Number/Representation Theory

In the early 1950s, N. Ankeny, E. Artin, and S. Chowla derived four congruence relations involving the arithmetic invariants of real quadratic fields $K$ with $\mathrm{disc}(K) = 1 \mod 4$. In particular, they obtained a congruence relating the ideal class number and fundamental unit of $K$ to generalized Bernoulli number for each of the odd ramified primes in $K$. In the course of their work, they posed a question regarding the arithmetic properties of fundamental units which, if true, would yield a local method for computing the ideal class number of $K$. Over time, this question became known as the Ankeny–Artin–Chowla (AAC) conjecture. Until recently, this question seemed to evade any resolution, but in 2024, counterexamples were finally found. In this talk, I will discuss how the restrictions on the discriminant and primes considered can be overcome using the Kubota–Leopoldt $p$-adic $L$-function. Additionally, we will discuss how counterexamples to the AAC conjecture correspond to well-studied objects, such as dihedral extensions with specified ramification and number field versions of Wieferich primes. Parts of this work are joint with M. Ram Murty.