Abstracts for Number theory lunch seminar

One says that the $p$-adic $L$-function $L_p(f,s)$ of a modular form $f$ has a trivial zero
if the interpolation property forces $L_p(f,s)$ to vanish at some integer $s=m.$
This phenomenon was first studied at 1980's by Mazur, Tate and Teitelbaum.
For modular forms of even weight they formulated a precise conjecture
about the value of the derivative $L_p'(f,s)$ at $s=m.$
This conjecture was proved by Greenberg-Stevens (using $p$-adic families of modular forms)
and by Kato-Kurihara-Tsuji (using Euler systems) around 1998.

We are interested in proving the triviality of rational points over
Atkin-Lehner's quotients of Shimura curves. In fact it is conjectured
that, except for finitely many exceptions, these quotients only have
special rational points.

Let p, q be prime numbers. We consider the quotient of the Shimura
curve X^{pq}, of discriminant pq,  by the Atkin-Lehner involution w_q.
Under certain explicit congruence conditions, known as the "cas non
ramifié de Ogg", Parent and Yafaev have found a criterion for the

ATR points were introduced by Henri Darmon as a conjectural construction of
algebraic points on certain elliptic curves for which the Heegner
point method is not available. They can be explicitly computed in
particular examples, and Darmon and Logan gathered some numerical evidence
supporting the conjecture. However, due to some restrictions on the
algorithm, they only numerically tested the construction on elliptic curves
which all happen to be also equipped with classical Heegner points. In a
joint work with Marc Masdeu we improve the algorithm used by Darmon and

Dessins d'Enfants are seemingly simple combinatorial objects that have been introduced by Grothendieck in order to encode coverings of the two-sphere ramified at three points. Such a covering defines an algebraic curve defined over a number field, and by a celebrated theorem of Belyi, every algebraic curve defined over a number field occurs in this way. As a consequence there is a faithful action of the absolute Galois group of the rationals on dessins d'enfants.