Galois theoretic study on simultaneous representation of primes by binary quadratic forms

Coincidence of $L$-functions

“…As another example, the decomposition law in K/Q is described by three different quadratic forms with distinct discriminants (see [8] and [17,Section 5]). This also leads to an application in [9] on simultaneous representations of primes by different quadratic forms. Furthermore, Shintani [22] proved his own conjecture on units of a certain abelian extension of a real quadratic field in the case where the coincidence of such L-functions occurs.…”