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Computing periods of cusp forms and modular elliptic curves. (English) Zbl 0894.11027

The author is well-known for his monumental “Algorithms for modular elliptic curves” [Cambridge Univ. Press (1992; Zbl 0872.14041)], where he gives exhaustive data for these curves for all conductors \(N\) up to 1000. Now he reports on the extension of his work for conductors up to 5077. He describes improved methods speeding up the computations without which the work could not have been done.
Let \(f\) be a newform of weight 2 on \(\Gamma_0 (N)\), and let \(f\) be rational, i.e., let its Hecke eigenvalues be rational. Then the set \(\Lambda_f\) of all periods of \(f\) is a lattice in \(\mathbb{C}\), and \(E_f= \mathbb{C}/\Lambda_f\) is the modular elliptic curve attached to \(f\) with conductor \(N\). The fact that \(f\) is an eigenform for the Fricke involution is used to establish a new formula for a period of \(f\), expressing this period as a power series in \(\exp (-2\pi/d\sqrt{N})\) for some small positive integer \(d\). The series converges much faster than the previously known power series expression in \(\exp(-2\pi/cN)\) with some other small positive integer \(c\).
In the former computations a major part of time and space had been used to find the exact imaginary period of \(f\), and this would be very costly for larger values of \(N\). Now for \(N> 3000\) an indirect approach is used: One quickly computes an integer multiple of the imaginary period and then guesses which multiple it is. The new method is fast, but it has the disadvantage that one might only find a sublattice \(\Lambda_f'\) of finite index in \(\Lambda_f\). Then \(E_f'= \mathbb{C}/\Lambda_f'\) and \(E_f\) are elliptic curves of conductor \(N\) with the same \(L\)-functions, but \(E_f'\) might only be isogenous to \(E_f\). The author worked up to \(N= 5077\) since he wished to verify that there was no curve of rank 3 with conductor below the known example of conductor 5077. The work will be continued.

MSC:

11G05 Elliptic curves over global fields
11F11 Holomorphic modular forms of integral weight
14Q05 Computational aspects of algebraic curves
11Y16 Number-theoretic algorithms; complexity

Citations:

Zbl 0872.14041

Software:

ecdata
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References:

[1] Birch B. J., Modular functions of one variable IV (1975)
[2] Cohen H., A course in computational algebraic number theory (1993) · Zbl 0786.11071
[3] Cremona J. E., Algorithms for modular elliptic curves (1992) · Zbl 0758.14042
[4] Cremona J. E., Math. Comp. 64 pp 1235– (1995)
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[6] Edixhoven B., Arithmetic algebraic geometry pp 25– (1991)
[7] Tingley D. J., Ph.D. thesis, in: Elliptic curves uniformised by modular functions (1975)
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