# A Gross-Kohnen-Zagier type theorem for higher-codimensional Heegner cycles

`2013-11-27 10:30:00``2013-11-27 10:30:00``A Gross-Kohnen-Zagier type theorem for higher-codimensional Heegner cycles``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 the coeffcients of a modular form of weight 3/2. The same holds for certain CM (or Heegner) divisors on Shimura curves. 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, can 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.``אוניברסיטת בר-אילן - Department of Mathematics``mathoffice@math.biu.ac.il``Asia/Jerusalem``public`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

the coeffcients of a modular form of weight 3/2. The same holds for

certain CM (or Heegner) divisors on Shimura curves.

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, can 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.

Last Updated Date : 14/11/2013