Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-24T14:03:41.432Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing overconvergent forms for small primes

Part of: Computational number theory Number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 April 2015

Jan Vonk
Jan Vonk*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct explicit bases for spaces of overconvergent $p$-adic modular forms when $p=2,3$ and study their interaction with the Atkin operator. This results in an extension of Lauder’s algorithms for overconvergent modular forms. We illustrate these algorithms with computations of slope sequences of some $2$-adic eigencurves and the construction of Chow–Heegner points on elliptic curves via special values of Rankin triple product L-functions.

MSC classification

Primary: 11F03: Modular and automorphic functions 11Y16: Algorithms; complexity 11-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 250 - 257
Copyright
© The Author 2015 

References

Buzzard, K., ‘Questions about slopes of modular forms’, Astérisque 298 (2005).Google Scholar
Buzzard, K. and Calegari, F., ‘A counterexample to the Gouvêa–Mazur conjecture’, C. R. Math. Acad. Sci. Paris 338 (2004) 751–753.Google Scholar
Buzzard, K. and Calegari, F., ‘Slopes of 2-adic overconvergent forms’, Compos. Math. 141 (2005) 591–604.Google Scholar
Calegari, F., ‘The Coleman–Mazur eigencurve is proper at integral weights’, Algebra Number Theory 2 (2008) no. 2.Google Scholar
Coleman, R., ‘Classical and overconvergent modular forms’, Invent. Math. 124 (1996) 215–241.Google Scholar
Coleman, R., ‘ p-Adic Banach spaces and families of modular forms’, Invent. Math. 127 (1997) 417–479.Google Scholar
Darmon, H., Daub, M., Lichtenstein, S. and Rotger, V., ‘Algorithms for Chow–Heegner points via iterated integrals’, Math. Comp., to appear, Preprint, available at http://www.math.mcgill.ca/darmon/pub/pub.html.Google Scholar
Darmon, H., Lauder, A. and Rotger, V., ‘Stark points and $p$ -adic iterated integrals attached to modular forms of weight one’, Preprint, available at http://people.maths.ox.ac.uk/lauder/papers/DLR.pdf.Google Scholar
Darmon, H. and Rotger, V., ‘Diagonal cycles and Euler systems I: a p-adic Gross–Zagier formula’, Ann. Sci. Éc. Norm. Supér. 47 (2014) 779–832.Google Scholar
Darmon, H., Rotger, V. and Sols, I., ‘Iterated integrals, diagonal cycles, and rational points on elliptic curves’, Publ. Math. Besançon Algèbre Théorie Nr. 2 (2012) 19–46.Google Scholar
Gouvêa, F., Arithmetic of p-adic modular forms , Lecture Notes in Mathematics 1304 (Springer, 1988).Google Scholar
Gouvêa, F. and Mazur, B., ‘Families of modular eigenforms’, Math. Comp. 58 (1992).Google Scholar
Katz, N., ‘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, 1973) 69–190.Google Scholar
Lauder, A., ‘Computations with classical and p-adic modular forms’, LMS J. Comput. Math. 14 (2011) 214–231.Google Scholar
Lauder, A., ‘Efficient computation of Rankin p-adic L-functions’, Computations with modular forms (eds Böckle, G. and Wiese, G.; Springer, 2014).Google Scholar
Mazur, B., ‘Modular curves and the Eisenstein ideal’, Publ. Math. Inst. Hautes Études Sci. 47 (1977) 33–186.Google Scholar
Wan, D., ‘Dimension variation of classical and p-adic modular forms’, Invent. Math. 133 (1998) 449–463.Google Scholar