scholarly journals Twisted Gross–Zagier Theorems

2009 ◽  
Vol 61 (4) ◽  
pp. 828-887 ◽  
Author(s):  
Benjamin Howard
Keyword(s):  
Complex Multiplication ◽  
Modular Curves ◽  
Shimura Curve ◽  
Modular Curve ◽  
Hida Theory ◽  
Companion Article ◽  
Cm Points ◽  
Hida Families ◽  
Derivatives Of

Abstract.The theorems of Gross–Zagier and Zhang relate the Néron–Tate heights of complex multiplication points on the modular curve X0(N) (and on Shimura curve analogues) with the central derivatives of automorphic L-function. We extend these results to include certain CM points on modular curves of the form X (Ⲅ0(M ) ∩ Ⲅ1(S)) (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article “Central derivatives of L -functions in Hida families”,Math. Ann. 399(2007), 803–818.

scholarly journals Singular Moduli of Shimura Curves

2011 ◽  
Vol 63 (4) ◽  
pp. 826-861 ◽  
Author(s):  
Eric Errthum
Keyword(s):  
Modular Forms ◽  
Complex Multiplication ◽  
Shimura Curves ◽  
Algebraic Integers ◽  
Quaternion Algebras ◽  
Shimura Curve ◽  
Modular Curve ◽  
Singular Moduli ◽  
Cm Points ◽  
Vector Valued

Abstract The j-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over Q. This paper shows that the coordinate maps given by N. Elkies for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points with arbitrarily large negative discriminant.

Introduction and Statement of Main Results

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Abelian Varieties ◽  
Main Idea ◽  
Shimura Curves ◽  
Modular Curves ◽  
Heegner Points ◽  
Cm Points ◽  
Totally Real ◽  
Totally Real Fields ◽  
Derivatives Of ◽  
Original Formula

This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves over totally real fields. It begins with a discussion of the original formula proved by Benedict Gross and Don Zagier, which relates the Néeron–Tate heights of Heegner points on X⁰(N) to the central derivatives of some Rankin–Selberg L-functions under the Heegner condition. In particular, it considers the Gross–Zagier formula on modular curves and abelian varieties parametrized by Shimura curves. It then decribes CM points and the Waldspurger formula before concluding with an outline of our proof, along with the notation and terminology.

scholarly journals Central derivatives of L-functions in Hida families

2007 ◽  
Vol 339 (4) ◽  
pp. 803-818 ◽  
Author(s):  
Benjamin Howard
Keyword(s):  
Hida Families ◽  
Derivatives Of

Zeta functions of twisted modular curves

2006 ◽  
Vol 80 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Cristian Virdol
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Modular Curves ◽  
Absolute Galois Group ◽  
Modular Curve ◽  
The Absolute

AbstractIn this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a modprepresentation of the absolute Galois group.

Rational torsion of generalized Jacobians of modular and Drinfeld modular curves

2019 ◽  
Vol 31 (3) ◽  
pp. 647-659
Author(s):  
Fu-Tsun Wei ◽  
Takao Yamazaki
Keyword(s):  
Prime Power ◽  
Analogous Result ◽  
Power Level ◽  
Modular Curves ◽  
Generalized Jacobian ◽  
Torsion Points ◽  
Modular Curve ◽  
Generalized Jacobians ◽  
Drinfeld Modular Curves ◽  
Primary Torsion

Abstract We consider the generalized Jacobian {\widetilde{J}} of the modular curve {X_{0}(N)} of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the {\mathbb{Q}} -rational torsion points on {\widetilde{J}} up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on {\widetilde{J}} and its Eisenstein property.

scholarly journals Quadratic Chabauty for modular curves and modular forms of rank one

2020 ◽  
Author(s):  
Netan Dogra ◽  
Samuel Le Fourn
Keyword(s):  
Modular Forms ◽  
Sufficient Conditions ◽  
Rational Points ◽  
Modular Curves ◽  
Type Result ◽  
Heegner Points ◽  
Modular Curve ◽  
Rank One ◽  
Finite Set ◽  
Set Of Points

