Computational aspects of modular forms and p-adic triple symbol

Thesis

In this thesis we study a p-adic symbol for triples of modular forms which was proposed to the author by Henri Darmon. Our main achievements are as follows. We prove the various symmetry properties of this p-adic triple product. We develop and successfully implement an efficient algorithm for calculating it; in the case in which all the forms have weight greater than two we require an auxiliary non-vanishing hypothesis. And we illustrate the application of our algorithm with numerous examples. A curious consequence of our work, relating to our non-vanishing hypothesis, is an efficient method to calculate certain Poincare pairings in higher weight.

Our symbol is intimately related to p-adic L-functions for triples of modular forms. However, we do not study at all continuity properties of our p-adic triple symbol. Indeed, any well-behaved variation of our symbol in the first variable is likely to be extremely subtle to study, and there may well be no way at all of making sense of this.

Authors: Ghantous, W

• DOI: 10.5287/ora-yjaqjkmrg
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: slope projection; p-adic Abel-Jacobi map; nearly overconvergent projection; ordinary projection; Poincaré pairing; p-adic triple symbol
• Subjects: Computational number theory