In this note, we will apply the results of Gross–Zagier, Gross–Kohnen–Zagier and their generalizations to give a short proof that the differences of singular moduli are not units. As a consequence, we obtain a result on isogenies between reductions of CM elliptic curves.

We explicitly construct cusp forms on the orthogonal group of signature (1, 8n + 1) for an arbitrary natural number n as liftings from Maass cusp forms of level one. In our previous works [28] and [21] the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds [4] instead.We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maass cusp forms are Hecke eigenforms. The standard L-functions of the cusp forms are proved to be products of symmetric square L-functions of the Hecke-eigen Maass cusp forms with shifted Riemann zeta functions.

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The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application, we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.

Abstract. In the present paper we prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagier's modularity of traces of singular moduli, together with new geometric interpretations of the traces with non-positive index.