# A $p$-adic Stark conjecture in the rank one setting

## TL;DR

This paper introduces a new $p$-adic $L$-function for certain characters of quadratic fields, states a related Stark conjecture, proves it in specific cases, and supports it with numerical evidence.

## Contribution

It defines a novel $p$-adic $L$-function for mixed signature characters and establishes the $p$-adic Stark conjecture in the split prime case, linking to Katz's $p$-adic $L$-function.

## Key findings

- Proved the $p$-adic Stark conjecture when $p$ is split in the imaginary quadratic field.
- Constructed a new $p$-adic $L$-function for mixed signature characters.
- Provided numerical evidence supporting the conjecture in three examples.

## Abstract

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 our 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.