Everand Logo

Welcome to Everand!

  • Discover millions of ebooks, audiobooks, and so much more with a free trial

    Only $11.99/month after trial. Cancel anytime.

    The Gross-Zagier Formula on Shimura Curves: (AMS-184)
    The Gross-Zagier Formula on Shimura Curves: (AMS-184)
    The Gross-Zagier Formula on Shimura Curves: (AMS-184)
    Ebook349 pages3 hours

    The Gross-Zagier Formula on Shimura Curves: (AMS-184)

    By Xinyi Yuan, Shou-wu Zhang and Wei Zhang

    ()

    Read preview

    About this ebook

    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.



    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.

    LanguageEnglish
    PublisherPrinceton University Press
    Release dateNov 11, 2012
    ISBN9781400845644
    The Gross-Zagier Formula on Shimura Curves: (AMS-184)

    Related to The Gross-Zagier Formula on Shimura Curves

    Titles in the series (100)

    View More
    View More

    Related ebooks

    Mathematics For You

    View More
    View More

    Related podcast episodes

    Related articles

    • Article

      A Supermassive Lens On The Constants Of Nature

      Nov 25, 2020

      The 2020 Nobel Prize in Physics went to three researchers who confirmed that Einstein’s general relativity predicts black holes, and established that the center of our own galaxy houses a supermassive black hole with the equivalent of 4 million suns

      9 min read

    • Article

      The Math That Takes Newton Into the Quantum World

      Feb 28, 2019

      In my 50s, too old to become a real expert, I have finally fallen in love with algebraic geometry. As the name suggests, this is the study of geometry using algebra. Around 1637, René Descartes laid the groundwork for this subject by taking a plane,

      9 min read

    • Article

      The Universe Began With a Big Melt, Not a Big Bang

      Oct 5, 2017

      There are two tantalizing mysteries about our universe, one dealing with its final fate and the other with its beginning, that have intrigued cosmologists for decades. The community has always believed these to be independent problems—but what if the

      10 min read

    • Article

      How Physics Is Like Three-Chord Rock: Like a set of common chords, the same math appears in diverse fields of physics.

      Jun 12, 2014

      A few years back, 12 million of us clicked over to watch the “Pachelbel Rant” on YouTube. You might remember it. Strumming repetitive chords on his guitar, comedian Rob Paravonian confessed that when he was a cellist, he couldn’t stand the Pachelbel

      6 min read

    • Article

      Heavyweight Particle

      Jun 12, 2022

      Heavyweight Particle
      13 min read

    • Article

      Erwin Schrödinger

      Dec 27, 2019

      Erwin Schrödinger
      2 min read

    • Article

      How Math’s Most Famous Proof Nearly Broke

      Dec 20, 2018

      Andrew Wiles gave a series of lectures cryptically titled “Modular Forms, Elliptic Curves, and Galois Representations” at a mathematics conference in Cambridge, England, in June 0f 1993. His argument was long and technical. Finally, 20 minutes into t

      8 min read

    • Article

      How Math’s Most Famous Proof Nearly Broke: Andrew Wiles thought he had a solution to an age-old puzzle. Until it began to unravel.

      May 28, 2015

      Andrew Wiles gave a series of lectures cryptically titled “Modular Forms, Elliptic Curves, and Galois Representations” at a mathematics conference in Cambridge, England, in June 0f 1993. His argument was long and technical. Finally, 20 minutes into t

      8 min read

    • Article

      Fine Tuning Is Just Fine

      Sep 20, 2018

      It is hard to overstate the anticipation that preceded the opening of the Large Hadron Collider (LHC) 10 years ago. Smashing protons together at energies well above those produced at any previous particle accelerator, the LHC seemed capable of vindic

      9 min read

    • Article

      The Deepest Uncertainty

      May 16, 2019

      Georg Cantor died in 1918 in a sanatorium in Halle, Germany. A pre-eminent mathematician, he had laid the foundation for the theory of infinite numbers in the 1870s. At the time, his ideas received hostile opposition from prominent mathematicians in

      4 min read

    • Article

      A Universe Made of Tiny, Random Chunks: The space-time that makes up our universe is inherently uncertain.

