Advanced topics in computational number theory

By H. Cohen. 2000

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Finiteness results for modular curves of genus at least 2 (2002, methodology): “5 respectively of [14], we can compute the relative discriminant ideal and factor it to obtain the ramified primes.”
Worst-case to average-case reductions for module lattices (2014, background): “Note that it is equivalent to give a basis of the module lattice and a pseudo-basis of the module because from the first representation, the second representation is computable in polynomial time [6,9].”
On the Arithmetic of Del Pezzo Surfaces of Degree 2 (2004, methodology): “Algorithms from [2] and [9] have been implemented in MAGMA.”
Counting nilpotent Galois extensions (2001, background): “In order to treat the case of controlled ramification we need the following result of Hecke (see [2], Thm.”
Computations in non‐commutative Iwasawa theory (2005, unknown): “ere is a standard procedure for computing values of so-called motivic Lfunctions. This method applies to L-functions of twists of elliptic curves, and we will brieﬂy outline it in our case. See [51], [13] 10.3 and [18] for details. Let Ebe an elliptic curve over Q, and let τbe a d-dimensional Artin ”
COHEN-LENSTRA HEURISTIC AND ROOTS OF UNITY (2008, methodology): “Using the efficient algorithm of Belabas for enumerating S3-fields of bounded discriminant, see [4], we have computed analogous data as above for the 2-part of the class group in both signatures.”