Modp Hecke operators and congruences between modular forms

Ribet, Kenneth A.

doi:10.1007/bf01393341

Cited by 66 publications

(42 citation statements)

References 13 publications

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“…We say that a prime is a congruence prime for E if it divides the congruence number r E . Congruence primes have been studied by Doi, Hida, Ribet, Mazur and others (see, e.g., [Rib83,§1]), and played an important role in Wiles's work [Wil95] on Fermat's last theorem. Frey and Mai-Murty have observed that an appropriate asymptotic bound on the modular degree is equivalent to the abc-conjecture (see [Fre97,p.544] and [Mur99,p.180]).…”

Section: S2(z)mentioning

confidence: 99%

The modular degree, congruence primes, and multiplicity one

Agashe¹,

Ribet

Stein

2011

Number Theory, Analysis and Geometry

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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.

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Section: S2(z)mentioning

confidence: 99%

The modular degree, congruence primes, and multiplicity one

Agashe¹,

Ribet

Stein

2011

Number Theory, Analysis and Geometry

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“…Surᏼ 0 /ᏼ 0 qui s'identifie au groupe des composantes de la fibre en p de J 0 ( p) ‫ޚ‬ , l'accouplement , n'est autre que l'accouplement de monodromie (voir [Illusie 1991] ou l'appendice de [Bertolini et Darmon 1997]). [Ribet 1983, théorème 2.2 ; Emerton 2002, proposition 1.3]). Les homomorphismes de ‫-ޔ‬modules et ‫-ޔ‬modules …”

Section: Introductionunclassified

Module supersingulier, formule de Gross–Kudla et points rationnels de courbes modulaires

Rebolledo¹

2008

Pacific J. Math.

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We show how the Gross-Kudla formula about triple product L-functions allows us to construct degree-zero elements of the supersingular module annihilated by the winding ideal. Using the method of Parent, we apply those results to the study of rational points on modular curves, determining a set of primes of analytic density 1−9/2 10 for which the quotient of X 0 ( p r ) (r > 1) by the Atkin-Lehner operator w p r has no rational points other than the cusps and the CM points.

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“…This condition is equivalent to assuming there are no other normalized eigenforms g ∈ S 2κ−n (Γ 1 ; O) that are eigenvalue equivalent to f modulo l. One can see [13,28] for further discussion. For a particular f this condition can be easily checked using [6] or [36].…”

Section: Theorem 14 Let κ and N Be Positive Even Integers With κ > Nmentioning

confidence: 99%

Congruence primes for Ikeda lifts and the Ikeda ideal

Brown

Keaton

2015

Pacific J. Math.

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Abstract. Let f be a newform of level 1 and weight 2κ − n for κ and n positive even integers. In this paper we study congruence primes for the Ikeda lift of f . In particular, we consider a conjecture of Katsurada stating that primes dividing certain L-values of f are congruence primes for the Ikeda lift. Instead of focusing on a congruence to a single eigenform, we deduce a lower bound on the number of all congruences between the Ikeda lift of f and forms not lying in the space spanned by Ikeda lifts.

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Modp Hecke operators and congruences between modular forms

Cited by 66 publications

References 13 publications

The modular degree, congruence primes, and multiplicity one

The modular degree, congruence primes, and multiplicity one

Module supersingulier, formule de Gross–Kudla et points rationnels de courbes modulaires

Congruence primes for Ikeda lifts and the Ikeda ideal

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