      Jun 6, 2013

      One of science’s most crucial yet underappreciated achievements is the description of the physical universe using mathematics—in particular, using continuous, smooth mathematical functions, like how a sine wave describes both light and sound. This is

      6 min read

    • Article

      The Strangeness of Black Holes

      Apr 11, 2019

      This essay is one of the five winners in the 2019 writing competition held by the Black Hole Initiative at Harvard University. “The Black Hole Initiative offers a unique environment for thinking about the topic of black holes more creatively and comp

      6 min read

    • Article

      Introduction To Shading Theory

      Nov 22, 2022

      Artist Charlie Pickard LOCATION: England Charlie Pickard is a classically trained fine artist and illustrator. Recently awarded the Philip de László Award for Excellence, Charlie continues to work, exhibit and teach out of his studio based in Londo

      5 min read

    • Article

      How to Build a Search Engine for Mathematics: The Surprising Power of Neil Sloane’s Encyclopedia of Integer Sequences.: The surprising power of Neil Sloane’s Encyclopedia of Integer Sequences.

      Oct 22, 2015

      On the average summer Saturday, the mathematician Neil Sloane woke up to a crisis. “There are always crises,” he said— albeit crises of the teapot tempest variety. One Saturday over breakfast, he faced an inbox message titled “edits from outer space.

      10 min read

    • Article

      Blossom Theory

      Dec 2, 2020

      Blossom Theory
      6 min read

    • Article

      The Deepest Uncertainty: When a hypothesis is neither true nor false.

      Jun 6, 2013

      Georg Cantor died in 1918 in a sanatorium in Halle, Germany. A pre-eminent mathematician, he had laid the foundation for the theory of infinite numbers in the 1870s. At the time, his ideas received hostile opposition from prominent mathematicians in

      4 min read

    Related categories

    Reviews for The Gross-Zagier Formula on Shimura Curves

    0 ratings

    0 ratings0 reviews

    What did you think?

