#### Title

The Relationship between Conductor and Discriminant of an Elliptic Curve over Q

#### School Name

Heathwood Hall Episcopal School

#### Grade Level

9th Grade

#### Presentation Topic

Mathematics

#### Presentation Type

Non-Mentored

#### Oral Presentation Award

1st Place

#### Written Paper Award

1st Place

#### Abstract

Saito (1988) establishes a relationship between two invariants associated with a smooth projective curve, the conductor and discriminant. Saito defined the conductor of an arbitrary scheme of finite type using p-adic etale cohomology. He used a definition of Deligne for the discriminant as measuring defects in a canonical isomorphism between powers of relative dualizing sheaf of smooth projective curves. The researcher in this paper uses the fact that this relationship is analogous to that of conductor to discriminant in the case of elliptic curves, Saito's result, as well as analysis of data on conductors and discriminants to determine whether patterns exist between discriminant and conductor of elliptic curves. The researcher finds such patterns do in fact exist and discusses two main patterns: that of the conductor dividing the discriminant and that of the conductor "branching" in a predictable way. These patterns also allow for easier algorithms for computing conductors.

#### Recommended Citation

Adamo, Nico, "The Relationship between Conductor and Discriminant of an Elliptic Curve over Q" (2018). *South Carolina Junior Academy of Science*. 122.

https://scholarexchange.furman.edu/scjas/2018/all/122

#### Location

Neville 206

#### Start Date

4-14-2018 11:45 AM

#### Presentation Format

Oral and Written

The Relationship between Conductor and Discriminant of an Elliptic Curve over Q

Neville 206

Saito (1988) establishes a relationship between two invariants associated with a smooth projective curve, the conductor and discriminant. Saito defined the conductor of an arbitrary scheme of finite type using p-adic etale cohomology. He used a definition of Deligne for the discriminant as measuring defects in a canonical isomorphism between powers of relative dualizing sheaf of smooth projective curves. The researcher in this paper uses the fact that this relationship is analogous to that of conductor to discriminant in the case of elliptic curves, Saito's result, as well as analysis of data on conductors and discriminants to determine whether patterns exist between discriminant and conductor of elliptic curves. The researcher finds such patterns do in fact exist and discusses two main patterns: that of the conductor dividing the discriminant and that of the conductor "branching" in a predictable way. These patterns also allow for easier algorithms for computing conductors.