Let $X$ be a smooth genus one curve over $k$. I don't call it elliptic curve because it will have no rational points.


By **index** of $X$ we mean the *smallest degree* of a closed point on $X$; equivalently by Riemann-Roch that's the same as the smallest positive degree of a divisor, or the greatest common divisor of all degrees of closed points.


Of course, the index of a curve equals one if the field is algebraically closed, so we consider non-closed fields here. For example, a smooth cubic curve in $\mathbb{P^2}$ can have index one or three.


**Here is the **question****: can one characterize fields $k$ which admit genus one curves of index $d$?


I am especially interested in the $d = 5$ case; then the model of such curve over algebraic closure is a linear section of the Grassmannian $\mathrm{Gr}(2,5)$. For which fields do we have genus one curves of index $5$? Is there a way to figure this out without writing explicit equations?


*UPDATE:* I actually don't fully understand the case of number fields. Are all number fields allowed?



degree five genus one curves without rational points?
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 $H^1(k,E[5])$, where $E$ is the Jacobian. I assume the converse is not true (presumably some 5-torsion classes come from torsors of index 25), but I'll give examples of cohomology classes which are visibly of index 5.


Over *any* field there exists at least one elliptic curve with a rational point of order 5, or by duality, a subgroup (scheme) isomorphic to $\mu\_5$. Over $\mathbb{Q}$ there is even an elliptic curve (of conductor 50 I believe) that contains both. In the first case, consider the induced map
$$\mathrm{Hom}(G\_k,\mathbb{Z}/5\mathbb{Z}) = H^1(k,\mathbb{Z}/5\mathbb{Z}) \stackrel{i}{\to} H^1(k, E[5])$$
In the dual case, consider $H^1(k, \mu\_5) \simeq k^{\times}/k^{\times 5}$. The kernel of $i$ comes from the $k$-rational points in the kernel of the dual isogeny, so it has $\mathbb{F}\_5$-dimension at most 1, in either case. The image of $i$ gives you torsors which have index 1 or 5 (since they are twists of a 5-isogeny). The index is 1 precisely if the cohomology class "comes from rational points", i.e if its in the image of $E(k)/5E(k)$ under the coboundary map. If $k$ is a number field, then the latter is a finite group, whereas $H^1(k, \mathbb{Z}/5\mathbb{Z})$ or $H^1(k, \mu\_5)$ are both infinite (see the explicit descriptions above). So there are infinitely many index 5 PHS $X$.


