Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of squares and two cone points with angle each, we set up and parametrize the classification into four diagrams. Our main result is to provide formulae for enumeration of square-tiled surfaces in these four diagrams, completing the detailed count for genus two. The formulae are in terms of various well-studied arithmetic functions, enabling us to give asymptotics for each diagram. Interestingly, two of the four cylinder diagrams occur with asymptotic density 1/4, but the other diagrams occur with different (and irrational) densities.
Mots clés : Square-tiled surface, translation surface, counting, primitive
Sunrose T. Shrestha 1
CC-BY 4.0
@article{AMBP_2020__27_1_83_0,
author = {Sunrose T. Shrestha},
title = {Counting {Formulae} for {Square-tiled} {Surfaces} in {Genus} {Two}},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {83--123},
publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
volume = {27},
number = {1},
year = {2020},
doi = {10.5802/ambp.392},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.392/}
}
TY - JOUR AU - Sunrose T. Shrestha TI - Counting Formulae for Square-tiled Surfaces in Genus Two JO - Annales mathématiques Blaise Pascal PY - 2020 SP - 83 EP - 123 VL - 27 IS - 1 PB - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.392/ DO - 10.5802/ambp.392 LA - en ID - AMBP_2020__27_1_83_0 ER -
%0 Journal Article %A Sunrose T. Shrestha %T Counting Formulae for Square-tiled Surfaces in Genus Two %J Annales mathématiques Blaise Pascal %D 2020 %P 83-123 %V 27 %N 1 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.392/ %R 10.5802/ambp.392 %G en %F AMBP_2020__27_1_83_0
Sunrose T. Shrestha. Counting Formulae for Square-tiled Surfaces in Genus Two. Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 83-123. doi : 10.5802/ambp.392. https://ambp.centre-mersenne.org/articles/10.5802/ambp.392/
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