    Tap to rate

    Review must be at least 10 words

      Book preview

      The Gross-Zagier Formula on Shimura Curves - Xinyi Yuan

      3^book_preview_excerpt.html=MoɕB(Y$vh$[mdqJ$"X'9n !)!9_ｪj6-RCl^zrzyȞ܇?9+$|wX?szء}>z|Ŏf$|9^GNĻ܋fc7YzrDȚEvٹp]HxY4͝q?]VZ(4oihD;~{x'"+:@t∭Ck#xRZ(uekz6 ;<`#^?zQ=fϜ hA7v!n<yKPDϙ"ށKmuJaa\6y/nWp]Cѳz{K;1+-.~θE#σH4wqE{̺ro1Ǎ#98G릿 Xqs,+s\՝գܥͥJ ߆.ٻi am{,(pHgI<ܓ\`CD|j^m"= H4 똻 tXq_i볰ac2KXwgr@  /vh͑O t{.`=8-6Nvr#H,3N+aw8^"r;4- G7&ھz%ۜw<jKBk) .XN0'9`-XÀ#x4Y8O;*!<%}B0#@DSG[*5S)bȤtew=)-f#{>pܐty~{ e=Wgg;;&E )*lLZzO$ejsjOP~^TPL#-o1%huD:`KӇifk99:M! B|S`q=~=eKoH?ѐ @#ӿA5.{+{|OyDh<ؒRx`;Y=<< 3BS4M{g7r8X~-wɃBʐ&q@Ǘ[ߠ-4pOG}n9Cgd&/=G@ې=i3`mq 0?{dm('64ZB8"<>{mCͣͣҵ"(@]Y(#oQ-o<۪s5hZ<σ7z/IV=@OCqBlҡLI-()$$8L?]pHu 3sa+ތAb=c)fGN: ";(3'Ы@LNLp$s-\uNIq~&`cZ Og碃:eAy2+zljo}:]c}b1R4Z`O/ )\R]'x˺\8~'>`K|)(_a LĹtzbp X׉sic8-φp.ɛ 6(*M!BÙh̪{m-!Ah緁gaNT:oan˛kOS?쑒%f{}>FSǥO?6p|7=tDMxS铃':XtpyQB|E~#rXq?%oƚT& Μ>ds /@B\ ȿۉ(mCyAqNmQNMy>5LA]8|#?L2HZ-2X,p{:rf3~C˓ޣ+?8liښ™z| "0INC1H`ǁF~h8½ 2/\Xً=㷔{PzH7v# )EWV<1@R Dk6;ufꊰPz?`i]fC-na-nEK2$#M taUNtok \ԸéٜgNN7&^A#B^"C<B&E \2L42)!.:҄s1'-Nv#O|Zz᫭\SŚ|ro,7`;𷥅’z"Eef/X,DNH9k(Edy(t@L  ri렶IfhtOLNbw\/#q60\ GdD zc?{6ٌ]ho;ZN7)ˈϬvp ~2aEԊcWQ蠼_8>_GҫV/^}oLNW?\ cA;,329i W'ƿ'9od}@@ EFid8<;w⺙}g8~TZ)%˖VɊ@aćѪ$ q:ì 65glr{(C$ZE9 8a駡fH#(d'y}wՒe谡V+Z3~S _}b{/ںhԔRYX*O3N%4=\0JvP  |0ٍec 弅,Yd'6h@vaR rG!5k,~Ø5H]!tu.]m+|c̤ƻ]&"0d+jdleBI$Mϝ8œ,ClRqbxz\N}@h~WקVSt|i'?UV( ?h!tZK2P$d_"I6 :oüs(n#SKGZ҉F 6(%ISdNv_N]ln!M&뾙'b ݮ4c6;l%M`:5Jc'6̅!qbŽe&_IZk_%`z-04 ? VzUX-طm)X@t!]cfNP!Ƥ- OTpѠc;ZH.*%9UTL;l<\~#hq>#if(kZP5 c휪\|7ua&ljau3n>(V_&Z݃ mGRbVw6`)wcd6L=V3д&Jmdd۞G峖ZnL0vNAr_O"4[j .6Ҙ!e-iSiL6-oQQUeOJjsjV4fD4CDK61IOzIژI[e6.XLl*xN&mqd-ச6;P̀ztTFrϗTq#NZrQ>&/;YsN7jC].if0wxeǵf'R~7䧞4&I\%"S.%ڝװ?Op |}-Ա?ClߏKl}=;98mz\;W/&{'YSNj]ɶ}:zI㾣~0i,73pNk/_˧UeBODIC7쑑tdѽj Oq@S Ӵ+fo}o6]|.Y"&Agh&_S\^4^*&ંLlB]o3yݢ%i8tˌSBXi9fwNɭW8WdqW$n RxWr6ȵ;I= Hۉ[fSQZSܸH3d1v!h2/q4pJo̥%M.DGwҕuvG6ڛOiEI46iN IEvJVN-Uʾˁ6 Qzx'.Jj ]844VK#lGH/RXufi޵!%Q3)1\u}Td¤S-_ 2M^|2 ,8K6:*C59&n߂R+bfIz*ӷe"=mD#%&dl^NqrhW^()V[,,S)U*^vH@ ԜK/mV& "1va:0U9VRi  ZCL\Ie ";"wUKlYx2^#wNG7q fMbets[M]xş!%~pByBŸ#ozuPЏCTפd>{o6lz5yy\߹~C#ܼjAKvnuX~mo7Jl[IV9B]lC?pSZ0QLX3 #LDYcXr3J 1kS[:-ѕ_7&+*;~*{)C]f#tWh֯3`jgמ?\Z,Urlv4טnLL-<"܊Eۥnj~fխ+I78MT&[q3J+@hj׃Ž당;~`N\p.Izvd9ǤFd KEd7æ%W}S)ý4ʳ 1fV'qsjS1N 1O[~?]b4JmUdh,ƫ8茡ct$ʼnt2/)؇J"ð "ʙe ! HmSQ\A.&\Ș^6/dfy&Uڕ#'*xӡaNV&Ȯ#QD1Quaᯌj(VNҁtq,YJ$/ffsHU1nf[(MY 9TSmiwNsA =I9m.)˦iTm6Q}0sjqg뫱KdkL ZiCNU*0U*X@YLh0~5n,haLNХ[1 0li<)Tݬ:&w/q(UeL]qB> @XA05`f&R\4rf@u\X`'W`ȼR.MeG&%$t1/QݟL%#OR"}o[qZ<3*~, "!FEU;LSV}"-12 BEFU#"ҙbia )U KJV +Nʟ
      Enjoying the preview?
      Page 1 of 1