Congruence class bias and the Lang-Trotter conjecture for families of elliptic curves (Q6631529)

Let \(E\) be an elliptic curve over \(\mathbb{Q}\). If \(p\) is a prime of good reduction, then the reduction modulo \(p\) of \(E\) is an elliptic curve over \(\mathbb{F}_p\). For such \(p\), we define. \N\[\Na_p(E) = p+1 - |E(\mathbb{F}_p)|.\N\]\NThe Lang-Trotter conjecture, for a non-CM elliptic curve \(E\) and an integer \(r\), state that the function \N\[\N\pi(x, E, r) = \# \{ p \leq x \, \colon \, a_p = r \} \sim C_{E, r} \dfrac{\sqrt{x}}{\log(x)},\N\]\Nas \(x \rightarrow \infty\), for some constant \(C_{E, r}\).\N\NSome approach to the conjecture has been to prove that it holds on average for certain families of elliptic curves.\N\NFor example, for the family \N\[\NS(A,B) = \{ E(a,b) \colon |a| \leq A, |b| \leq B, a, b \in \mathbb{Z}, \Delta(a, b) \neq 0 \},\N\]\Nof elliptic curves \(E(a,b)\), with equation \(E(a,b) \colon y^2 =x^3 +ax+b\), we can obtain \N\[\N\dfrac{1}{4AB} \sum_{E \in S(A,B)} \pi(x, E, r) \sim C(r) \dfrac{\sqrt{x}}{\log(x)},\N\]\Nas \(x \rightarrow \infty\), for \(A, B > x^{1+\epsilon}\) and some constant \(C(r)\).\N\NAveraging results have also been carried out under a restriction of primes to congruence classes. Suppose, for example, that \(r\) is an odd positive integer, the integers \(c,m\) are co-prime and we define \N\[\N\pi(x, E, r, c, m) = \# \{ p \leq x \, \colon \, a_p = r , \, p \equiv c\, (\text{mod } m)\}.\N\]\NThen again, we can obtain results in average like\N\[\N\dfrac{1}{4AB} \sum_{E \in S(A,B)} \pi(x, E, r, c, m) \sim C(r, c, m) \dfrac{\sqrt{x}}{\log(x)},\N\]\Nas \(x \rightarrow \infty\), for \(A, B > x^{1+\epsilon}\) and some constant \(C(r, c, m)\).\N\NThe paper in question consider families of elliptic curves based on a combination of Weierstrass equations, congruent classes and some exponentials and polynomial functions as growth of the coefficients. Let us define, for functions \(f, g \colon \mathbb{Z} \longrightarrow \mathbb{Z}\), the family \N\[\N\mathcal{F}_{A,B}(f, g) =\{ E(f(a), g(b)) \colon |a| \leq A, |b| \leq B, a, b \in \mathbb{Z}, \Delta(f(a), g(b)) \neq 0\}\N\]\Nand for a subset \(P\) of the prime numbers, the counting function \N\[\N\pi(x, E(a,b), r, P) = \# \{ p \leq x \, \colon \, a_p = r, \, p \in P\}.\N\]\NIn this paper, the authors are able to obtain the average asymptotic \N\[\N\dfrac{1}{|\mathcal{F}_{A, B}(f, g)|} \sum_{|a| < A, |b| < B} \pi(x, E(f(a), g(b)), r, P) \sim C(r, f, g, P) \dfrac{\sqrt{x}}{\log(x)},\N\]\Nas \(x \rightarrow \infty\), for any integer \(r\), in the following cases:\N\begin{itemize}\N\item[1.] For functions \(f = f_{k_1}(n)=(-1)^n n^{k_1}\), \(g = g_{k_2}(n) = (-1)^n n^{k_2}\), the subset \(P\) of prime numbers given by the conditions \N\[\Np \equiv -1\, (\text{mod } 2k_1), \qquad \qquad p \equiv -1\, (\text{mod } 2k_2),\N\]\Nand values of \(A, B > x^{1+ \epsilon}\).\N\item[2.] For functions \(f = f_{a_1}(n) = (a_1)^n n^2\), \(g = g_{a_2}(n) = (a_2)^n n^2\), the subset \(P\) of prime numbers given by the conditions \N\[\N\left(\frac{a_1}{p}\right) =-1 \qquad \qquad \left(\frac{a_2}{p}\right) =-1\N\]\Nand values of \(A, B > x^{2+ \epsilon}\).\N\end{itemize}\N\NThe proof uses the Kronecker-Hurwitz class number relation. Given a prime number \(p>3\) and an integer \(r\) in the range \((-2\sqrt{p}, 2\sqrt{p})\), the number of isomorphism classes of elliptic curves over \(\mathbb{F}_p\) with \(a_p=r\) is equal to the Kronecker class number \N\[\NH(r^2 -4p) = 2 \sum_{ f^2 | r^2 -4p, \, d \equiv 0, 1\, (\text{mod } 4)} \frac{h(d)}{w(d)},\N\]\Nhere \(h(d)\) is the class number of the order \(\mathbb{Z}[(d + \sqrt{d})/2]\) with discriminant \(d\) and \(w(d)\) is the number of units in the order, \(f\) is an integer and \(d= \frac{r^2-4p}{f^2}\). Since, except some special cases, the number of elliptic curves in each class is \(\frac{p-1}{2}\), we get that the number of elliptic curves \(E(a, b)\) with \(0 \leq a, b \leq p\) and \(a_p = r\) is given by \(H(r^2 -4p) \frac{p-1}{2} + O(p)\).\N\NThe particular form of our functions \(f, g\) in cases (1) and (2) allow us to express the average sums in terms of sums of \(H(r^2 -4p)\) over congruence classes. This makes possible to estimate \N\[\N\dfrac{1}{|\mathcal{F}_{A, B}(f, g)|} \sum_{|a| < A, |b| < B} \pi(x, E(f(a), g(b)), r, P)\N\]\Nas \N\[\N\frac{1}{2} \sum_{p \leq x, \, \\\Np \equiv c \, (\text{mod } m)} \frac{H(r^2 -4p)}{p} = \frac{2}{\pi} K_r(c, m) \pi_{1/2}(x) + O \left( \frac{\sqrt{x}}{\log^2(x)} \right),\N\]\Nwhere the function \(\pi_{1/2}(x)\) is defined as \(\pi_{1/2}(x) = \int_2^x \frac{dt}{2 \sqrt{t} \log(t)}\) and the constants \(K_r(c, m)\) can be obtained as product over primes. The precise determination of some of the constants \(K_r(c, m)\) in the results, shows the existence of congruence class bias on average in terms of the occurrence of primes \(p\) such that \(a_p=r\). This bias, expressed as the ratio \(\frac{K_r(c_1, m)}{K_r(c_2, m)}\), depends on the conditions for the set \(P\) and was already observed before for supersingular primes.