Extremal elasticity of quadratic orders

We study how large and small elasticity can be for orders belonging to a fixed quadratic field, in terms of the corresponding conductors. For example, we show that if K K is an imaginary quadratic field, then the order of conductor f f in K K has elasticity exceeding ( log ⁡ f ) c 1 log ⁡ log ⁡ log ⁡ f (\log {f})^{c_{1} \log \log \log {f}} for all f f that are sufficiently large. On the other hand, this elasticity is smaller than ( log ⁡ f ) c 2 log ⁡ log ⁡ log ⁡ f (\log {f})^{c_{2}\log \log \log {f}} for infinitely many f f . Here c 1 , c 2 c_{1}, c_{2} are universal positive constants. The proofs borrow methods from analytic number theory previously employed to study statistics of the multiplicative groups ( Z / m Z ) ×