The modular degree, congruence primes, and multiplicity one

Amod Agashe (), Kenneth A. Ribet () and William A. Stein ()
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Amod Agashe: Florida State University, Department of Mathematics
Kenneth A. Ribet: UC Berkeley, Department of Mathematics
William A. Stein: University of Washington

A chapter in Number Theory, Analysis and Geometry, 2012, pp 19-49 from Springer

Abstract: Abstract The modular degree and congruence number are two fundamental invariants of an elliptic curve over the rational field. Frey and Müller have asked whether these invariants coincide. We find that the question 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 divisible 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.

Keywords: elliptic curves; abelian varieties; modular degree; congruence primes; multiplicity one (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-1-4614-1260-1_2

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