School of Mathematics and Statistics

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Quaternionic modular forms mod p

Fam, Yiannis Heijun ( 2023-08)

In a 1987 letter, Serre proves that the systems of Hecke eigenvalues arising from mod p modular forms are the same as those arising from certain functions on the adelic points of D^*, where D is the unique quaternion algebra over Q ramified at p and infinity. We give a detailed account of this proof, the key idea of which is to restrict our study of mod p modular forms to the supersingular locus of the modular curve using the Hasse invariant, and then we extend the result to other level structures. We then incorporate additional ramification into the quaternion algebra D, and this correlates with the study of modular forms on Shimura curves.
A shape theorem for the half-orthant model and the features of its shape

Huang, Xin ( 2022)

We study degenerate random environments which are site-based models of random media involving a parameter p in [0,1]. We are focused on a particular case, the half-orthant model, and will prove a shape theorem for this model when pp_c(2), which is not addressed in this thesis. At last, we prove the general case when p