Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp012227ms681
 Title: Borcherds products for $$O(2,2)$$ and the $$\theta$$ operator on $$p$$-adic Hilbert modular forms Authors: Lin, Alice Advisors: Skinner, Christopher Department: Mathematics Class Year: 2020 Abstract: A result of Bruinier and Ono shows that under certain conditions on the zeroes and poles of a meromorphic elliptic modular form $$f$$ with respect to a prime $$p$$, the quotient $$\theta f / f$$ is a $$p$$-adic modular form of weight 2 in the sense of Serre, where $$\theta$$ is the Ramanujan differential operator. We give another proof of the same result for $$p$$-adic modular forms in the sense of Katz, using the geometric interpretation of modular forms as sections of a line bundle over the modular curve. We also prove a new, analogous result for Hilbert modular forms. For a given prime $$p$$, we characterize which Hirzebruch-Zagier divisors lie in the supersingular locus of the Hilbert modular surface modulo $$p$$, yielding an application of the analogous result for Borcherds products. URI: http://arks.princeton.edu/ark:/88435/dsp012227ms681 Type of Material: Princeton University Senior Theses Language: en Appears in Collections: Mathematics, 1934-2020