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100   $a20120424h20122013  uy             0
101 0 $aeng
135   $aur|n|---|||||
181   $ctxt
182   $cc
183   $acr
200 14$aThe Gross-Zagier formula on Shimura curves$b[electronic resource] /$fXinyi Yuan, Shou-wu Zhang, and Wei Zhang
205   $aCourse Book
210   $aPrinceton $cPrinceton University Press$d2012, c2013
215   $a1 online resource (267 p.)
225 0 $aAnnals of mathematics studies ;$vno. 184
300   $aDescription based upon print version of record.
311   $a0-691-15592-5 
311   $a0-691-15591-7 
320   $aIncludes bibliographical references and index.
327   $t Frontmatter -- $tContents -- $tPreface -- $tChapter One. Introduction and Statement of Main Results -- $tChapter Two. Weil Representation and Waldspurger Formula -- $tChapter Three. Mordell-Weil Groups and Generating Series -- $tChapter Four. Trace of the Generating Series -- $tChapter Five. Assumptions on the Schwartz Function -- $tChapter Six. Derivative of the Analytic Kernel -- $tChapter Seven. Decomposition of the Geometric Kernel -- $tChapter Eight. Local Heights of CM Points -- $tBibliography -- $tIndex
330   $aThis comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross-Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross-Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross-Zagier formula is reduced to local formulas. The Gross-Zagier Formula on Shimura Curves will be of great use to students wishing to enter this area and to those already working in it.
410  0$aAnnals of Mathematics Studies
606   $aShimura varieties
606   $aArithmetical algebraic geometry
606   $aAutomorphic forms
606   $aQuaternions
610   $aArakelov theory.
610   $aBenedict Gross.
610   $aDon Zagier.
610   $aEichler?himura theory.
610   $aEisenstein series.
610   $aGross?agier formula.
610   $aHeegner point.
610   $aHodge bundle.
610   $aHodge index theorem.
610   $aL-series.
610   $aMordell?eil group.
610   $aNeron?ate height.
610   $aRankin?elberg L-function.
610   $aSchwartz function.
610   $aShimizu lifting.
610   $aShimura curve.
610   $aShimura curves.
610   $aSiegel?eil formula.
610   $aWaldspurger formula.
610   $aWeil representation.
610   $aabelian varieties.
610   $aanalytic kernel function.
610   $aanalytic kernel.
610   $adegenerate Schwartz function.
610   $adiscrete series.
610   $agenerating series.
610   $ageometric kernel.
610   $aheight series.
610   $aholomorphic kernel function.
610   $aholomorphic projection.
610   $aincoherent Eisenstein series.
610   $aincoherent automorphic representation.
610   $aincoherent quaternion algebra.
610   $akernel function.
610   $akernel identity.
610   $alocal height.
610   $amodular curve.
610   $amodularity.
610   $amultiplicity function.
610   $anon-archimedean local field.
610   $anon-degenerate quadratic space.
610   $aordinary component.
610   $aorthogonal space.
610   $aprojector.
610   $apull-back formula.
610   $aramified quadratic extension.
610   $asupersingular component.
610   $asuperspecial component.
610   $atheta function.
610   $atheta liftings.
610   $atheta series.
610   $atrace identity.
610   $aun-normalized kernel function.
610   $aunramified quadratic extension.
615  0$aShimura varieties.
615  0$aArithmetical algebraic geometry.
615  0$aAutomorphic forms.
615  0$aQuaternions.
676   $a516.3/52
700   $aYuan$b Xinyi$f1981-$0521265
701   $aZhang$b Shouwu$01195550
701   $aZhang$b Wei$f1981-$01477874
801  0$bMiAaPQ
801  1$bMiAaPQ
801  2$bMiAaPQ
906   $aBOOK
912   $a9910790961403321
996   $aThe Gross-Zagier formula on Shimura curves$93693348
997   $aUNINA