Title: A $$p$$-adic Criterion for Quadratic Twists of CM Elliptic Curves to Have Rank $$1$$ Authors: Chandran, Kapil Advisors: Skinner, Christopher Department: Mathematics Class Year: 2020 Abstract: We present a $$p$$-adic criterion for a family of quadratic twists of an elliptic curve $$E/\mathbb{Q}$$ with complex multiplication (CM) by the full ring of integers to have analytic rank $$1$$. This criterion is obtained by studying the $$p$$-adic $$L$$-value $$\mathscr{L}_p(\psi_E^*)$$, where $$\psi_E^*$$ is the Hecke character associated $$E$$. By relating $$\psi_E^*$$ to a congruent Hecke character that lies in the range of classical interpolation, we reduce the required nonvanishing of $$\mathscr{L}_p(\psi_E^*)$$ to a calculation of the $$p$$-adic valuation of the central $$L$$-value of a classical modular form. A theorem of Waldspurger then provides a half-integer weight eigenform whose Fourier coefficients control whether a quadratic twist of $$E$$ has analytic rank $$1$$. URI: http://arks.princeton.edu/ark:/88435/dsp0108612r56c Type of Material: Princeton University Senior Theses Language: en Appears in Collections: Mathematics, 1934-2020