Arithmetic and geometry of the Kudla-Millson theta function

• Applicant: Dr. Riccardo Zuffetti
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 554793187

Modular forms play an important role in many areas of mathematics. For instance, given a rational elliptic curve E, there exists a modular form f whose Fourier coefficients at the primes p encode the number of solutions modulo p of the corresponding Weierstrass equation. We may consider f as the generating series of the solution-counts modulo p of E. This existence statement is a version of the celebrated Modularity Theorem. Its confirmation was the final step in completing the proof of Fermat's Last Theorem, provided by Wiles at the end of the last century. The approach of studying arithmetic properties of numbers through the modular properties of generating series has been vastly generalized, leading to remarkable results in algebraic geometry, particularly concerning Shimura varieties. In the 1980s, Kudla and Millson constructed a theta function of genus g, for every positive integer g, with several remarkable properties. This theta function is a (non-holomorphic) Siegel modular form of genus g with values in the space of closed differential 2g-forms on some orthogonal Shimura variety X. Its cohomology class is the celebrated generating series of cohomology classes of codimension g special cycles on X. The Kudla-Millson theta function has been a fundamental tool in several recent discoveries, including those made by the applicant and co-authors, yet its arithmetic and geometric properties are not fully understood. The proposed project aims to further investigate these properties, employing new methods developed by the applicant on various types of Shimura varieties. As an application, the project aims to deduce fundamental geometric invariants of Shimura varieties, such as dimensions of cohomology groups, and explore new properties of regularized theta lifts, such as the Borcherds lift. Specifically, the project aims to establish injectivity criteria for the theta lift associated with the Kudla-Millson theta function, compute the Lefschetz decomposition of this theta function in terms of non-holomorphic modular forms, and investigate the Lefschetz decomposition of the cohomology class associated with this theta function in cases of small weight.

• DFG Programme: Research Grants