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Mosc. Math. J., 2002, Volume 2, Number 2, Pages 403–431 (Mi mmj59)  

This article is cited in 26 scientific papers (total in 26 papers)

Very simple 2-adic representations and hyperelliptic Jacobians

Yu. G. Zarhin

Abstract: Let $K$ be a field of characteristic zero, $n\ge 5$ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $\mathrm{S}_n$ or the alternating group $\mathrm{A}_n$. Let $C\colon y^2 = f(x)$ be the corresponding hyperelliptic curve and $X = J(C)$ its Jacobian defined over $K$. For each prime $\ell$ we write $V_{\ell}(X)$ for the $\mathbf{Q}_{\ell}$-Tate module of $X$ and $e_{\lambda}$ for the Riemann form on $V_{\ell}(X)$ attached to the theta divisor. Let $\mathfrak{sp}(V_{\ell}(X),e_{\lambda})$ be the $\mathbf{Q}_{\ell}$-Lie algebra of the symplectic group of $e_{\lambda}$. Let $\mathfrak{g}_{\ell,X}$ be the $\mathbf{Q}_{\ell}$-Lie algebra of the image of the Galois group $\mathrm{Gal}(K)$ of $K$ in $\mathrm{Aut}(V_{\ell}(X))$. Assuming that $K$ is finitely generated over $\mathbb{Q}$, we prove that $\mathfrak{g}_{\ell,X}=\mathbf{Q}_{\ell}\operatorname{Id}\oplus \mathfrak{sp}(V_{\ell}(X),e_{\lambda})$ where $\operatorname{Id}$ is the identity operator.

Key words and phrases: Abelian varieties, $\ell$-adic representations, hyperelliptic Jacobians, very simple representations


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MSC: Primary 14H40; Secondary 14K05, 11G30, 11G10
Received: September 8, 2001; in revised form February 28, 2002

Citation: Yu. G. Zarhin, “Very simple 2-adic representations and hyperelliptic Jacobians”, Mosc. Math. J., 2:2 (2002), 403–431

Citation in format AMSBIB
\by Yu.~G.~Zarhin
\paper Very simple 2-adic representations and hyperelliptic Jacobians
\jour Mosc. Math.~J.
\yr 2002
\vol 2
\issue 2
\pages 403--431

