Algebraic equations represent an important subject in mathematics. The solution sets of equations are viewed as geometric objects such as curves, surfaces, etc. Among them, there are classical objects called elliptic curves. These types of curves can be deﬁned by cubic polynomials. Elliptic curves appear in many ﬁelds of mathematics such as number theory, algebraic geometry, complex analysis, etc.
This thesis studies two important arithmetic problems on curves of genus one, which are natural generalizations of elliptic curves. The ﬁrst problem is to study the density of points over the rational numbers on such curves. If there are inﬁnitely many rational points on a genus one curve, one might quantify the inﬁnite by considering the density of points in a bounded domain. We provide upper bounds for the number of rational points in a box of given size on smooth curves of genus one deﬁned by ternary cubics or intersections of two quadric surfaces, which are two typical ways of representing curves of genus one. The implicit constants in these bounds are moreover uniform which are independent of the equations deﬁning the curves.
When considering working with the density of rational points on genus one curves problem, there is a classical notion called the discriminant which naturally appears to characterize the smoothness of the curves. This discriminant is a classical example of what is called an invariant. Here an invariant of a genus one curve is a polynomial in the coeﬃcients of the polynomials deﬁning the curve, which is unchanged under certain natural transformations. The discriminant led the author to work on the second problem in this thesis which concerns invariants of genus one curves.
Among all invariants of genus one curves over a ﬁeld, there are two important ones which can be used to describe all the others. These two invariants were normalized by Fisher in such a way that they are primitive polynomials with integer coeﬃcients. In this thesis, we provide an alternative way to express these normalized invariants by naturally relating to modular forms. These modular forms are functions on the upperhalf plane satisfying a certain transformation and can be expressed in terms of Taylor series. If all coeﬃcients of the Taylor series are integral, the corresponding modular form is deﬁned over the integers and so is the associated invariant. The behavior of invariants will then be predicted by the transformation property of modular forms. The special case of the discriminant of ternary cubics over the complex numbers is also treated by an another approach.
MANH HUNG TRAN