The multiplicative Borcherds singular theta lift is a well-known tool for obtaining automorphic forms with known zeros and poles on quotients of orthogonal symmetric spaces. This has been used by Borcherds in order to prove a generalization of the Gross-Kohnen-Zagier Theorem, stating that certain combinations of Heegner points behave, in an appropriate quotient of the Jacobian variety of the modular curve, like coefficients of a modular form of weight 3/2. The same holds for certain $CM$ (or Heegner) divisors on Shimura curves. \nn The moduli interpretation of Shimura and modular curves yields universal families (Kuga-Sato varieties) over them, as well as variations of Hodge structures coming from these universal families. In these universal families one defines the $CM$ cycles which are vertical cycles of codimension larger than 1 in the Kuga-Sato variety. We will show how a variant of the additive lift, which was used by Borcherds in order to extend the Shimura correspondence, ca be used in order to prove that the (fundamental cohomology classes of) higher codimensional Heegner cycles become, in certain quotient groups, coefficients of modular forms as well. Explicitly by taking the $m$th symmetric power of the universal family, we obtain a modular form of the desired weight $3/2+m$. Along the way we obtain a new singular Shimura-type lift, from weakly holomorphic modular forms of weight $1/2-m$ to meromorphic modular forms of weight $2m+2$.
Mittwoch, den 15. Januar 2014 um 11.15 Uhr Uhr, in INF 288, HS5 Mittwoch, den 15. Januar 2014 at 11.15 Uhr, in INF 288, HS5
Der Vortrag folgt der Einladung von The lecture takes place at invitation by Prof. Kohnen