Geodesic
Homomorphisms of Hyperelliptic Jacobians
Informatics and Automation, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 90-104.

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Let $K$ be a field of characteristic different from $2$ and $K_a$ be its algebraic closure. Let $n\ge 5$ and $m\ge 5$ be integers. Assume, in addition, that if $K$ has positive characteristic, then $n\ge 9$. Let $f(x),h(x)\in K[x]$ be irreducible separable polynomials of degree $n$ and $m$, respectively. Suppose that the Galois group of $f$ is either the full symmetric group $\mathbf S_n$ or the alternating group $\mathbf A_n$ and the Galois group of $h$ is either the full symmetric group $\mathbf S_m$ or the alternating group $\mathbf A_m$. Let us consider the hyperelliptic curves $C_f\colon y^2=f(x)$ and $C_h\colon y^2=h(x)$. Let $J(C_f)$ be the Jacobian of $C_f$ and $J(C_h)$ be the Jacobian of $C_h$. Earlier, the author proved that $J(C_f)$ is an absolutely simple abelian variety without nontrivial endomorphisms over $K_a$. In the present paper, we prove that $J(C_f)$ and $J(C_h)$ are not isogenous over $K_a$ if the splitting fields of $f$ and $h$ are linearly disjoint over $K$. 
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Yu. G. Zarhin. Homomorphisms of Hyperelliptic Jacobians. Informatics and Automation, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 90-104. http://geodesic.mathdoc.fr/item/TRSPY_2003_241_a3/

[1] Abhyankar S. S., “Projective polynomials”, Proc. Amer. Math. Soc., 125:6 (1997), 1643–1650 | DOI | MR | Zbl

[2] Curtis Ch. W., Reiner I., Methods of representation theory, V. 1, J. Wiley Sons, Chichester Brisbane, New York, Toronto, 1981 | MR

[3] Dixon J. D., Mortimer B., Permutation groups, Springer, New York, Berlin, Heidelberg, 1996 | MR

[4] Huppert B., Endliche Gruppen, I, Springer, Berlin, Heidelberg, New York, 1967 | MR | Zbl

[5] Huppert B., Blackburn N., Finite groups, III, Springer, Berlin, Heidelberg, New York, 1982 | MR

[6] Janusz G. J., Algebraic number fields, 2nd ed., Amer. Math. Soc., Providence, RI, 1996 | MR

[7] Klemm M., “Über die Reduktion von Permutationsmoduln”, Math. Ztschr., 143 (1975), 113–117 | DOI | MR | Zbl

[8] Koblitz N., $p$-Adic numbers, $p$-adic analysis, and zeta-functions, Springer, Berlin, Heidelberg, New York, 1977 | MR | Zbl

[9] Ivanov A. A., Praeger Ch. E., “On finite affine 2-arc transitive graphs”, Europ. J. Comb., 14:5 (1993), 421–444 | DOI | MR | Zbl

[10] Lang S., Algebra, 3rd ed., Addison–Wesley, Reading, MA, 1993 | MR | Zbl

[11] Mori Sh., “The endomorphism rings of some abelian varieties, II”, Japan. J. Math., 3 (1977), 105–109 | MR | Zbl

[12] Mortimer B., “The modular permutation representations of the known doubly transitive groups”, Proc. London Math. Soc. Ser. 3, 41 (1980), 1–20 | DOI | MR | Zbl

[13] Mumford D., “Theta characteristics of an algebraic curve”, Ann. Sci. École Norm. Supér. Sér. 4, 4 (1971), 181–192 | MR | Zbl

[14] Mumford D., Abelian varieties, 2nd ed., Oxford Univ. Press, London, 1974 | MR | Zbl

[15] Schur I., “Gleichungen ohne Affect”, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl., 1930, 443–449 ; Ges. Abh., 3, 191–197 | Zbl

[16] Serre J.-P., Lectures on the Mordell–Weil theorem, 2nd ed., F. Vieweg Sohn, Braunschweig, Wiesbaden, 1989 | MR | Zbl

[17] Serre J.-P., Topics in Galois theory, Jones and Bartlett Publ., Boston, London, 1992 | MR | Zbl

[18] Serre J.-P., Représentations linéaires des groupes finis, 3me éd., Hermann, Paris, 1978 | MR | Zbl

[19] Suzuki M., Group theory, I, Springer, Berlin, Heidelberg, New York, 1982 | MR | Zbl

[20] Zarhin Yu. G., “Hyperelliptic Jacobians without complex multiplication”, Math. Res. Lett., 7:1 (2000), 123–132 | MR | Zbl

[21] Zarhin Yu. G., “Hyperelliptic Jacobians and modular representations”, Moduli of abelian varieties, Progr. Math., eds. C. Faber, G. van der Geer, F. Oort, Birkhäuser, Basel, Boston, Berlin, 2001, 473–490 | MR | Zbl

[22] Zarhin Yu. G., “Hyperelliptic Jacobians without complex multiplication in positive characteristic”, Math. Res. Lett., 8:4 (2001), 429–435 | MR | Zbl

[23] Zarhin Yu. G., “Very simple $2$-adic representations and hyperelliptic Jacobians”, Moscow Math. J., 2:2 (2002), 403–431 | MR | Zbl

[24] Zarhin Yu. G., “Hyperelliptic Jacobians and simple groups $\mathrm{U}_3(2^m)$”, Proc. Amer. Math. Soc., 131:1 (2003), 95–102 | DOI | MR | Zbl