The Gross-Zagier Formula on Shimura Curves

By Xinyi Yuan, Shou-wu Zhang, Wei Zhang. 2012

170 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f142c40be353bf77215f13ab4b3a6141be78a50d

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Faltings heights of abelian varieties with complex multiplication (2015, methodology): “The proof of Yuan-Zhang is based on the Gross-Zagier style results of [YZZ13] for Shimura curves over totally real fields.”
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.”
Selmer groups and the indivisibility of Heegner points (2014, unknown): “Let X be X1 = XN+,N− .”
Explicit Gross–Zagier and Waldspurger formulae (2014, methodology): “P0 χ(f)|2 hf,fi , With an approach close to Gross-Zagier’s original one, Shouwu Zhang [43], [44], and [45] obtained explicit formulas with some restrictions on ramiﬁcation. Recently, Yuan-Zhang-Zhang [41] generalizes the Gross-Zagier formula to the most general cases. However, many applications in a”
Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups (2014, background): “Zhang in the 1990’s, completed by Yuan–Zhang–Zhang ([53]) recently.”
The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One (2015, methodology): “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].”
Shtukas and the Taylor expansion of L-functions (II) (2015, background): “The positivity of the first derivative is known for the L-function appearing in the Gross–Zagier formula in [12], [27], for example the L-function of an elliptic curve over Q.”
A converse to a theorem of Gross, Zagier, and Kolyvagin (2014, methodology): “After the first version of this paper was completed, Wei Zhang released a preprint [28] in which he proves many cases of a conjecture of Kolyvagin, showing that the padic Selmer group of Af is often spanned by classes derived from Heegner points.”