To every k-dimensional modular invariant vector space we associate a modular form on SL(2,Z) of weight 2k. We explore number theoretic properties of this form and found a sufficient condition for its vanishing which yields modular identities (e.g., Ramanujan-Watson’s modular identities). Furthermore, we focus on a family of modular invariant spaces coming from suitable two-dimensional spaces vi...

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let’s mention : − the control of the image of the Galois representation modulo p [37][35], − Hida’s congruence criterion outside an explicit set of primes p [21], − the freeness of the integral cohomology of the Hilbert modular variety over certain local...

Let p be a prime. We prove that if a modular unit has a pth root that is again a modular unit, then the level of that root is at most p times the level of the original unit.

For a fixed odd prime p and a representation ̺ of the absolute Galois group of Q into the projective group PGL2(Fp), we provide the twisted modular curves whose rational points supply the quadratic Q-curves of degree N prime to p that realize ̺ through the Galois action on their p-torsion modules. The modular curve to twist is either the fiber product of the modular curves X0(N) and X(p) or a cer...