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I imagine this is very much open. Even special cases of this question seem hard. Consider weight two modular forms of level $\Gamma_0(N)$ with real quadratic coefficient field $K$. Galois orbits of …

The answer is yes, and it's fairly elementary. By the usual 2-descent, the curve $C$ gives a class $c$ in $H^1(\mathbb{Q},E[2])$, where $E$ is the Jacobian you wrote down. As you vary $d$, the groups …

I'll address the case $d = 5$ over any number field, without recourse to Gross-Zagier formulas and Tate-Shafarevich groups. If $X$ has index 5, then it has order 5 in $H^1(k,E)$, hence comes from $H^ …

One way to get a counterexample is to take any elliptic curve $E$ over a quadratic field $K$ whose conductor is a prime ideal $\mathfrak{p}$ lying over a split prime $p$ in $\mathbb{Q}$. This means t …

The reference is his 1995 Math Annalen paper "On the $p$-adic height of Heegner cycles". See e.g. the discussion on page 6. The fact that the $q$-expansion of height pairings is modular is also used i …

Bhargava and Shankar have conjectured that the average size of the $n$-Selmer group $S_n(E)$ is the sum of the divisors of $n$. They proved this for $n \leq 5$. If you assume Equidistribution of …