Armando Gutiérrez Terradillos defended his PhD thesis Theta correspondences and arithmetic intersections on May 26, 2023.
The thesis was produced within the UPC doctoral program on Applied Mathematics and his advisors were Víctor Rotger and Gerard Freixas.
Starting January 2024, he will be a postdoctoral researcher at the Morningside Center of Mathematics of the Chinese Academy of Sciences working in the group of Yichao Tian.
Thesis summary
Automorphic representations are an evolution of the classical notion of modular forms going back to Hecke in the 1900’s. These objects are central in number theory and arithmetic grometry, providing the theoretical framework of deep conjectures due to Langlands. The theory of automorphic representations is formulated in terms of representations of Hecke algebras in spaces of L2−functions defined over the adelic points of a group. Fundamental for this framework is the theory of representation of Lie algebras and algebraic groups.
The questions adressed in this thesis are motivated by the special values of L-functions, which are fundamental invariants of automorphic representations. Following general conjectural principles one expects deep connections with the geometry of relevant spaces in arithmetic geometry, known as Shimura varieties. A key tool to deal with this kind of problems is the so called theta correspondence. It allows us to relate automorphic representations for different groups, transferring certain properties from one representation to another.
The thesis is mainly divided into two parts. In essence, the first one is an extension of the paper [5]. The integrals of the logarithm of the Borcherds forms have been related to zeta and L-values in a wide variety of papers. In [2], the author studies the integral of the logarithm of the Borcherds forms for certain quasi-projective Shimura varieties associated to the group GSpin, obtaining an expression involving certain Fourier coefficients of Eisenstein series. One of the main tools in [2] is the Siegel-Weil formula in the convergent range of Weil and for anisotropic quadratic spaces. On account of the eventual divergence of the integral of the theta function over the modular curve, the integral of the logarithm of the Borcherds forms over the modular curve was not addressed in [2]. Along this chapter, using the regularized Siege-Weil formula of [3], we obtain an explicit expression for the truncated integral of the Siegel theta function. The main application of this result is an explicit formula for the integral of the logarithm of the Borcherds forms. The final result involves different zeta values and coefficients of Eisenstein series, completing the work of [2].
In chapter two, the analytic properties of L-functions are analyzed from a representation theoretic perspective. It is an extension of the work with Antonio Cauchi in [4]. First, we consider a zeta integral of GU2,2 which unfolds to a unitary Shalika functional. In order to compute this function we proceed from local to global, leading us to perform a detailed analysis of local Shalika models for unramified representations of GU2,2. On the one hand, under some local conditions, we show that the multiplicity of the Shalika model of unramified representations for the group GU2,2 is one. Using this result and following the ideas of [1], we are able to find an expression of the Shalika functional in terms of the Satake parameter of a representation in GSp4. Similarly to the classic Casselman-Shalika formula for Whittaker functionals, the above result can be used to explicitly calculate Z-integrals. In fact, it allows us to establish a relationship between the zeta integral for the group GU(2, 2) and a twisted standard L-function of GSp4, where the relation between the involved automorphic representations is given by the theta correspondence.
Selected Pre-publication: [5]
References
[1] Yiannis Sakellaridis, A Casselman-Shalika formula for the Shalika model of GLn, Canadian Journal of Mathematics 58 (2006), no. 5, 1095–1120.
[2] Stephen S. Kudla, Integrals of Borcherds forms, Compositio Mathematica 137 (2003), no. 3, 293–349.
[3] Wee Teck Gan, Yannan Qiu, and Shuichiro Takeda, The regularized Siegel-Weil formula (the second term identity) and the Rallis inner product formula, Inventiones Mathematicae 198 (2014), no. 3, 739–831.
[4] Antonio Cauchi and Armando Gutierrez Terradillos, Shalika models on GU2,2 and the degree 5 L-function of GSp4, 2023. arXiv pdf (44 pages).
[5] Armando Gutierrez Terradillos, A truncated Siegel-Weil formula and Borcherds forms, 2022. arXiv pdf (36 pages).