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The Mordell-Weil groups E(Q) of elliptic curves influence the structures of their quadratic twists E −D (Q) and the ideal class groups CL(−D) of imaginary quadratic fields. We define a family of homomorphisms Φ u,v : E(Q) → CL(−D) for particular negative fundamental discriminants −D := −D E (u, v), which we use to simultaneously address questions related to Goldfeld's Conjecture on ranks of quadratic twists, the Cohen-Lenstra heuristics, and Gauss's class number problem. Specifically, let r be the rank of E(Q), and define Ψ E to be the set of suitable fundamental discriminants −D < 0 with positive integers u, v such that (i) the point, and (iii) for an explicit function f E (u, v) and a computable constant c(E), we have the lower boundwhereas D → ∞. Then for any ε > 0, we show that as X → ∞, we haveMoreover, assuming the Parity Conjecture, our results hold with the additional condition that r Q (E −D ) ≥ 2. Buell, Call, and Soleng [2,3,28]. Griffin, Ono, and Tsai [15,16] obtain an effective lower bound of the form h(−D) ≥ c 1 (E) log (D) Recent work improves on this bound by exploiting ideal class pairings

Let f (z) = ∞ n=1 a f (n)q n be a holomorphic cuspidal newform with even integral weight k ≥ 2, level N , trivial nebentypus, and no complex multiplication (CM). For all primes p, we may define θ p ∈ [0, π] such that a f (p) = 2p (k−1)/2 cos θ p . The Sato-Tate conjecture states that the angles θ p are equidistributed with respect to the probability measure µ ST (I) = 2 π I sin 2 θ dθ, where I ⊆ [0, π]. Using recent results on the automorphy of symmetric-power L-functions due to Newton and Thorne, we construct the first unconditional explicit bound on the error term in the Sato-Tate conjecture, which applies when N is squarefree as well as when f corresponds to an elliptic curve with arbitrary conductor. In particular, if π f,I (x) := #{p ≤ x : p ∤ N, θ p ∈ I}, and π(x) := #{p ≤ x}, we show the following bound:As an application, we give an explicit bound for the number of primes up to x that violate the Atkin-Serre conjecture for f .

Let $f(z) = \sum_{n=1}^\infty a_f(n)q^n$ be a holomorphic cuspidal newform with even integral weight $k\geq 2$, level N, trivial nebentypus and no complex multiplication. For all primes p, we may define $\theta_p\in [0,\pi]$ such that $a_f(p) = 2p^{(k-1)/2}\cos \theta_p$. The Sato–Tate conjecture states that the angles θp are equidistributed with respect to the probability measure $\mu_{\textrm{ST}}(I) = \frac{2}{\pi}\int_I \sin^2 \theta \; d\theta$, where $I\subseteq [0,\pi]$. Using recent results on the automorphy of symmetric power L-functions due to Newton and Thorne, we explicitly bound the error term in the Sato–Tate conjecture when f corresponds to an elliptic curve over $\mathbb{Q}$ of arbitrary conductor or when f has square-free level. In these cases, if $\pi_{f,I}(x) := \#\{p \leq x : p \nmid N, \theta_p\in I\}$ and $\pi(x) := \# \{p \leq x \}$, we prove the following bound: $$ \left| \frac{\pi_{f,I}(x)}{\pi(x)} - \mu_{\textrm{ST}}(I)\right| \leq 58.1\frac{\log((k-1)N \log{x})}{\sqrt{\log{x}}} \qquad \text{for} \quad x \geq 3. $$ As an application, we give an explicit bound for the number of primes up to x that violate the Atkin–Serre conjecture for f.