On the Lang-Trotter conjecture for two elliptic curves
Let $E$ be an elliptic curve over the rationals $\mathbb{Q}$ without complex multiplication. The Lang-Trotter conjecture [\textit{S. Lang} and \textit{H. Trotter}, Frobenius distributions in \(\text{GL}_2\)-extensions. Distribution of Frobenius automorphisms in \(\text{GL}_2\)-extensions of the rational numbers. Berlin-Heidelberg-New York: Springer (1976; Zbl 0329.12015)] postulates that the number of primes $p\leq x$ of good reduction of $E$ such that the Frobenius trace $a_p(E)$ of $E$ at $p$ equals a fixed integer $t$ is asymptotic with $c_{E,t}\cdot x^{\frac12}/\log x$. Here the constant $c_{E,t}$ may be split $c_{E, t}=r_{E,t}\cdot c_t$, where the ingredients $r_{E,t}$ (a rational number which depends only on the cokernel of the Galois representation associated with $E$) and $c_t$ (an infinite Euler product over the primes $\ell$) are well understood. The authors propose a similar conjecture (Conjecture 1.2) for the case of two elliptic curves $E_i/\mathbb{Q}$ ($i=1,2$), both non-CM and non-isogenous. Namely, letting $a_p(E_i)$ be the Frobenius trace of $E_i$ at the prime $p$ as above, then $\#\{p\leq x\mid a_p(E_i)=t_i\}$ ought to be asymptotic with $c_{E_1,E_2,t_1,t_2}\log\log x$. Again, the conjectural constant $c_{E_1,E_2,t_1,t_2}$ decomposes into factors $r_{E_1,E_2,t_1,t_2}$ and $c_{t_1,t_2}$, where the first one depends in a transparent fashion on the Galois representation of $E_1\times E_2$. The conjecture is compatible with some results ``on average'' of \textit{E. Fouvry} and \textit{M. R. Murty} [Lond. Math. Soc. Lect. Note Ser. 215, 91--102 (1995; Zbl 0852.11029)]. Besides stating and discussing the conjecture, the merits of the paper are as follows. The authors develop a probabilistic model similar to Lang-Trotter's, which predicts the precise form of Conjecture 1.2. Then they describe in their Theorem 1.4 the universal constant $c_{t_1,t_2}$ as an Euler product. Finally, their Theorem 1.5 gives an explicit description as a rational function in $\ell$ of the $\ell$-th Euler factor of $c_{t_1,t_2}$, provided that $t_2=\pm t_1$. The proof of Theorem 1.5 is through complicated case-by-case calculations, which apparently are too complex to carry out if $t_2\neq\pm t_1$. However, they offer a conjecture (Conjecture 1.9, based on some numerical evidence) about the precise form of the $\ell$-th factor even for $t_2\neq\pm t_1$.
Lang-Trotter conjecture for two elliptic curves
Lang-Trotter constant for two elliptic curves
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