Distinguishing Siegel eigenforms from Hecke eigenvalues

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 determine
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.