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Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian. (English) Zbl 0974.14021
According to Torelli’s theorem any smooth projective curve over an algebraically closed field is determined by its polarized Jacobian variety. G. Humbert in [Journ. de Math. (5) 6, 279-386 (1900; JFM 31.0455.04)] gave the first example of two non-isomorphic curves (of genus 2) with the same unpolarized Jacobian. Later many more such examples have been given. The present paper constructs for any positive integer \(n\) distinct smooth plane quartics and one hyperelliptic curve of genus 3 such that all \(n+1\) of these curves have isomorphic unpolarized Jacobian variety. The construction produces explicit equations for curves over \(\mathbb{C}\) whose Jacobians are isomorphic as unpolarized abelian varieties.

MSC:
14H40 Jacobians, Prym varieties
14H45 Special algebraic curves and curves of low genus
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[1] B. W. BROCK: Superspecial curves of genera two and three, doctoral dissertation, Princeton University, 1993.
[2] E. CIANI: I varii tipi possibili di quartiche piane più volte omologico-armoniche, Rend. Circ. Mat. Palermo 13 (1899), 347-373. · JFM 30.0527.07
[3] Ciro Ciliberto and Gerard van der Geer, Nonisomorphic curves of genus four with isomorphic (nonpolarized) Jacobians, Classification of algebraic varieties (L’Aquila, 1992) Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 129 – 133. · Zbl 0823.14011 · doi:10.1090/conm/162/01530 · doi.org
[4] Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. · Zbl 0786.11071
[5] Tsuyoshi Hayashida, A class number associated with the product of an elliptic curve with itself, J. Math. Soc. Japan 20 (1968), 26 – 43. · Zbl 0186.26501 · doi:10.2969/jmsj/02010026 · doi.org
[6] Tsuyoshi Hayashida and Mieo Nishi, Existence of curves of genus two on a product of two elliptic curves, J. Math. Soc. Japan 17 (1965), 1 – 16. · Zbl 0132.41701 · doi:10.2969/jmsj/01710001 · doi.org
[7] Everett W. Howe, Constructing distinct curves with isomorphic Jacobians, J. Number Theory 56 (1996), no. 2, 381 – 390. · Zbl 0842.14019 · doi:10.1006/jnth.1996.0026 · doi.org
[8] Everett W. Howe, Constructing distinct curves with isomorphic Jacobians in characteristic zero, Internat. Math. Res. Notices 4 (1995), 173 – 180. · Zbl 0832.14019 · doi:10.1155/S1073792895000134 · doi.org
[9] E. W. HOWE, F. LEPRÉVOST, AND B. POONEN: Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math. 12 (2000), 315-364. CMP 2000:10
[10] G. HUMBERT: Sur les fonctiones abéliennes singulières (deuxième mémoire), J. Math. Pures Appl. (5) 6 (1900), 279-386.
[11] Tomoyoshi Ibukiyama, Toshiyuki Katsura, and Frans Oort, Supersingular curves of genus two and class numbers, Compositio Math. 57 (1986), no. 2, 127 – 152. · Zbl 0589.14028
[12] Akikazu Kuribayashi and Kaname Komiya, On Weierstrass points and automorphisms of curves of genus three, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 253 – 299. · Zbl 0494.14012
[13] Akikazu Kuribayashi and Eitaro Sekita, On a family of Riemann surfaces. I, Bull. Fac. Sci. Engrg. Chuo Univ. 22 (1979), 107 – 129. · Zbl 0447.30033
[14] Herbert Lange, Abelian varieties with several principal polarizations, Duke Math. J. 55 (1987), no. 3, 617 – 628. · Zbl 0657.14023 · doi:10.1215/S0012-7094-87-05531-1 · doi.org
[15] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. · Zbl 0221.10029
[16] Alexius Maria Vermeulen, Weierstrass points of weight two on curves of genus three, Universiteit van Amsterdam, Amsterdam, 1983. Dissertation, University of Amsterdam, Amsterdam, 1983; With a Dutch summary. · Zbl 0534.14010
[17] Paul van Wamelen, Examples of genus two CM curves defined over the rationals, Math. Comp. 68 (1999), no. 225, 307 – 320. · Zbl 0906.14025
[18] Paul van Wamelen, Proving that a genus 2 curve has complex multiplication, Math. Comp. 68 (1999), no. 228, 1663 – 1677. · Zbl 0936.14033
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