On class groups of imaginary quadratic fields

In this paper, it is proved that one can find imaginary quadratic fields with class number not divisible by a specified prime l and with certain specified splitting conditions at a finite number of primes. Such existence theorems are useful in the arithmetic of elliptic curves and, potentially, also in certain lifting problems for reducible two-dimensional Galois representations. The methods used are a blend of geometry and the theory of modular forms, especially the trace formula.