The average size of Ramanujan sums over cubic number fields

Stark's conjecture over complex cubic number fields

Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark’s conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) an...

No 17-torsion on elliptic curves over cubic number fields

Consider, for d an integer, the set S(d) of prime numbers p such that: there exists a number field K of degree d, an elliptic curve E overK, and a point P in E(K) of order p. It is a well-known theorem of Mazur, Kamienny, Abramovich and Merel that S(d) is finite for every d; moreover S(1) and S(2) are known. In [7], we tried to answer a question of Kamienny and Mazur by determining S(3), and we...

Class-number Problems for Cubic Number Fields

The Euclidean Algorithm in Cubic Number Fields

In this note we present algorithms for computing Euclidean minima of cubic number fields; in particular, we were able to find all normEuclidean cubic number fields with discriminants −999 < d < 104.

Ramanujan Sums as Supercharacters

The theory of supercharacters, recently developed by DiaconisIsaacs and André, can be used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since s...