Multiplicity one theorem and modular symbols1)

A Weak Multiplicity-one Theorem for Siegel Modular Forms

In a recent paper by Breulmann and Kohnen [BK99], the authors obtain a weak multiplicity-one result on (integral weight) Siegel-Hecke cuspidal eigenforms of degree 2, showing that such forms are completely determined by their coefficients on matrices of the form mS, where S is primitive and m is square-free. To show this, they twist Andrianov’s identity relating the Maaß-Koecher series and the ...

A Theorem on Analytic Strong Multiplicity One

Let K be an algebraic number field, and π = ⊗πv an irreducible, automorphic, cuspidal representation of GLm(AK) with analytic conductor C(π). The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant c depending on ε > 0,m, and K only, such that π can be decided completely by its local components πv with norm N(v) < c · C...

The modular number , the congruence number , and multiplicity one ∗

The Modular Degree, Congruence Primes and Multiplicity One

We answer a question of Frey and Müller about whether or not the modular degree and congruence number of elliptic curves are equal. We give examples in which they are not, prove a theorem relating them, and make a conjecture about the extent to which they differ. We also obtain relations between analogues of the modular degree and congruence number for modular abelian varieties, and give new ex...

On a Spectral Analogue of the Strong Multiplicity One Theorem

We prove spectral analogues of the classical strong multiplicity one theorem for newforms. Let Γ1 and Γ2 be uniform lattices in a semisimple group. Suppose all but finitely many irreducible unitary representations (resp. spherical) of G occur with equal multiplicities in L(Γ1\G) and L(Γ2\G). Then L(Γ1\G) ∼= L(Γ2\G) as G modules (resp. the spherical spectra of L(Γ1\G) and L(Γ2\G) are equal).