Congruences between modular forms modulo prime powers

  • Maximiliano Camporino

    Universidad de Buenos Aires, Argentina
  • Ariel Pacetti

    Universidad Nacional de Córdoba, Argentina

Abstract

Given a prime and an abstract odd representation with coefficients modulo (for some ) and big image, we prove the existence of a lift of to characteristic whenever local lifts exist (under minor technical conditions). Moreover, our results allow to chose the lift's inertial type at all primes but finitely many (where the lift is of Steinberg type).

We apply this result to the realm of modular forms, proving a level lowering theorem modulo prime powers and providing examples of level raising. An easy application of our main result proves that given a modular eigenform whose Galois representation is not induced from a character (i.e., has no inner twists), for all primes but finitely many, and for all positive integers , there exists an eigenform , which is congruent to modulo .

Cite this article

Maximiliano Camporino, Ariel Pacetti, Congruences between modular forms modulo prime powers. Rev. Mat. Iberoam. 34 (2018), no. 4, pp. 1609–1643

DOI 10.4171/RMI/1037