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    This publication is cited in the following articles:
    1. Yu. G. Zarhin, “Endomorphism rings of certain Jacobians in finite characteristic”, Sb. Math., 193:8 (2002), 1139–1149  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Yu. G. Zarhin, “Homomorphisms of Hyperelliptic Jacobians”, Proc. Steklov Inst. Math., 241 (2003), 79–92  mathnet  mathscinet  zmath
    3. Zarhin Yu.G., “The endomorphism rings of Jacobians of cyclic covers of the projective line”, Math. Proc. Cambridge Philos. Soc., 136:2 (2004), 257–267  crossref  mathscinet  zmath  adsnasa  isi
    4. Zarhin Yu.G., “Non-supersingular hyperelliptic Jacobians”, Bull. Soc. Math. France, 132:4 (2004), 617–634  crossref  mathscinet  zmath  isi
    5. Vasiu A., “Surjectivity criteria for p-adic representations, part II”, Manuscripta Math, 114:4 (2004), 399–422  crossref  mathscinet  zmath  isi
    6. Zarhin Y.G., “Endomorphism algebras of superelliptic Jacobians”, Geometric Methods in Algebra and Number Theory, Progress in Mathematics, 235, 2005, 339–362  crossref  mathscinet  zmath  isi
    7. Zarhin Y.G., “Very simple representations: Variations on a theme of Clifford”, Progress in Galois Theory, Developments in Mathematics, 12, 2005, 151–168  crossref  mathscinet  zmath  isi
    8. Elkin A., “Hyperelliptic Jacobians with real multiplication”, J. Number Theory, 117:1 (2006), 53–86  crossref  mathscinet  zmath  isi  elib
    9. Zarhin Yu.G., “Superelliptic Jacobians”, Diophantine Geometry, Proceedings, CRM Series, 4, 2007, 363–390  mathscinet  zmath  isi
    10. Vasiu A., “Some cases of the Mumford-Tate conjecture and Shimura varieties”, Indiana Univ. Math. J., 57:1 (2008), 1–75  crossref  mathscinet  zmath  isi
    11. Rohde J.Ch., Cyclic coverings, Calabi-Yau manifolds and complex multiplication, Lecture Notes in Math., 1975, Springer-Verlag, Berlin, 2009, x+228 pp.  crossref  mathscinet  zmath  isi
    12. Zarhin Yu.G., “Endomorphisms of superelliptic Jacobians”, Math. Z., 261:3 (2009), 691–707  crossref  mathscinet  zmath  isi  elib
    13. Katz N.M., “Lang-Trotter Revisited”, Bull Amer Math Soc, 46:3 (2009), 413–457  crossref  mathscinet  zmath  isi
    14. Zarhin Yu.G., “Families of absolutely simple hyperelliptic Jacobians”, Proc. Lond. Math. Soc. (3), 100:1 (2010), 24–54  crossref  mathscinet  zmath  isi
    15. Xue J., Zarhin Yu.G., “Hodge Groups of Certain Superelliptic Jacobians”, Math Res Lett, 17:2 (2010), 371–388  crossref  mathscinet  zmath  isi
    16. Xue J., Zarhin Yu.G., “Centers of Hodge groups of superelliptic Jacobians”, Transform Groups, 15:2 (2010), 449–482  crossref  mathscinet  zmath  isi  elib
    17. Xue J., “Hodge Groups of Certain Superelliptic Jacobians II”, Math Res Lett, 18:4 (2011), 579–590  crossref  mathscinet  zmath  isi  elib
    18. Zarhin Yu.G., “Hodge Classes on Certain Hyperelliptic Prymians”, Arithmetic, Geometry, Cryptography and Coding Theory, Contemporary Mathematics, 574, eds. Aubry Y., Ritzenthaler C., Zykin A., Amer Mathematical Soc, 2012, 171–183  crossref  mathscinet  zmath  isi  elib
    19. Dietmann R., “On the Distribution of Galois Groups”, Mathematika, 58:1 (2012), 35–44  crossref  mathscinet  zmath  isi
    20. Katz N.M., “Equidistribution Questions For Universal Extensions”, Exp. Math., 23:4 (2014), 452–464  crossref  mathscinet  zmath  isi
    21. Katz N.M., “Wieferich Past and Future”, Topics in Finite Fields, Contemporary Mathematics, 632, eds. Kyureghyan G., Mullen G., Pott A., Amer Mathematical Soc, 2015, 253–270  crossref  mathscinet  zmath  isi
    22. Zarhin Yu.G., “Two-Dimensional Families of Hyperelliptic Jacobians With Big Monodromy”, Trans. Am. Math. Soc., 368:5 (2016), 3651–3672  crossref  mathscinet  zmath  isi
    23. Banaszak G., Kedlaya K.S., “Motivic Serre Group, Algebraic Sato-Tate Group and Sato-Tate Conjecture”, Frobenius Distributions: Lang-Trotter and Sato-Tate Conjectures, Contemporary Mathematics, 663, eds. Kohel D., Shparlinski I., Amer Mathematical Soc, 2016, 11–44  crossref  mathscinet  zmath  isi
    24. St. Petersburg Math. J., 29:1 (2018), 81–106  mathnet  crossref  mathscinet  isi  elib
    25. Zarhin Yu.G., “Endomorphism Algebras of Abelian Varieties With Special Reference to Superelliptic Jacobians”, Geometry, Algebra, Number Theory, and Their Information Technology Applications, Springer Proceedings in Mathematics & Statistics, 251, eds. Akbary A., Gun S., Springer, 2018, 477–528  crossref  mathscinet  isi  scopus
    26. Katz N.M., “a Note on Galois Representations With Big Image”, Enseign. Math., 65:3-4 (2019), 271–301  crossref  mathscinet  isi
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