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# International Seminar on Automorphic Forms

## June 7 at 9:00 – 10:00 CEST

Determination of modular forms is one of the fundamental and interesting problems in number theory. It is known that if the Hecke eigenvalues of two newforms agree for all but finitely many primes, then both the forms are the same. In other words, the set of Hecke eigenvalues at primes determines the newform uniquely and this result is known as the multiplicity one theorem. In the case of Siegel cuspidal eigenforms of degree two, the multiplicity one theorem has been proved only recently in 2018 by Schmidt. In this talk, we refine the result of Schmidt by showing that if the Hecke eigenvalues of two Siegel eigenforms of level 1 agree at a set of primes of positive density, then the eigenforms are the same (up to a constant). We also distinguish Siegel eigenforms from the signs of their Hecke eigenvalues. The main ingredient to prove these results are Galois representations attached to Siegel eigenforms, the Chebotarev density theorem and some analytic properties of associated L-functions.