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.
