Doctoral thesis, 2019

The present thesis contains three papers dealing with two arithmetic problems on curves of genus one, which are closely related to elliptic curves.

The first problem is to study the density of rational points presented in Papers I and II. We give uniform upper bounds for the number of rational points of bounded height on smooth curves of genus one given by ternary cubics or complete intersections of two quadratic surfaces. The main tools used in these two papers are descent on elliptic curves and determinant methods. While working with the rational points counting problem, one need to deal with the smoothness of geometric objects and the bad reduction of polynomials. To characterize these properties, there is a classical object called the discriminant which naturally appears.

The above discriminant gives an inspiration to the study of the next problem in the thesis concerning invariants of models of genus one curves presented in Paper III. Here an invariant of a genus one curve is a polynomial in the coefficients of the model defining the curve that is stable under certain linear transformations. The discriminant is a classical example of an invariant. Besides that, there are two more important invariants which generate the ring of invariants of genus one models over a field. Fisher considered these invariants over the field of rational numbers and normalized them such that they are moreover defined over the integers. We provide an alternative way to express these normalized invariants using a natural connection to modular forms. In the case of the discriminant of ternary cubics over the complex numbers, we also present another approach using determinantal representations. This latter approach produces a natural connection to theta functions.

The common idea in the thesis is to link a smooth genus one curve to a Weierstrass form, which is a more well-understood object.

The first problem is to study the density of rational points presented in Papers I and II. We give uniform upper bounds for the number of rational points of bounded height on smooth curves of genus one given by ternary cubics or complete intersections of two quadratic surfaces. The main tools used in these two papers are descent on elliptic curves and determinant methods. While working with the rational points counting problem, one need to deal with the smoothness of geometric objects and the bad reduction of polynomials. To characterize these properties, there is a classical object called the discriminant which naturally appears.

The above discriminant gives an inspiration to the study of the next problem in the thesis concerning invariants of models of genus one curves presented in Paper III. Here an invariant of a genus one curve is a polynomial in the coefficients of the model defining the curve that is stable under certain linear transformations. The discriminant is a classical example of an invariant. Besides that, there are two more important invariants which generate the ring of invariants of genus one models over a field. Fisher considered these invariants over the field of rational numbers and normalized them such that they are moreover defined over the integers. We provide an alternative way to express these normalized invariants using a natural connection to modular forms. In the case of the discriminant of ternary cubics over the complex numbers, we also present another approach using determinantal representations. This latter approach produces a natural connection to theta functions.

The common idea in the thesis is to link a smooth genus one curve to a Weierstrass form, which is a more well-understood object.

genus one

discriminant

modular form

Elliptic curve

height

descent

determinantal representation

invariant

rational point

theta function.

determinant method

Chalmers, Mathematical Sciences, Algebra and geometry

Journal of Number Theory,; Vol. 189(2018)p. 138-146

**Journal article**

Acta Arithmetica,; Vol. 186(2018)p. 301-318

**Journal article**

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

Mathematics

Geometry

978-91-7905-204-1

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4671

Chalmers University of Technology

Pascal, Hörsalsvägen 1

Opponent: Professor Fabien Pazuki, University of Copenhagen, Denmark.