Masanari Kida

Reduction of elliptic curves over certain real quadratic number fields

The main result of this paper is that an elliptic curve having good reduction everywhere over a real quadratic field has a 2-rational point under certain hypotheses (primarily on class numbers of related fields). It extends the earlier case in which no ramification at 2 is allowed. Small fields satisfying the hypotheses are then found, and in four cases the non-existence of such elliptic curves can be shown, while in three others all such curves have been classified.

Primality Tests Using Algebraic Groups

We introduce primality tests using algebraic groups. Some previously known tests are naturally interpreted as special cases of these algebraic group tests. Moreover, in this framework we can generalize Lucas and n + 1 tests in a natural way.

Cyclic polynomials arising from kummer theory of norm algebraic tori

We study cyclic extensions arising from Kummer theory of norm algebraic tori. In particular, we compute quintic cyclic polynomials defining ‘Kummer extension’. The polynomials do not only give all the quintic cyclic extensions over the rationals by choosing the parameters but also classify all such extensions. Some arithmetic properties of the polynomials are also derived.

Nonexistence of elliptic curves having good reduction everywhere over certain quadratic fields

Abstract. The purpose of this paper is to show the nonexistence of elliptic curves having good reduction everywhere over certain quadratic fields.

Variation of the reduction type of elliptic curves under small base change with wild ramification

We study the variation of the reduction type of elliptic curves under base change. A complete description of the variation is given when the base field is the p-adic field and the base change is of small degree.

Potential Good Reduction of Elliptic Curves

We show that there is no elliptic curve defined over the field of rational numbers that attains good reduction at every finite place under quadratic base change. We also give some examples of elliptic curves that acquire good reduction everywhere under cubic or quartic base changes.

Ramification in the division fields of an elliptic curve

We study the ramification in the division fields of an elliptic curve. Applications on the class number of quadratic fields and on elliptic curves with prime power conductor are given.