Evaluating Igusa functions

Bröker, Reinier; Lauter, Kristin

doi:10.1090/S0025-5718-2014-02816-0

Evaluating Igusa functions
HTML articles powered by AMS MathViewer

by Reinier Bröker and Kristin Lauter PDF

Math. Comp. 83 (2014), 2977-2999 Request permission

Abstract:

The moduli space of principally polarized abelian surfaces is parametrized by three Igusa functions. In this article we investigate a new way to evaluate these functions by using the Siegel Eisenstein series. We explain how to compute the Fourier coefficients of certain Siegel modular forms using classical modular forms of half-integral weight. One of the results in this paper is an explicit algorithm to evaluate the Igusa functions to a prescribed precision.

References

Harald Baier, Efficient computation of singular moduli with application in cryptography, Fundamentals of computation theory (Riga, 2001) Lecture Notes in Comput. Sci., vol. 2138, Springer, Berlin, 2001, pp. 71–82. MR 1914098, DOI 10.1007/3-540-44669-9_{9}
S. Böcherer and S. Raghavan, On Fourier coefficients of Siegel modular forms, J. Reine Angew. Math. 384 (1988), 80–101. MR 929979, DOI 10.1515/crll.1988.384.80
Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
H. Cohen, Number Theory, volume II: analytic and modern tools, Springer Graduate Texts in Mathematics, vol. 240, 2007.
Henri Cohen, Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. MR 382192, DOI 10.1007/BF01436180
John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR 503901, DOI 10.1007/978-1-4612-6313-5
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
Erhard Gottschling, Explizite Bestimmung der Randflächen des Fundamentalbereiches der Modulgruppe zweiten Grades, Math. Ann. 138 (1959), 103–124 (German). MR 107020, DOI 10.1007/BF01342938
G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, Oxford University Press, 1938.
Jun-ichi Igusa, Modular forms and projective invariants, Amer. J. Math. 89 (1967), 817–855. MR 229643, DOI 10.2307/2373243
Jun-ichi Igusa, On Siegel modular forms of genus two, Amer. J. Math. 84 (1962), 175–200. MR 141643, DOI 10.2307/2372812
Jun-ichi Igusa, On Siegel modular forms genus two. II, Amer. J. Math. 86 (1964), 392–412. MR 168805, DOI 10.2307/2373172
Neal Koblitz, Introduction to elliptic curves and modular forms, 2nd ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993. MR 1216136, DOI 10.1007/978-1-4612-0909-6
Max Koecher, Zur Theorie der Modulformen $n$-ten Grades. I, Math. Z. 59 (1954), 399–416 (German). MR 59946, DOI 10.1007/BF01180269
Winfried Kohnen, Estimates for Fourier coefficients of Siegel cusp forms of degree two. II, Nagoya Math. J. 128 (1992), 171–176. MR 1197036, DOI 10.1017/S0027763000004268
W. Kohnen and D. Zagier, Values of $L$-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175–198. MR 629468, DOI 10.1007/BF01389166
H. W. Lenstra Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), no. 3, 649–673. MR 916721, DOI 10.2307/1971363
H. W. Lenstra Jr. and Carl Pomerance, A rigorous time bound for factoring integers, J. Amer. Math. Soc. 5 (1992), no. 3, 483–516. MR 1137100, DOI 10.1090/S0894-0347-1992-1137100-0
K. Mahler, On a class of non-linear functional equations connected with modular functions, J. Austral. Math. Soc. Ser. A 22 (1976), no. 1, 65–118. MR 441867, DOI 10.1017/s1446788700013367
H. L. Resnikoff and R. L. Saldaña, Some properties of Fourier coefficients of Eisenstein series of degree two, J. Reine Angew. Math. 265 (1974), 90–109. MR 352009
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216, DOI 10.1007/978-1-4684-9884-4
Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46, Princeton University Press, Princeton, NJ, 1998. MR 1492449, DOI 10.1515/9781400883943
Paul van Wamelen, Examples of genus two CM curves defined over the rationals, Math. Comp. 68 (1999), no. 225, 307–320. MR 1609658, DOI 10.1090/S0025-5718-99-01020-0
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
Annegret Weng, Constructing hyperelliptic curves of genus 2 suitable for cryptography, Math. Comp. 72 (2003), no. 241, 435–458. MR 1933830, DOI 10.1090/S0025-5718-02-01422-9