Joint IAS/Princeton University Number Theory Seminar

The modular Jacobians decompose, up to isogeny, into the abelian varieties $X_f$ cut out by cuspforms $f$ of weight 2, and a conjecture attributed to Serre posits that $X_f$ has infinitely many ordinary primes. Similarly for the André motives in the Hilbert modular varieties for totally real fields, cut out by cuspforms of parallel weight 2.We report on some methods for verifying the conjecture that are effective for some cuspforms, together with some examples where they fall short.