Manin–Drinfeld cycles and derivatives of -functions

  • Ari Shnidman

    Hebrew University of Jerusalem, Israel
Manin–Drinfeld cycles and derivatives of $L$-functions cover

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We study algebraic cycles in the moduli space of -shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner–Drinfeld cycle is the product of the -th central derivative of an automorphic -function and Waldspurger's toric period integral. When , this gives a new geometric interpretation for the Taylor series expansion. When , the pairing vanishes, suggesting higher order analogues of the vanishing of cusps in the modular Jacobian, as well as other new phenomena.

Our proof sheds new light on the algebraic correspondence introduced by Yun and Zhang, which is the geometric incarnation of “differentiating the -function”. We realize it as the Lie algebra action of on . The comparison of relative trace formulas needed to prove our formula is then a consequence of Schur–Weyl duality.

Cite this article

Ari Shnidman, Manin–Drinfeld cycles and derivatives of -functions. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1343