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Yes, that's a basis for NS$(A)$. It's enough to show that $L := 2E + 2E'$ is $\pi^*M$ for some line bundle $M$ on $A$. This is true if and only if $\ker \pi$ is contained in $K(L)$ and is isotropic f …

I believe that's right, at least when $S$ is finitely generated over $R$. See Serre's Local Fields page 21-22 (in the English translation); he states his assumptions on page 13.

A tight bound for simple $A/\mathbb{C}$ is $\rho(A) \leq 3n/2$. This follows from Proposition 5.5.7 in Birkenhake-Lange. If $A$ does not have indefinite quaternionic multiplication, the stronger bou …

No, not without some extra assumptions. Take for example $A = E \times E'$ where $E$ and $E'$ are generic elliptic curves connected by an isogeny $E \to E'$ with kernel $\mathbb{Z}/m\mathbb{Z}$, for s …

You might want to look at Ogus' A p-adic analogue of the Chowla-Selberg Formula. There he defines p-adic periods for CM motives $X$ of rank 1 over a CM field $E$. Instead of using the dR-etale compa …

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^ …

The action of $L$ on global 1-forms would give an embedding of $L$ into the algebra $M$ of $g$-by-$g$ matrices over $\mathbb{Q}$ (since we're in characteristic 0). But any maximal commutative $\mathb …

There are some papers of van Luijk, where he computes the ranks of some K3s over number fields. The trick is to note that $NS(X) \hookrightarrow NS(X_p)$, where $X_p$ is the reduction of $X$ modulo a …

I think the following should give a counterexample. Let $\mathcal{O}$ be an order in an imaginary quadratic field $K$ and $\mathcal{O}_K$, the ring of integers. Then it's not too hard to find a (non …

When $n < 4$, $A$ is isogenous to the Jacobian of a stable curve because the Torelli locus is dense. This should imply that $A$ is isogenous to the Jacobian of a smooth curve because $A$ is simple (a …

Any curve of genus greater than two, whose Jacobian $J$ is simple, will do. If it were a divisor on an abelian surface $S$, then there would be a surjection $J\to S$ with positive dimensional kernel, …

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 …

Technically speaking, an elliptic curve is a genus 1 curve with a choice of rational point. The automorphism group of an elliptic curve is the subgroup of automorphisms of the genus 1 curve that fix t …