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Seshadri constants on abelian surfaces are fully understood in the case ofPicard number one. Little is known so far for simple abelian surfaces of higherPicard number. In this paper we investigate principally polarized abeliansurfaces with real multiplication. They are of Picard number two and might beconsidered the next natural case to be studied. The challenge is to not onlydetermine the Seshadri constants of individual line bundles, but to understandthe whole \emph{Seshadri function} on these surfaces. Our results show on theone hand that this function is surprisingly complex: On surfaces with realmultiplication in $\mathbb Z[\sqrt e]$ it consists of linear segments that arenever adjacent to each other -- it behaves like the Cantor function. On theother hand, we prove that the Seshadri function it is invariant under aninfinite group of automorphisms, which shows that it does have interestingregular behavior globally.

Seshadri constants on principally polarized abelian surfaces with real multiplication