Half-integral weight modular forms and modular forms for Weil representations
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© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Yichao Zhang
Half-integral weight modular forms and modular forms for Weil representations Received: 20 June 2018 / Accepted: 16 November 2019 Abstract. We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for 0 (4) with Kohnen’s plus condition and modular forms for the Weil representation associated to the lattice with Gram matrix (2). With such an isomorphism, we prove the Zagier duality and express the Borcherds lifts in the case of O(2, 1) explicitly.
Introduction The theory of modular forms is of fundamental importance in many parts of modern number theory and many other related fields. Weakly holomorphic modular forms, namely those with possible poles at cusps, received less attention historically than the holomorphic ones. One of a few exceptions is the modular j-function, whose Fourier coefficients possess deep geometric, representation-theoretic and arithmetic information. The inverse of the Dedekind eta function, η(τ )−1 , is another exception because of its direct connection with the partition function. Things changed when Borcherds, in his seminal papers [1,2], constructed a multiplicative theta lifting, which sends weakly holomorphic vector-valued modular forms of full level to modular forms, called Borcherds products, on orthogonal groups. The Borcherds lift is in general a meromorphic modular form in the form of an infinite product, and Borcherds’ theory shows precisely the location of its divisors. Remarkably, in his work on traces of singular moduli, Zagier [21] proved a duality for Fourier coefficients of modular forms weights k and 2 − k for six small half-integral k, with which he gave a different proof of Borcherds’s theorem for 0 (4). Such duality is now known as Zagier duality, and extensions of which to various types of modular forms, of integral or half-integral weight, have been proved since then (see [25] for a list of reference on this research). Many important works have been built on Borcherds lifts by directly employing vector-valued modular forms ever since. However, the vector-valued condition is Yichao Zhang: partially supported by an HIT start-up grant and a Grant RC2016XK001001 of Technology Division of Harbin. Y. Zhang (B): Institute for Advanced Study in Mathematics and Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China. e-mail: [email protected] Mathematics Subject Classification: Primary: 11F37 · 11F30 · 11F27
https://doi.org/10.1007/s00229-019-01169-y
Y. Zhang
not convenient to work with computationally. To overcome such difficulty, in case of integral weights, Bruinier and Bundschuh [4] constructed an isomorphism between prime-level complex-valued modular forms and full-level vector-valued modular forms. Such an isomorphism proves to be useful, with which Choi [8] proved the Zagier duality for level 5, 13, 17, and Kim and Lee [10] provided au
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