Heights of Heegner points on Shimura curves

By Shou-wu Zhang. 2001

210 citations recorded by the source corpus.

The purpose of this paper is to generalize some results of Gross-Zagier [20] and Kolvyvagin [28] to totally real fields.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/31a44a94453adc204d013ffba0a0247115be6c49

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In Algebra and Number Theory

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Citations in the corpus

Automorphic period and the central value of Rankin--Selberg L-function (2012, background): “Zhang [16], [46],[47],[44] play an important role in the spectacular development in application to the Birch and Swinnerton-Dyer conjecture for elliptic curves in the past 30 years.”
Explicit Gross–Zagier and Waldspurger formulae (2014, methodology): “0 χ(f)|2 〈f, f〉 , With an approach close to Gross-Zagier’s original one, Shouwu Zhang [43], [44], and [45] obtained explicit formulas with some restrictions on ramification.”
Equidistribution, L-functions and ergodic theory: on some problems of Yu. Linnik (2006, background): “Thanks to the work of many people ([12], [36], [37], [47], [50], [66], [78], [80], [81]) notably Gross, Zagier and Zhang such an expression is by now available in considerable generality.”
Hida families and rational points on elliptic curves (2007, methodology): “Proof. The first statement follows from the work of Zhang [Zh2] extending the formula of Gross and Zagier to Shimura curve parametrisations (attached to a quaternion algebra over a totally real field F, although only the case F = Q is needed here.) The second statement follows from the methods of Ko”
Iwasawa theory of Heegner points on abelian varieties of GL2 type (2004, background): “In Section 1 we summarize some of the work of Shimura and Zhang, closely following [25] and [26], to which we refer the reader for proofs.”
The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II (2006, methodology): “In particular, for ρ an odd dihedral representation, the Gross-Zagier type formulae which have now been established in many cases [GZ], [G], [Z1], [ Z2 ], [Z3] interpret L(f ⊗ gρ, 1/2) or its first derivative in terms of the height of Heegner divisors.”
Faltings heights of CM cycles and derivatives of L-functions (2008, background): “Zhang considered heights of Heegner type cycles on Kuga-Sato fiber varieties over modular curves in [46], and the heights of Heegner points on compact Shimura curves over totally real fields in [47].”
L -functions and Galois Representations: Nontriviality of Rankin-Selberg L -functions and CM points (2007, background): “The extra assumptions in the indefinite case are due to the fact that these formulae are not yet known in full generality, although great progress has been made by Zhang [32, 31, 33] in extending the original work of Gross and Zagier.”
Selmer groups and the indivisibility of Heegner points (2014, unknown): “When the analytic rank of E/Q is one, combining with the general Gross–Zagier formula on Shimura curves [17, 45, 46] and Kolyvagin’s theorem [23], we are able to prove the p-part of the refined Birch–SwinnertonDyer conjecture.”
Gross-Zagier Formula for GL(2), II (2004, methodology): “The aim of this paper is to review the proofs in our previous papers [16; 17].”
Root numbers and parity of ranks of elliptic curves (2009, unknown): “Since Eβ also has a prime of multiplicative reduction, by Zhang’s theorem ([32] Thm.”
Unearthing the visions of a master: harmonic Maass forms and number theory (2008, background): “For instance, Zhang considered heights of Heegner type cycles on Kuga-Sato fiber varieties over modular curves in [214], and the heights of Heegner points on compact Shimura curves over totally real fields in [215].”
The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One (2015, methodology): “ng). To pass from the inequality (1.3.a) to one where the right-hand side is replaced with an L-value, we combine the inequality with the general Gross–Zagier formula for the point z′ K, due to Zhang [Zha01a] and Yuan, Zhang, and Zhang [YZZ13]. The result is the exact lower bound for #X(E/K′)[p∞] pr”