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Galois Representations and Automorphic Forms
Hilado, Anton
Hilado, Anton
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Abstract
This thesis is divided into four parts, all tied together by the common theme of modular forms on higher-dimensional locally symmetric spaces. In the first part, we describe the irregular loci on the moduli stack of mod p Galois representations associated to certain Hilbert modular forms. This is based on joint work with Rebecca Bellovin, Neelima Borade, Kalyani Kansal, Heejong Lee, Brandon Levin, David Savitt, and Hanneke Wiersema. In the second part, we study generalizations of Bianchi modular forms to higher-dimensional hyperbolic spaces. This is based on joint work with Taylor Dupuy, Colin Ingalls, and Adam Logan, and subsequent joint work with Spencer Backman, Taylor Dupuy, and Veronika Potter. In the third part, we look at quaternionic modular forms associated to the unitary group U(2, 1), which live on Picard modular surfaces. This is based on joint work with Finn McGlade and Pan Yan. Finally, in the fourth part, we look at modular forms again associated to U(2, 1) but which are obtained as Borcherds products, and describe the decomposable locus on the moduli space of abelian threefolds with complex multiplication.
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2025-01-01