2
$\begingroup$

Consider the Fermat quartic surface $$ x^4 + y^4 + z^4 + t^4 = 0 $$ over an algebraically closed field $k$ of characteristics $p$, where $p \equiv 3$ ($\mathrm{mod}$ $4$). Is there the full classification of elliptic fibrations (up to an isomorphism) on it? In particular, can you give examples of isotrivial elliptic fibrations (j-invariants are constants)?

$\endgroup$
  • 6
    $\begingroup$ For $p \equiv 3 \bmod 4$ this K3 surface is the "supersingular" surface of Artin invariant $1$; elliptic fibrations correspond to even lattices of rank $20$ with $L^*/L \cong ({\bf Z}/p{\bf Z})^2$, which is a "full classification" of sorts but the number of such lattices grows rapidly with $p$ (there are $52$ for $p=3$, and I recently counted $2683$ for $p=7$). $\endgroup$ – Noam D. Elkies May 20 '18 at 14:05
  • 5
    $\begingroup$ For isotrivial fibrations, there's at least $Y^2 = X^3 - (t^3-t)^2 X$, or more generally $y^2 = Q(t)^3 P(x/Q(t))$ where $u^2 = P(t)$ and $u^2 = Q(t)$ are supersingular elliptic curves in characteristic $p$. These K3 surfaces must all be isomorphic with the Fermat quartic, because the $e=1$ surface is unique; but finding explicit isomorphisms can be tricky. $\endgroup$ – Noam D. Elkies May 20 '18 at 15:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.