Breadcrumb
On the denominator of modular symbols for elliptic curves
Wed 01 May 2013, 16:00
Christian Wuthrich
Nottingham
Organisers: Tim Dokchitser, Daniel Loughran
ABSTRACT
Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ to which we can attach modular symbols. For each rational $r$, we may integrate the associated newform along the path from $r$ to $\infty$ in the upper half plane and compare this complex number with the Neron lattice of $E$. The theorem of Manin and Drinfel'd shows that we extract two rational numbers with a very good bound on the possible denominator. The main question I wish to answer is if we may find in the isogeny class of $E$ over $\mathbb{Q}$ an elliptic curve such that its modular symbols are all integers. Even partial answers to this, will give interesting integrality properties of special $L$-values and a strengthening of Kato's result on the main conjecture in Iwasawa theory of $E$.