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In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the quotient of the modular curve of level $n$ by Fricke involution.

Consider a modular parametrization $\mu:X_0(n)^+ \to E$ for some rational elliptic curve $E$, say of rank two. A rational point $p$ on $E$ gives rise to a collection of points $\mu^{-1}(p)$ on $X_0(n)^+$ and therefore to a finite set of the aforementioned K3 surfaces.

Question: What is special about these K3 surfaces? For example, is their Picard group of rank 19 or does it jump to 20? Do they have some other algebra-geometric characterization, such as existence of some sections of some line bundle on them? Has this been studied at all?

P.S. My guess is that the jump in the rank of K3 would be some CM condition, and thus not terribly interesting. But then what is special about these K3-s that correspond to points of rank 2 curve?

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You can ask similar questions for elliptic curves, and as far as I know, the answer to your question of "What is special..." is not much. Thus let $\mu:X_0(n)\to E$ be a modular parametrization of an elliptic curve of conductor $n$, and let $P\in E(\mathbb{Q})$. Then is there anything special about the elliptic curves parametrized by the points in $\mu^{-1}(P)$? (Aside from the obvious fact that they are Galois conjugates of one another and have marked cyclic subgroups of order $n$ that respect the Galois conjugation.) For something further to ponder, here's something that I've thought about over the years and never come up with anything interesting: How are the elliptic curves parametrized by $\mu^{-1}(kP)$ related (if at all) for $k=1,2,\ldots,\,$?

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  • $\begingroup$ Right. I am somehow hoping that by looking at a more complicated object, namely the associated K3 surface, one might discover something that is not visible at the elliptic curve level. $\endgroup$
    – Lev Borisov
    Commented Jul 30, 2014 at 19:06
  • $\begingroup$ I don't think that these K3-s are all that hard to construct from the elliptic curves. My best guess is that if $E_1\to E_2$ is the corresponding cyclic isogeny, then the K3 is the resolution of $E_1\times E_2/<(-1,-1)>$, but I have not tried to verify it. $\endgroup$
    – Lev Borisov
    Commented Jul 30, 2014 at 20:53
  • $\begingroup$ Okay, that would be interesting, to geometrically relate the K3s and the elliptic curves. But I don't see how either is related to the point on $E$, and especially, how would the group law on $E$ relate to the $E_i$'s or the K3's. $\endgroup$
    – Joe Silverman
    Commented Jul 30, 2014 at 21:00
  • $\begingroup$ It seems that the K3 is not the Kummer, but is is a double cover (i.e. related by a Shioda-Inose structure to the product abelian surface). See theorem 7.6 of that paper. $\endgroup$
    – Abhinav Kumar
    Commented Jul 30, 2014 at 23:35
  • $\begingroup$ The field of definition of the point $P$ on $X_0(n)^+$ should probably correspond to the field of the definition of the corresponding divisor on the K3 surface. So saying that it comes from a $\bf{Q}$ point just means that the Picard group of the K3 can be fully realized over a small degree number field (here, probably the $2$-torsion field of the elliptic curves). $\endgroup$
    – Abhinav Kumar
    Commented Jul 30, 2014 at 23:55

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