AbstractIn this paper, we provide refined sufficient conditions for the quadratic Chabauty method on a curve X to produce an effective finite set of points containing the rational points $$X({\mathbb {Q}})$$ X ( Q ) , with the condition on the rank of the Jacobian of X replaced by condition on the rank of a quotient of the Jacobian plus an associated space of Chow–Heegner points. We then apply this condition to prove the effective finiteness of $$X({\mathbb {Q}})$$ X ( Q ) for any modular curve $$X=X_0^+(N)$$ X = X 0 + ( N ) or $$X_\mathrm{{ns}}^+(N)$$ X ns + ( N ) of genus at least 2 with N prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell–Weil rank is equal to its dimension (and at least 2), which is proven via analytic estimates for orders of vanishing of L-functions of modular forms, thanks to a Kolyvagin–Logachev type result.

scholarly journals ON THE EISENSTEIN IDEAL OF DRINFELD MODULAR CURVES

2007 ◽  
Vol 03 (04) ◽  
pp. 557-598 ◽  
Author(s):  
AMBRUS PÁL
Keyword(s):  
Prime Ideal ◽  
Hecke Algebra ◽  
General Characteristic ◽  
Drinfeld Modules ◽  
Modular Curves ◽  
Characteristic P ◽  
Modular Curve ◽  
Drinfeld Modular Curves ◽  
Eisenstein Ideal

Let 𝔈(𝔭) denote the Eisenstein ideal in the Hecke algebra 𝕋(𝔭) of the Drinfeld modular curve X0(𝔭) parameterizing Drinfeld modules of rank two over 𝔽q[T] of general characteristic with Hecke level 𝔭-structure, where 𝔭 ◃ 𝔽q[T] is a non-zero prime ideal. We prove that the characteristic p of the field 𝔽q does not divide the order of the quotient 𝕋(𝔭)/𝔈(𝔭) and the Eisenstein ideal 𝔈(𝔭) is locally principal.

scholarly journals A MODULI INTERPRETATION FOR THE NON-SPLIT CARTAN MODULAR CURVE

2017 ◽  
Vol 60 (2) ◽  
pp. 411-434 ◽  
Author(s):  
MARUSIA REBOLLEDO ◽  
CHRISTIAN WUTHRICH
Keyword(s):  
Elliptic Curves ◽  
Half Plane ◽  
London Math ◽  
Number Fields ◽  
Arithmetic Geometry ◽  
Modular Curves ◽  
Torsion Points ◽  
Modular Curve ◽  
Upper Half Plane ◽  
Cartan Subgroups

AbstractModular curves likeX0(N) andX1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here byXnsp(p) andXnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures ofp-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc.(3)77(1) (1998), 1–38;J. Algebra231(1) (2000), 414–448).

scholarly journals Modular curves and the refined distance conjecture

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Daniel Kläwer
Keyword(s):  
Moduli Space ◽  
Congruence Subgroup ◽  
K3 Surfaces ◽  
Heterotic String ◽  
Complete Intersections ◽  
Modular Curves ◽  
Vector Multiplet ◽  
Modular Curve ◽  
Space Geometry ◽  
Weakly Coupled

Abstract We test the refined distance conjecture in the vector multiplet moduli space of 4D $$ \mathcal{N} $$ N = 2 compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations. This is reflected in the type IIA frame through the existence of a K3 fibration. We identify the degree d = 2N of the K3 fiber as a parameter that could potentially lead to large distances, which is substantiated by studying several explicit models. The moduli space geometry degenerates into the modular curve for the congruence subgroup Γ0(N)+. In order to probe the large N regime, we initiate the study of Calabi-Yau threefolds fibered by general degree d > 8 K3 surfaces by suggesting a construction as complete intersections in Grassmann bundles.

scholarly journals Intersection theory on Shimura surfaces

2009 ◽  
Vol 145 (2) ◽  
pp. 423-475 ◽  
Author(s):  
Benjamin Howard
Keyword(s):  
Fourier Coefficients ◽  
Complex Multiplication ◽  
Intersection Theory ◽  
Integral Model ◽  
Shimura Curve ◽  
Codimension One ◽  
Integral Weight ◽  
Intersection Multiplicities ◽  
General Program ◽  
Shimura Surfaces

AbstractKudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla, Rapoport and Yang. In the present paper we prove results in a higher dimensional setting. On the integral model of a Shimura surface we consider the intersection of a Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersection to certain cycle classes constructed by Kudla, Rapoport and Yang. As a corollary we deduce that our intersection multiplicities appear as Fourier coefficients of a Hilbert modular form of half-integral weight.

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