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

Seminar
Speaker
Dr. Shaul Zemel (Technische Universitaet Darmstadt)
Date
27/11/2013 - 10:30Add to Calendar 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 - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Abstract

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.

תאריך עדכון אחרון : 14/11/2013