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The Modular Degree, Congruence Primes, and Multiplicity One Amod Agashe 1
 

Summary: 1
The Modular Degree, Congruence Primes, and Multiplicity One
Amod Agashe 1
Kenneth A. Ribet William A. Stein
Abstract.
The modular degree and congruence number are two fundamental in-
variants of an elliptic curve over the rational field. Frey and M¨uller
have asked whether these invariants coincide. We find that the ques-
tion has a negative answer, and show that in the counterexamples,
multiplicity one (defined below) does not hold. At the same time, we
prove a theorem about the relation between the two invariants: the
modular degree divides the congruence number, and the ratio is divis-
ible only by primes whose squares divide the conductor of the elliptic
curve. We discuss the ratio even in the case where the square of a
prime does divide the conductor, and we study analogues of the two
invariants for modular abelian varieties of arbitrary dimension.
1 Introduction
Let E be an elliptic curve over Q. By [BCDT01], we may view E as an
abelian variety quotient over Q of the modular Jacobian J0(N), where N is
the conductor of E. We assume that the kernel of the map J0(N) E is

  

Source: Agashe, Amod - Department of Mathematics, Florida State University

 

Collections: Mathematics