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In case anyone's still interested, it turns out one can prove the following: if $p= 3 \,\mathrm{mod}\,4 $ is a prime, then $v_p(H_k) \geq v_p(k!)-\frac{k}{p^2-1}$ (nb. the RHS here is $\sim k \cdot \f …

It's fairly easy to write down families of elliptic curves over $\mathbb{Q}(t)$ such that almost every (i.e. when the "height" of $t$ is sufficiently large) curve in the family has positive rank over …

Suppose $V/K$ is a smooth projective variety. Let $\mathrm{Ch}^{j}(V)_{0}$ be the group of codimension-$j$ homologically trivial $K$-rational cycles on $V$, modulo rational equivalence. A conjecture …

Define numbers $H_k$ for integers $k\geq 4$ by $\sum_{x \in \mathbf{Z}[i]}x^{-k}=\frac{H_k}{k!} \omega^k$, where $\omega=\frac{\Gamma(1/4)^2}{\sqrt{2\pi}}$. These are nonzero when $4|k$, and Hurwitz …

To partly answer your last two questions: It's not too hard to write down a sequence of abelian varieties $A_i/\mathbf{Q}$ such that $\mathrm{rank}A_i(\mathbf{Q})=0$ but $\mathrm{rank}A_i(\mathbf{Q}( …

Suppose X is a smooth projective variety, say over $\mathbb{Q}$ for simplicity. Let $F$ be a finite extension of $\mathbb{Q}$. Let $\mathrm {Ch}^{r}(X/F)$ denote the Chow group of codimension $r$ alg …

Suppose $E/\mathbb{Q}$ is an elliptic curve with rank zero. According to the conjecture of Birch and Swinnerton-Dyer, the special value $L(1,E_{/\mathbb{Q}})$ should be equal (up to some harmless fac …

In Swinnerton-Dyer's charming paper "An application of computing to classfield theory", in Cassels-Frohlich, he discusses the genesis of the Birch/Swinnerton-Dyer conjecture and numerical tests of it …

Heuristic arguments due to (I believe) Delauney predict that every prime divides the order of the Tate-Shafarevich group of infinitely many elliptic curves over $\mathbf{Q}$. However, is it even know …

This is a very naive question. Fix a prime $p$ and consider the forgetful map from varieties over $\mathbf{Q}$ to varieties over $\mathbf{Q}_p$. Is there a conjectural "purely $p$-adic" characteriz …

Consider a finite-dimensional $\mathbf{Q}_p$-vector space $V$ and a continuous representation $\rho : G_{\mathbf{Q}_p} \to \mathrm{GL}(V)$. Fontaine introduced various $\mathbf{Q}_p$-algebras with $G …

Suppose $E/ \mathbf{Q}$ is an elliptic curve with additive reduction at a prime $p$. Is there an easy way to tell if $E$ is a quadratic twist of an elliptic curve $E'/\mathbf{Q}$ with good reduction …

Suppose $K$ is a finite extension of $\mathbf{Q}_p$, $A=K[[T_1,\dots,T_n]]$, $V$ a finite-rank free $A$-module, and $\rho:G_{\mathbf{Q}} \to \mathrm{GL}(V)$ a continuous Galois representation. Are th …

A well-known mathematician once explained the following conjecture to me, as "an example of how little we know about cycles of codimension $\geq 2$." Let $C$ be a curve defined over a number field $k …

(edited for clarity) In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 curve …