scholarly journals The -adic Gross–Zagier formula on Shimura curves

2017 ◽  
Vol 153 (10) ◽  
pp. 1987-2074 ◽  
Author(s):  
Daniel Disegni
Keyword(s):  
General Formula ◽  
Analytic Functions ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
Root Number ◽  
Heegner Points ◽  
Semistable Reduction

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.

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.

The Gross-Zagier Formula on Shimura Curves

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-wu Zhang ◽  
Wei Zhang
Keyword(s):  
Diophantine Equations ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
Heegner Points ◽  
Quaternion Algebras ◽  
Complete Proof ◽  
Weil Representations ◽  
Arakelov Theory ◽  
Generating Series ◽  
New Applications

This comprehensive account of the Gross–Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross–Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross–Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross–Zagier formula is reduced to local formulas. This book will be of great use to students wishing to enter this area and to those already working in it.

scholarly journals Heights of Heegner Points on Shimura Curves

2001 ◽  
Vol 153 (1) ◽  
pp. 27 ◽  
Author(s):  
Shouwu Zhang
Keyword(s):  
Shimura Curves ◽  
Heegner Points

scholarly journals Shimura curves in the Prym loci of ramified double covers

2021 ◽  
Author(s):  
Paola Frediani ◽  
Gian Paolo Grosselli
Keyword(s):  
Computer Algebra ◽  
Group Action ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
One Dimensional ◽  
Fixed Group ◽  
Base Curve ◽  
Double Covers ◽  
Ramification Points

We study Shimura curves of PEL type in the space of polarized abelian varieties [Formula: see text] generically contained in the ramified Prym locus. We generalize to ramified double covers, the construction done in [E. Colombo, P. Frediani, A. Ghigi and M. Penegini, Shimura curves in the Prym locus, Commun. Contemp. Math. 21(2) (2019) 1850009] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is [Formula: see text]. Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Prym loci.

Trace of the Generating Series

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Eisenstein Series ◽  
Abelian Varieties ◽  
Discrete Series ◽  
Theta Series ◽  
Shimura Curves ◽  
Trace Identity ◽  
Shimura Curve ◽  
Schwartz Functions ◽  
Generating Series ◽  
New Space

This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the chapter considers the new space of Schwartz functions and constructs theta series and Eisenstein series from such functions. It proceeds by discussing discrete series at infinite places, modularity of the generating series, degree of the generating series, and the trace identity. It also presents the pull-back formula for the compact and non-compact cases. In particular, it describes CM cycles on the Shimura curve, pull-back as cycles, degree of the pull-back, and some coset identities.

Mordell-Weil Groups and Generating Series

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Kernel Function ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
Generating Series ◽  
Analytic Kernel

This chapter deals with Mordell–Weil groups and generating series. It first provides an overview of the basics on Shimura curves and abelian varieties parametrized by Shimura curves before introducing a theorem, which is an identity between the analytic kernel and the geometric kernel. It then defines the generating series and uses it to describe the geometric kernel. It also presents a theorem, which is an identity formulated in terms of projectors, and reviews some basic notations and results on Shimura curves. Other topics covered include the Eichler–Shimura theory for abelian varieties parametrized by Shimura curves, normalization of the geometric kernel, and the analytic kernel function. The chapter concludes with an analysis of the kernel identity implied in the first theorem.

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

2014 ◽  
Vol 66 (5) ◽  
pp. 1167-1200 ◽  
Author(s):  
Victor Rotger ◽  
Carlos de Vera-Piquero
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Quadratic Field ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Main Idea ◽  
Galois Representation ◽  
Rational Points ◽  
Shimura Curves ◽  
Fields Of Moduli

AbstractThe purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

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