Borcherds products and Chern classes of Hirzebruch-Zagier divisors.

*(English)*Zbl 1011.11027The present paper is a short version of the author’s thesis (Heidelberg, 1998; Zbl 0936.11028).

From the introduction: In [Invent. Math. 120, 161-213 (1995; Zbl 0932.11028); ibid. 132, 491-562 (1998; Zbl 0919.11036)] R. E. Borcherds constructed a lifting from elliptic modular forms of weight \(1-n/2\) with poles in the cusps to automorphic forms on the orthogonal group \(O(2,n)\) with known zeros and poles along Heegner divisors. These can be written as infinite products, so-called Borcherds products. The present paper was motivated by the question of whether every principal Heegner divisor can be obtained as the divisor of such a Borcherds product.

We give an affirmative answer in the special \(O(2,2)\)-case of Hilbert modular forms (under a technical condition on the underlying real quadratic field). We construct for each Heegner divisor \(H\) a certain “generalized Borcherds product” \(\Psi_H\). Considering suitable finite products of these \(\Psi_H\) we find a new proof of Theorem 13.3 in [R. E. Borcherds (1998), loc. cit.] and further results of his in [Duke Math. J. 97, 219-233 (1999; Zbl 0967.11022)] in this particualr case. Moreover, it turns out that the function \(\Psi_H\) can be used to determine the Chern class of \(H\) explicitly. One obtains a lifting from elliptic modular forms into the cohomology. We show that it coincides with the Doi-Naganuma lifting.

From the introduction: In [Invent. Math. 120, 161-213 (1995; Zbl 0932.11028); ibid. 132, 491-562 (1998; Zbl 0919.11036)] R. E. Borcherds constructed a lifting from elliptic modular forms of weight \(1-n/2\) with poles in the cusps to automorphic forms on the orthogonal group \(O(2,n)\) with known zeros and poles along Heegner divisors. These can be written as infinite products, so-called Borcherds products. The present paper was motivated by the question of whether every principal Heegner divisor can be obtained as the divisor of such a Borcherds product.

We give an affirmative answer in the special \(O(2,2)\)-case of Hilbert modular forms (under a technical condition on the underlying real quadratic field). We construct for each Heegner divisor \(H\) a certain “generalized Borcherds product” \(\Psi_H\). Considering suitable finite products of these \(\Psi_H\) we find a new proof of Theorem 13.3 in [R. E. Borcherds (1998), loc. cit.] and further results of his in [Duke Math. J. 97, 219-233 (1999; Zbl 0967.11022)] in this particualr case. Moreover, it turns out that the function \(\Psi_H\) can be used to determine the Chern class of \(H\) explicitly. One obtains a lifting from elliptic modular forms into the cohomology. We show that it coincides with the Doi-Naganuma lifting.

##### MSC:

11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |

11F55 | Other groups and their modular and automorphic forms (several variables) |