4 problems

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…

Conjecture on simultaneous supersingular reduction. There is a polynomial, with degree and coefficients depending on $E$ and $E'$, such that

Let $E$ and $E'$ be two elliptic curves defined over a number field $K$. Assume that $E$ and $E'$ are non-CM and non-isogenous over $\overline{\mathbb{Q}}$. Let … where…

Let $E_1$ and $E_2$ be as in the preceding product setting, let $T\in\mathbb{Z}$ be fixed, and let $\Phi$ denote the relevant Sato–Tate trace density. Integral heuristic. The heuri…