A $p$-adic Stark conjecture in the rank one setting
Volume 193 / 2020
Acta Arithmetica 193 (2020), 369-417 MSC: Primary 11F67; Secondary 11R42. DOI: 10.4064/aa190212-15-8 Published online: 17 February 2020
We give a new definition of a $p$-adic $L$-function for a mixed signature character of a real quadratic field and for a nontrivial ray class character of an imaginary quadratic field. We then state a $p$-adic Stark conjecture for this $p$-adic $L$-function. We prove the conjecture in the case when $p$ is split in the imaginary quadratic field by relating our construction to Katz’s $p$-adic $L$-function. We also provide numerical evidence for our conjecture in three examples.