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Register a new userA converse theorem for Borcherds products on $X_0(N)$
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We show that every Fricke invariant meromorphic modular form for $Gamma_0(N)$ whose divisor on $X_0(N)$ is defined over $mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight $1/2$. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of $L$-function of certain weight $2$ newforms. We also prove similar results for twisted Borcherds products.
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Given an infinite set of special divisors satisfying a mild regularity condition, we prove the existence of a Borcherds product of non-zero weight whose divisor is supported on these special divisors. We also show that every meromorphic Borcherds product is the quotient of two holomorphic ones. The proofs of both results rely on the properties of vector valued Eisenstein series for the Weil representation.
In this paper, we reprove a global converse theorem of Cogdell and Piatetski-Shapiro using purely global methods.
Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujans mock theta functions f and {omega} in detail.
For a positive integer $N$, let $mathscr C(N)$ be the subgroup of $J_0(N)$ generated by the equivalence classes of cuspidal divisors of degree $0$ and $mathscr C(N)(mathbb Q):=mathscr C(N)cap J_0(N)(mathbb Q)$ be its $mathbb Q$-rational subgroup. Let also $mathscr C_{mathbb Q}(N)$ be the subgroup of $mathscr C(N)(mathbb Q)$ generated by $mathbb Q$-rational cuspidal divisors. We prove that when $N=n^2M$ for some integer $n$ dividing $24$ and some squarefree integer $M$, the two groups $mathscr C(N)(mathbb Q)$ and $mathscr C_{mathbb Q}(N)$ are equal. To achieve this, we show that all modular units on $X_0(N)$ on such $N$ are products of functions of the form $eta(mtau+k/h)$, $mh^2|N$ and $kinmathbb Z$ and determine the necessary and sufficient conditions for products of such functions to be modular units on $X_0(N)$.
We present some applications of the Kudla-Millson and the Millson theta lift. The two lifts map weakly holomorphic modular functions to vector valued harmonic Maass forms of weight $3/2$ and $1/2$, respectively. We give finite algebraic formulas for the coefficients of Ramanujans mock theta functions $f(q)$ and $omega(q)$ in terms of traces of CM-values of a weakly holomorphic modular function. Further, we construct vector valued harmonic Maass forms whose shadows are unary theta functions, and whose holomorphic parts have rational coefficients. This yields a rationality result for the coefficients of mock theta functions, i.e., harmonic Maass forms whose shadows lie in the space of unary theta functions. Moreover, the harmonic Maass forms we construct can be used to evaluate the Petersson inner products of unary theta functions with harmonic Maass forms, giving formulas and rationality results for the Weyl vectors of Borcherds products.
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