13 problems

Let $\mathcal E$ be an elliptic curve over $\mathbb{Q}(T)$ and let $A_{\mathcal E}(p)$ denote the average of the Frobenius traces over its fibers, as defined above. Nagao's conject…

Let $A$ be a generic abelian $g$-fold over $\mathbb{Q}$, and let $\pi_{A,T}(x)$ count primes $p\leq x$ of good reduction for which $a_p(A)=T$. Write $H_A$ for the adelic Galois ima…

Let $\mathcal{E}$ be a twist family, such as a quadratic, cubic, quartic, or sextic twist family. For every $E\in\mathcal{E}$ and prime $p$, let $\overline{a}_p(E)$ be the normaliz…

Let $p$ be prime and let $u\in\mathbb F_p$ be a parameter for which the 15 elliptic curves obtained from the five elliptic-curve factors and the shifts $u\mapsto u-22$ and…

Let $E$ be a nonsingular elliptic curve over the rational numbers $\mathbb{Q}$ without complex multiplication and conductor $N\geq 1$. Suppose that, for primes $p$ of good reductio…

Let $E$ be an elliptic curve over ${\mathbb Q}$, and let $a_p(E)$ denote the trace of Frobenius of $E$ at a prime $p$ of good reduction. An integer $a$ is subject to a congruence o…

Halved Frobenius trace conjecture. Similarly to the corresponding conjecture for $|a_p(E)|$, one should have

Let $E$ be an elliptic curve defined over $\mathbb Q$, with conductor $N_E$, without complex multiplication, and torsion conductor $m_E$. For a prime $p$ of good reduction, let…

Let $A$ be a principally polarized abelian variety over $\mathbb Q$ of dimension $g\geq 1$, with conductor $N_A$, and let $\rho_A$ be the product of its $\ell$-adic Galois represen…

Let $E/\mathbb{Q}$ be a non-CM elliptic curve, and let … count primes $p\leq x$ for which $a_p(E)=\pm\lfloor 2\sqrt{p}\rfloor$, allowing the trace to vary with $p$. Extremal-prime…

James–Trifković–Trifković–Weiss–Zaharescu conjecture. The number of extremal primes $p\leq x$ is asymptotically equal to

Akbary–Park fixed-traces conjecture. There exists a constant $C(E_1,E_2,t_1,t_2)\geq0$ such that

Refined Lang–Trotter conjecture. Uniformly in this range,