Tenth Algorithmic Number Theory Symposium ANTSX

Computing modular forms: a very explicit perspective
Nils Skoruppa
Abstract: Since Manin's work on modular forms in the 1970s it is theoretically well understood how to compute elliptic modular forms of integral weight (different from 1), and since two decades, based on work of various authors, this problem is also practically well understood. However, the current implementations of the underlying ideas lack certain properties. It would be desirable, for example, to have an algorithm, which, for a given elliptic curve over Q generates a closed formula for the associated newform and the associated modular forms of half integral weight.
In this talk, I shall review the main method for computing modular forms of integral weight, and I shall try to work out its beautiful simplicity. I shall explain what, to my understanding, is missing in current implementations, and I shall propose methods for filling the gaps. In particular, I shall describe how to generate algorithmically closed formulas for modular forms, and how to compute modular forms of half integral weight.
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