Some Non-Gorenstein Hecke Algebras Attached to Spaces of Modular Forms

Kilford, L. J. P.

doi:10.1006/jnth.2002.2803

Cited by 13 publications

(26 citation statements)

References 10 publications

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“…Interestingly, and perhaps contrary to general expectations, L. Kilford [14] has recently found examples of non-Eisenstein m for which T m is non-Gorenstein. (The results of [15] show that such m are necessarily of residue characteristic two and ordinary.)…”

Section: Theorem 04 the Natural Morphism T −→ End T (X ) Is An Isommentioning

confidence: 73%

“…Thus in this case Theorem 0.5 is genuinely a list of equivalent possibilities, that may or may not hold in any particular case. (It is mentioned in [14] that William Stein has implemented Theorem 0.5 as a means of testing whether or not T m is Gorenstein, by taking advantage of the fact that condition (ii) is amenable to being checked by computer algebra. )…”

Section: Theorem 04 the Natural Morphism T −→ End T (X ) Is An Isommentioning

confidence: 99%

“…induced by the connected-étale short exact sequence of the -divisible group of the Néron model of [14], and so our description applies in particular to these cases. …”

Section: Some Remarks On the Case When T M Is Non-gorensteinmentioning

confidence: 99%

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Supersingular elliptic curves, theta series and weight two modular forms

Emerton

2002

J. Amer. Math. Soc.

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This paper deals with two subjects and their interaction. The first is the problem of spanning spaces of modular forms by theta series. The second is the commutative algebraic properties of Hecke modules arising in the arithmetic theory of modular forms.Let p be a prime, and let B denote the quaternion algebra over Q that is ramified at p and ∞ and at no other places. If L is a left ideal in a maximal order of B, then L is a rank four Z-module equipped in a natural way with a positive definite quadratic form [6, §1]. (We shall say that L is a rank four quadratic space, and remark that the isomorphism class of L as a quadratic space depends only on the left ideal class of L in its maximal order.) Eichler [5] proved that the theta series of L is a weight two modular form on Γ 0 (p), and that as L ranges over a collection of left ideal class representatives of all left ideals in all maximal orders of B these theta series span the vector space of weight two modular forms on Γ 0 (p) over Q.In this paper we strengthen this result as follows: if L is as above, then the q-expansion of its theta series Θ(L) has constant term equal to one and all other coefficients equal to even integers. Suppose that f is a modular form whose qexpansion coefficients are even integers, except perhaps for its constant term, which we require merely to be an integer. It follows from Eichler's theorem that f may be written as a linear combination of Θ(L) (with L ranging over a collection of left ideals of maximal orders of B) with rational coefficients. We show that in fact these coefficients can be taken to be integers.Let T denote the Z-algebra of Hecke operators acting on the space of weight two modular forms on Γ 0 (p). The proof that we give of our result hinges on analyzing the structure of a certain T-module X . We can say what X is: it is the free Zmodule of divisors supported on the set of singular points of the (reducible, nodal) curve X 0 (p) in characteristic p. The key properties of X , which imply the above result on theta series, are that the natural map T −→ End T (X ) is an isomorphism, and that furthermore X is locally free of rank one in a Zariski neighbourhood of the Eisenstein ideal of T. We remark that it is comparatively easy to prove the analogous statements after tensoring with Q, for they then follow from the fact that X is a faithful T-module. Indeed, combining this with the semi-simplicity of the Q-algebra T⊗ Z Q, one deduces that X ⊗ Z Q is a free T⊗ Z Q-module of rank one, and in particular that the map T ⊗ Z Q −→ End T⊗ Z Q (X ⊗ Z Q) is an isomorphism.

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Section: Theorem 04 the Natural Morphism T −→ End T (X ) Is An Isommentioning

confidence: 73%

Section: Theorem 04 the Natural Morphism T −→ End T (X ) Is An Isommentioning

confidence: 99%

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Supersingular elliptic curves, theta series and weight two modular forms

Emerton

2002

J. Amer. Math. Soc.

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“…The reason one calls this "multiplicitly one" is that if the canonical two dimensional representation ρ m over T/m attached to m (e.g., see [Rib90,Prop. 5 [Kil02] found examples of failure of multiplicity one where N is prime and the residue characteristic of m is 2. See also [Wie07] and [KW08] for examples of failure of multiplicity one in the Γ 1 (N ) context.…”

Section: Multiplicity One and Its Failurementioning

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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“…Using results of [Kil02], Adam Joyce [Joy05] proves that there is a new optimal quotient of J 0 (431) with Manin constant 2. Joyce's methods also produce examples with Manin constant 2 at levels 503 and 2089.…”

Section: Examples Of Nontrivial Manin Constantsmentioning

confidence: 99%

The Manin Constant

Agashe¹,

Ribet²,

Stein³

2006

Pure and Applied Mathematics Quarterly

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The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and Swinnerton-Dyer. One conjectures that this constant is 1; it is known to be an integer. After surveying what is known about the Manin constant, we establish a new sufficient condition that ensures that the Manin constant is an odd integer. Next, we generalize the notion of the Manin constant to certain abelian variety quotients of the Jacobians of modular curves; these quotients are attached to ideals of Hecke algebras. We also generalize many of the results for elliptic curves to quotients of the new part of J 0 (N ), and conjecture that the generalized Manin constant is 1 for newform quotients. Finally an appendix by John Cremona discusses computation of the Manin constant for all elliptic curves of conductor up to 130000.

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Some Non-Gorenstein Hecke Algebras Attached to Spaces of Modular Forms

Cited by 13 publications

References 10 publications

Supersingular elliptic curves, theta series and weight two modular forms

Supersingular elliptic curves, theta series and weight two modular forms

The modular degree, congruence primes, and multiplicity one

The Manin Constant

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