Shtukas and the Taylor expansion of $L$-functions

Abstract

We define the Heegner–Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with $r$-modifications for an even integer $r$. We prove an identity between (1) the $r$-th central derivative of the quadratic base change $L$-function associated to an everywhere unramified cuspidal automorphic representation $\pi$ of $\mathrm{PGL}_{2}$, and (2)~the self-intersection number of the $\pi$-isotypic component of the Heegner–Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross–Zagier formula for higher derivatives of $L$-functions.

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      mrnumber = {2897697},
      mrreviewer = {Fabrizio Andreatta},
      doi = {10.1007/s00222-011-0348-1},
      url = {http://dx.doi.org/10.1007/s00222-011-0348-1},
      zblnumber = {1247.14031},
      }
  • [Z14] Go to document W. Zhang, "Automorphic period and the central value of Rankin-Selberg $L$-function," J. Amer. Math. Soc., vol. 27, iss. 2, pp. 541-612, 2014.
    @ARTICLE{Z14,
      author = {Zhang, Wei},
      title = {Automorphic period and the central value of {R}ankin-{S}elberg {$L$}-function},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {27},
      year = {2014},
      number = {2},
      pages = {541--612},
      issn = {0894-0347},
      mrclass = {11F67 (11F70 11G40 22E55)},
      mrnumber = {3164988},
      mrreviewer = {Neven Grbac},
      doi = {10.1090/S0894-0347-2014-00784-0},
      url = {http://dx.doi.org/10.1090/S0894-0347-2014-00784-0},
      zblnumber = {1294.11069},
      }

Authors

Zhiwei Yun

Yale University, New Haven, CT

Wei Zhang

Columbia University, New York, NY

Current address:

Massachusetts Institute of Technology, Cambridge, MA