• Corpus ID: 244709641

Permutations on Weierstrass Prym eigenforms

@inproceedings{GutierrezRomo2021PermutationsOW,
  title={Permutations on Weierstrass Prym eigenforms},
  author={Rodolfo Guti'errez-Romo and Angel Pardo},
  year={2021}
}

• Rodolfo Guti'errez-Romo, Angel Pardo
• Published 26 November 2021
• Mathematics

Let X ∈ H(2) be a Veech surface of discriminant D and let G (X) be the permutation group induced by the a ne group of X on the set of Weierstrass points of X . We show thatG (X) Dih4 if D≡4 0,G (X) Dih5 if D≡8 5, andG (X) Dih6 if D≡81, whereDihn is the dihedral group of order 2n. Thus,G (X) is a weak invariant, as it can distinguish the residue class ofDmod 8, but it cannot tell di erent spin invariants apart whenD≡81. Moreover, we show that the same groups arise whenwe only consider the action of the parabolic elements of the Veech group of X on the Weierstrass points. We prove a similar result in the Weierstrass Prym locus in genus three. Indeed, we show that the permutation group induced by the a ne group on the set of xed points for the Prym involution of a Weierstrass Prym eigenform of discriminant D is isomorphic to Sym2 when D is an even quadratic residue modulo 16, and it is isomorphic to Sym3 otherwise. In genus three, the same group is also obtained when we consider only parabolic elements. Since the permutation action of the a ne group on Prym xed points is trivial in genus four and no Weierstrass Prym eigenforms exist for genus ve or larger, we completely classify this action for all Weierstrass Prym eigenforms.