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Computations with classical and p-adic modular forms

  • Alan G. B. Lauder (a1)

Abstract

We present p-adic algorithms for computing Hecke polynomials and Hecke eigenforms associated to spaces of classical modular forms, using the theory of overconvergent modular forms. The algorithms have a running time which grows linearly with the logarithm of the weight and are well suited to investigating the dimension variation of certain p-adically defined spaces of classical modular forms.

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References

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[1]Bernstein, D., ‘Fast multiplication and its applications’, Algorithmic number theory: lattices, number fields, curves and cryptography, Mathematical Sciences Research Institute Publications 44 (eds Buhler, J. P. and Stevenhagen, P.; Cambridge University Press, Cambridge, 2008) 325384.
[2]Buzzard, K., ‘Questions about slopes of modular forms’, Astérique 298 (2005) 115.
[3]Buzzard, K. and Calegari, F., ‘A counterexample to the Gouvéa–Mazur conjecture’, C. R. Math. Acad. Sci. Paris 338 (2004) no. 10, 751753.
[4]Cohen, H., A course in computational algebraic number theory, Graduate Texts in Mathematics 138 (Springer, Berlin, 1993).
[5]Coleman, R., ‘Classical and overconvergent modular forms’, Invent. Math. 124 (1996) 215241.
[6]Coleman, R., ‘p-adic Banach spaces and families of modular forms’, Invent. Math. 127 (1997) 417479.
[7]Coleman, R., Stevens, G. and Teitelbaum, J., ‘Numerical experiments on families of p-adic modular forms’, Computational perspectives on number theory, AMS/IP Studies in Advanced Mathematics 7 (American Mathematical Society, Providence, RI, 1998) 143158.
[8]Darmon, H. and Pollack, R., ‘Efficient calculation of Stark–Heegner points via overconvergent modular symbols’, Israel J. Math. 153 (2006) 319354.
[9]Edixhoven, B., ‘Introduction, main results, context’, Computational aspects of modular forms and galois representations, Annals of Mathematics Studies 176 (eds Couveignes, J.-M. and Edixhoven, B.; Princeton University Press, Princeton, NJ, 2011).
[10]von zur Gathen, J. and Gerhard, J., Modern computer algebra (Cambridge University Press, Cambridge, 1999).
[11]Gouvêa, F., ‘Where the slopes are’, J. Ramanujan Math. Soc. 16 (2001) no. 1, 7599.
[12]Gouvêa, F. and Mazur, B., ‘Families of modular eigenforms’, Math. Comp. 58 (1992) no. 198, 793805.
[13]Gouvêa, F. and Mazur, B., ‘On the characteristic power series of the U operator’, Ann. Inst. Fourier (Grenoble) 43 (1993) no. 2, 301312.
[14]Gouvêa, F. and Mazur, B., ‘Searching for p-adic eigenfunctions’, Math. Res. Lett. 2 (1995) 515536.
[15]Katz, N. M., ‘p-adic properties of modular schemes and modular forms’, Modular forms in one variable III, Lecture Notes in Mathematics 350 (eds Deligne, P. and Kuyk, W.; Springer, New York, 1973) 69190.
[16]Kilford, L., Modular forms: a classical and computational introduction (Imperial College Press, London, 2008).
[17]Kilford, L., ‘On the U p operator acting on p-adic overconvergent modular forms when X 0(p) has genus 1’, J. Number Theory 130 (2010) 586594.
[18]Loeffler, D., ‘Spectral expansions of overconvergent modular functions’, Int. Math. Res. Not. IMRN 2007 (2007) doi:10.1093/imrn/rnm050.
[19]Pollack, R. and Stevens, G., ‘‘Overconvergent modular symbols and p-adic L-functions’,’, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011) 142.
[20]Stein, W., Modular forms, a computational approach, Graduate Studies in Mathematics 79 (American Mathematical Society, Providence, RI, 2007).
[21]Wan, D., ‘Dimension variation of classical and p-adic modular forms’, Invent. Math. 133 (1998) 449463.
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LMS Journal of Computation and Mathematics
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  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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