Abstract
The purpose of this partly expository paper is to give an introduction to modular forms on G 2. We do this by focusing on two aspects of G 2 modular forms. First, we discuss the Fourier expansion of modular forms, following work of Gan-Gross-Savin and the author. Then, following Gurevich-Segal and Segal, we discuss a Rankin-Selberg integral yielding the standard L-function of modular forms on G 2. As a corollary of the analysis of this Rankin-Selberg integral, one obtains a Dirichlet series for the standard L-function of G 2 modular forms; this involves the arithmetic invariant theory of cubic rings. We end by analyzing the archimedean zeta integral that arises from the Rankin-Selberg integral when the cusp form is an even weight modular form.
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Notes
- 1.
The 2 here is 1J × 1J = 21J.
- 2.
The 3 here is \( \operatorname {\mathrm {tr}}_{J}(1_J)\).
- 3.
In the paper [Pol18] we generalize slightly the definition of modular forms on G 2; for example, one can also define modular forms of weight 1.
- 4.
The Levi subgroup M of P is conjugate in G 2(C) to the short root \( \operatorname {\mathrm {SU}}(2)\), and through this conjugation, W ⊗C becomes identified with the W C appearing in the K-type decomposition of π n. This is why we use the same letter for both spaces.
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Acknowledgements
We thank Takuya Yamauchi for helpful comments on an earlier version of this manuscript. We also thank the anonymous referee for his or her comments and corrections, which have improved the readability of this paper.
This work supported by the Schmidt fund at IAS. We thank the IAS for its hospitality and for providing an excellent working environment. We also thank the Simons Foundation for its support of the symposium on the Relative Trace Formula, where some aspects of this work were discussed.
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Pollack, A. (2021). Modular forms on G 2 and their Standard L-Function. In: Müller, W., Shin, S.W., Templier, N. (eds) Relative Trace Formulas. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-030-68506-5_12
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