We establish Ramanujan-style congruences modulo certain primes $$\ell $$ between an Eisenstein series of weight k, prime level p and a cuspidal newform in the $$\varepsilon -eigenspace Atkin–Lehner operator inside space cusp forms k for $$\Gamma _0(p)$$ . Under mild assumption, this refines result Gaba–Popa. use these recent work Ciolan, Languasco third author on Euler–Kronecker constants, to q...

These are the (more detailed) notes accompanying a talk that I am to give at the University of Pennsylvania on July 21, 2006. The topic is rational points on Atkin-Lehner twists of the modular curves X0(N). Apart from being an interesting Diophantine problem in its own right, there is an ulterior motive: Q-rational points correspond to “elliptic Q-curves” and thus to projective Galois represent...

We describe an algorithm that we used to compute the q-expansions of all weight 2 cusp forms prime level at most $$2,\!000,\!000$$ and dimension 6. also present verify there was only one form 7 or more per Atkin-Lehner eigenspace for levels between $$10,\!000$$ $$1,\!000,\!000$$ . Our is based on Mestre’s Méthode des Graphes, involves supersingular isogeny graphs Wiedemann’s finding minimal pol...

We describe an algorithm that computes explicit models of hyperelliptic Shimura curves attached to an indefinite quaternion algebra over Q and Atkin-Lehner quotients of them. It exploits Cerednik-Drinfeld’s nonarchimedean uniformisation of Shimura curves, a formula of Gross and Zagier for the endomorphism ring of Heegner points over Artinian rings and the connection between Ribet’s bimodules an...

Let A/Q be a modular abelian variety. We establish criteria to prevent a given quaternion algebra over a totally real number field to be the endomorphism algebra of A over Q̄. We accomplish this by analyzing the representation of Gal (Q̄/Q) on the points of N-torsion of A at primes N which ramify in B and by applying descent techniques to certain covers of Shimura varieties. Our result also appli...

Let X 0 (N) be the Atkin-Lehner quotient of the modular curve X0(N) associated to the Fricke involution wN . Assume N ≥ 5 prime and endow the real locus X 0 (N)(R) with the real topology. In this paper we revisit a special case of a result due to Ogg on the connected components of X 0 (N)(R). Then we obtain a formula for the homology class of each connected component of X 0 (N)(R) in terms of M...

Let C be the image of a canonical embedding φ of the Atkin-Lehner quotient X 0 (N) associated to the Fricke involution wN . In this note we exhibit some relations among the rational points of C. For each g = 3 (resp. the first g = 4) curve C we found that there are one or more lines (resp. planes) in P whose intersection with C consists entirely of rational Heegner points or the cusp point, whe...

Let A be an elliptic curve over Q of square free conductor N . Suppose A has a rational torsion point of prime order r such that r does not divide 6N . We prove that then r divides the order of the cuspidal subgroup C of J0(N). If A is optimal, then viewing A as an abelian subvariety of J0(N), our proof shows more precisely that r divides the order of A ∩ C. Also, under the hypotheses above, we...