L-function of Skinner–Urban [SU14].

A genus two arithmetic Siegel-Weil formula on 𝑋₀(𝑁)

We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X 0 ( N ) X_0(N) , for N > 3 N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q q -coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.

Rational points on Atkin–Lehner quotients of geometrically hyperelliptic Shimura curves

Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves X0(D,N). In this paper we compute the Q-rational points on the Atkin–Lehner quotients of these curves using a variety of techniques. We also determine which rational points are CM for many of these curves.

Counting intersection numbers of closed geodesics on Shimura curves

Counting intersection numbers of closed geodesics on Shimura curves

Higher dimensional Shimura varieties in the Prym loci of ramified double covers

AbstractIn this paper, we construct Shimura subvarieties of dimension bigger than one of the moduli space of δ‐polarized abelian varieties of dimension p, which are generically contained in the Prym loci of (ramified) double covers. The idea is to adapt the techniques already used to construct Shimura curves in the Prym loci to the higher dimensional case, namely, to use families of Galois covers of . The case of abelian covers is treated in detail, since in this case, it is possible to make explicit computations that allow to verify a sufficient condition for such a family to yield a Shimura subvariety of .

Arithmetic level raising on triple product of Shimura curves and Gross–Schoen Diagonal cycles, I : Ramified case

In this article we study the Gross-Schoen diagonal cycle on a triple product of Shimura curves at a place of bad reduction. We relate the image of the diagonal cycle under the Abel-Jacobi map to certain period integral that governs the central critical value of the Garrett-Rankin type triple product $L$-function via level raising congruences. As an application we prove certain rank $0$ case of the Bloch-Kato conjecture for the symmetric cube motive of a weight $2$ modular form.

$p$-adic equidistribution of CM points

Let X be a modular curve and consider a sequence of Galois orbits of CM points in X , whose p -conductors tend to infinity. Its equidistribution properties in X(\mathbf{C}) and in the reductions of X modulo primes different from p are well understood. We study the equidistribution problem in the Berkovich analytification X_{p}^{\operatorname{an}} of X_{\operatorname{Q}_{p}} . We partition the set of CM points of sufficiently high conductor in X_{\operatorname{Q}_{p}} into finitely many explicit basins B_{V} , indexed by the irreducible components V of the \bmod\text{-}p reduction of the canonical model of X . We prove that a sequence z_{n} of local Galois orbits of CM points with p -conductor going to infinity has a limit in X_{p}^{\operatorname{an}} if and only if it is eventually supported in a single basin B_{V} . If so, the limit is the unique point of X_{p}^{\operatorname{an}} whose mod- p reduction is the generic point of V . The result is proved in the more general setting of Shimura curves over totally real fields. The proof combines Gross's theory of quasi-canonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin–Tate space.

On the anticyclotomic Iwasawa main conjecture for Hilbert modular forms of parallel weights

In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove an one-sided divisibility result toward the Iwasawa main conjecture in this setting. The proof relies on the first and second reciprocity laws relating theta elements to Heegner point Euler systems on Shimura curves. As a by-product we also prove a result towards the rank 0 case of certain Bloch–Kato conjecture and a parity conjecture.

Hodge cycles and the Leray filtration

This is loosely a continuation of the author's previous paper arXiv:1802.09496. In the first part, given a fibered variety, we pull back the Leray filtration to the Chow group, and use this to give some criteria for the Hodge and Tate conjectures to hold for such varieties. In the second part, we show that the Hodge conjecture holds for a good desingularization of a self fibre product of a non-isotrivial elliptic surface under appropriate conditions. We also show that the Hodge and Tate conjectures hold for natural families of abelian varieties parameterized by certain Shimura curves. This uses Zucker's description of the mixed Hodge structure on the cohomology of a variation of Hodge structures on a curve, along with appropriate "vanishing" theorems.

Special cycles on unitary Shimura curves at ramified primes

In this paper, we study special cycles on the Krämer model of \(\mathrm {U}(1,1)(F/F_0)\)-Rapoport–Zink spaces where \(F/F_0\) is a ramified quadratic extension of p-adic number fields with the assumption that the 2-dimensional hermitian space of special quasi-homomorphisms is anisotropic. We write down the decomposition of these special cycles and prove a version of Kudla–Rapoport conjecture in this case. We then apply the local results to compute the intersection numbers of special cycles on unitary Shimura curves and relate these intersection numbers to Fourier coefficients of central derivatives of certain Eisenstein series.

A p-adic approach to singular moduli on Shimura curves

A p-adic approach to singular moduli on Shimura curves