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A new unicity theorem and Erdös’ problem for polarized semi-abelian varieties

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Abstract

In 1988 Erdös asked if the prime divisors of x n − 1 for all n = 1, 2, … determine the given integer x; the problem was affirmatively answered by Corrales-Rodrigáñez and Schoof (J Number Theory 64:276–290, 1997) [but a solution could also be deduced from an earlier result of Schinzel (Bull Acad Polon Sci 8:307–309, 2007)] together with its elliptic version. Analogously, Yamanoi (Forum Math 16:749–788, 2004) proved that the support of the pulled-back divisor f * D of an ample divisor on an abelian variety A by an algebraically non-degenerate entire holomorphic curve f : C → A essentially determines the pair (A, D). By making use of the main theorem of Noguchi (Forum Math 20:469–503, 2008) we here deal with this problem for semi-abelian varieties; namely, given two polarized semi-abelian varieties (A 1, D 1), (A 2, D 2) and algebraically non-degenerate entire holomorphic curves f i : C → A i , i = 1, 2, we classify the cases when the inclusion \({{\rm{Supp}}\, f_1^*D_1\subset {\rm{Supp}}\, f_2^* D_2}\) holds. We shall remark in §5 that these methods yield an affirmative answer to a question of Lang formulated in 1966. Our answer is more general and more geometric than the original question. Finally, we interpret the main result of Corvaja and Zannier (Invent Math 149:431–451, 2002) to provide an arithmetic counterpart in the toric case.

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References

  1. Ax, J.: Transcendence and differential algebraic geometry, Actes Congrès intern. math 1970. Tome 1, pp. 483–485, Gauthier-Villars, Paris (1971)

  2. Ax J.: Some topics in differential algebraic geometry II. Am. J. Math. 94, 1205–1213 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barsky D., Bézivin J.P., Schinzel A.: Une caractérisation arithmétique de suites récurrentes linéaires. J. Reine Angew. Math. 494, 73–84 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Corrales-Rodrigáñez C., Schoof R.: The support problem and its elliptic analogue. J. Number Theory 64, 276–290 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Corvaja P., Zannier U.: Diophantine equations with power sums and universal Hilbert sets. Indag. Math. N.S. 9(3), 317–332 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Corvaja P., Zannier U.: Finiteness of integral values for the ratio of two linear recurrences. Invent. Math. 149, 431–451 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Corvaja P., Zannier U.: A lower bound for the height of a rational function at S-unit points. Monatsh. Math. 144(3), 203–224 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Corvaja P., Zannier U.: Some cases of Vojta’s conjecture on integral points over function fields. J. Algebraic Geom. 17(2), 295–333 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fujimoto H.: The uniqueness problem of meromorphic maps into the complex projective space. Nagoya Math. J. 58, 1–23 (1975)

    MathSciNet  MATH  Google Scholar 

  10. Lang S.: Introduction to Transcendental Numbers. Addison-Wesley, Reading (1966)

    MATH  Google Scholar 

  11. Larsen M.: The support problem for abelian varieties. J. Number Theory 101, 398–403 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Laurent M.: Equations diophantiennes exponentielles. Invent. Math. 78, 299–327 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nevanlinna R.: Le Théorème de Picard-Borel et la théorie des fonctions méromorphes. Gauthier-Villars, Paris (1939)

    Google Scholar 

  14. Noguchi J.: Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties. Nagoya Math J. 83, 213–233 (1981)

    MathSciNet  MATH  Google Scholar 

  15. Noguchi, J.: Introduction to Complex Analysis (Translated from the 1993 Japanese original), Transl. Math. Monog. vol. 168, American Mathematical Society, Providence (1998)

  16. Noguchi J.: On holomorphic curves in semi-abelian varieties. Math. Z. 228, 713–721 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Noguchi, J., Ochiai, T.: Geometric function theory in several complex variables, Japanese edition, Iwanami, Tokyo, 1984; English Translation, Transl. Math. Mono. vol. 80. American Mathematical Society, Providence (1990)

  18. Noguchi J., Winkelmann J., Yamanoi K.: The second main theorem for holomorphic curves into semi-Abelian varieties. Acta Math. 188(1), 129–161 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Noguchi J., Winkelmann J., Yamanoi K.: The second main theorem for holomorphic curves into semi-Abelian varieties II. Forum Math. 20, 469–503 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. van der Poorten A.J.: Solution de la conjecture de Pisot sur le quotient de Hadamard de deux fractions rationnelles. C. R. Acad. Sci. Sér. I Math. 306, 97–102 (1988)

    MATH  Google Scholar 

  21. Rumely, R.: Notes on van der Poorten’s proof of the Hadamard Quotient Theorem: Part I, II, Séminaire de Théorie des Nombres de Paris 1986–87, pp. 349–382, pp. 383–409, Progr. Math. vol. 75, Birkhäuser, Boston (1988)

  22. Schinzel, A.: On the congruence \({a^x\equiv b}\) (mod p), Bull. Acad. Polon. Sci. 8 , 307–309 (1960); Selecta vol. 2, 909–911, EMS 2007

  23. Schinzel A.: On power residues and exponential congruences. Acta Arith. XXVII, 397–420 (1975)

    MathSciNet  Google Scholar 

  24. Schinzel, A.: On exponential congruences, Selecta vol. 2, 996–1000, Heritage of European Mathematics, EMS 2007, originally published in Mat. Zapiski (in Russian, 1996)

  25. Schmid E.M.: Some theorems on value distributions of meromorphic functions. Math. Z. 120, 61–92 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schmidt, W.M.: Linear recurrence sequences. In: Diophantine approximations. Lecture Notes in Mathematics, vol. 1819, pp. 171–247. Springer, Berlin (2003)

  27. Yamanoi K.: Holomorphic curves in abelian varieties and intersection with higher codimensional subvarieties. Forum Math. 16, 749–788 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, University of Udine, Via delle Scienze, 206-33100, Udine, Italy

    Pietro Corvaja

  2. Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan

    Junjiro Noguchi

Authors
  1. Pietro Corvaja
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  2. Junjiro Noguchi
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Correspondence to Pietro Corvaja.

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Research supported in part by Grant-in-Aid for Scientific Research (S) 17104001.

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Corvaja, P., Noguchi, J. A new unicity theorem and Erdös’ problem for polarized semi-abelian varieties. Math. Ann. 353, 439–464 (2012). https://doi.org/10.1007/s00208-011-0692-x

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  • DOI: https://doi.org/10.1007/s00208-011-0692-x

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