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Group actions on the tropical Hesse pencil. (English) Zbl 1362.14066
Summary: Addition of points on the tropical Hesse curve is realized via its intersections with two tropical lines. Then the addition formula for the points on the curve is reduced from the one for the level-three theta functions through the ultradiscretization procedure. In addition, a tropical analogue of the Hessian group $$G_{216}$$, the group of linear automorphisms acting on the Hesse pencil, is investigated; it is shown that the dihedral group $$\mathcal {D}_3$$ of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.

##### MSC:
 14T05 Tropical geometry (MSC2010) 14H52 Elliptic curves 37P99 Arithmetic and non-Archimedean dynamical systems
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##### References:
 [1] Artebani, M., Dolgachev, I.L The Hesse pencil of plane cubic curves. (2006). arXiv:math/0611590v3 · Zbl 1192.14024 [2] Gathmann, A.: Tropical algebraic geometry. (2006). arXiv:math/0601322v1 · Zbl 1109.14038 [3] Gathmann, A., Kerber, M.: A Riemann-Roch theorem in tropical geometry (2006). arXiv:math/0612129v2 · Zbl 1187.14066 [4] Hesse, O, Über die elimination der variabeln aus drei algebraischen gleichungen vom zweiten, grade mit zwei variabeln, J. Reine Angew. Math., 28, 68-96, (1844) · ERAM 028.0817cj [5] Hesse, O, Über die wendepunkte der curven dritter ordnung, J. Reine Angew. Math., 28, 97-102, (1844) · ERAM 028.0818cj [6] Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical Algebraic Geometry. Birkhäuser, Basel (2007) · Zbl 1162.14300 [7] Kajiwara, K; Kaneko, M; Nobe, A; Tsuda, T, Ultradiscretization of a solvable two-dimensional chaotic map associated with the Hesse cubic curve, Kyushu J. Math., 63, 315-338, (2009) · Zbl 1177.14061 [8] Kajiwara, K; Nobe, A; Tsuda, T, Ultradiscretization of solvable one-dimensional chaotic maps, J. Phys. A: Math. Theoret., 41, 395202 (13pp), (2008) · Zbl 1149.37023 [9] Mikhalkin, G., Zharkov, I.: Tropical curves, their Jacobians and theta functions. (2006). arXiv:math/0612267v1 · Zbl 1152.14028 [10] Nobe, A, Ultradiscrete QRT maps and tropical elliptic curves, J. Phys. A: Math. Theoret., 41, 125205 (12pp), (2008) · Zbl 1145.35097 [11] Nobe, A, On the addition formula for the tropical Hesse pencil, RIMS Kokyuroku, 1765, 188-208, (2011) [12] Nobe, A.: A tropical analogue of the Hessian group. (2011). arXiv:1104.0999v1 · Zbl 1254.37049 [13] Nobe, A, An ultradiscrete integrable map arising from a pair of tropical elliptic pencils, Phys. Lett. A, 375, 4178-4182, (2011) · Zbl 1254.37049 [14] Quispel, GRW; Roberts, AG; Thompson, CJ, Integrable mappings and soliton equations II, Physica, 34D, 183-192, (1989) · Zbl 0679.58024 [15] Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. (2003). arXiv:math/0306366v2 · Zbl 1093.14080 [16] Shaub, HC; Schoonmaker, HE, The Hessian configuration and its relation to the group of order 216, Am. Math. Mon., 38, 388-393, (1931) · Zbl 0002.34703 [17] Schröder, E, Über iterierte functioned, Math. Ann., 3, 296-322, (1871) · JFM 02.0200.01 [18] Tsuda, T, Integrable mappings via rational elliptic surfaces, J. Phys. A: Math. Gen., 37, 2721-2730, (2004) · Zbl 1060.14051 [19] Umeno, K, Method of constructing exactly solvable chaos, Phys. Rev. E, 55, 5280-5284, (1997) [20] Vigeland, M.D.: The group law on a tropical elliptic curve. (2004). arXiv:math/0411485v1 · Zbl 1167.14018
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