2018/09/15 by Beckwith, Olivia
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1809.05750
In this note we revisit classic work of Soundararajan on class groups of imaginary quadratic fields. Let A,B,g ≥ 3 be positive integers such that gcd(A,B) is square-free. We refine Soundararajan's result to show that if 4 \nmid g or if A and B satisfy certain conditions, then the number of negative square-free D ≡ A \pmodB down to -X such that the ideal class group of ℚ (√(D)) contains an element of order g is bounded below by X(1)/(2) + ε(g) - ε, where the exponent is the same as in Soundararajan's theorem. Combining this with a theorem of Frey, we give a lower bound for the number of quadratic twists of certain elliptic curves with p-Selmer group of rank at least 2, where p ∈ \3,5,7\.