Some non-Gorenstein Hecke algebras attached to spaces of modular forms.^{1}

L. J. P. Kilford, Imperial College, London.

Imperial College, Department of Mathematics 180 Queen’s Gate London SW7 2AZ United Kingdom.

l.kilford@ic.ac.uk

(landline) 01502 566564. (mobile) 07752 340186. (fax) 0207 5948517.

May 25, 2001

1This work was partially carried out at the Institut Henri Poincare in Paris; the author is grateful for their hospitality.

Contents

1 Non-Gorenstein Hecke Algebras attached to classical cuspidal modular forms 2 1.1 Definitions and preamble . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 𝕋_{431} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 𝕋_{503} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Abelian Intersections . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 𝕋_{2089} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Further Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

A Modular forms defined over ℤ_{2} 9

Chapter 1

Non-Gorenstein Hecke Algebras attached to classical cuspidal modular forms

1.1 Definitions and preamble

In this paper we exhibit some examples of non-Gorenstein Hecke algebras, and hence some modular forms for which mod 2 multiplicity one does not hold.

Define S_{2}(Γ_{0}(q)) to be the space of classical cuspidal modular forms of weight 2, level q, trivial character ε = 1. We consider the Hecke algebras 𝕋_{q} generated over ℤ by the Hecke operators T_{p} for p ≠ q and U_{q}, for q ∈ 𝕊 := {431, 503, 2071, 2089}. The main theorem of this paper shows that these are not Gorenstein when localised at a certain maximal ideal 𝔪 over 2.

There has been a great deal of work done on proving that the localizations of these rings at certain of their maximal ideals 𝔪 are Gorenstein; the first work was that of Mazur [12], who proves that if the residue characteristic of the finite field 𝕋_{q}/𝔪 is not 2 then the localization is Gorenstein. He also showed that, for residue characteristic 2, if the associated modular form, f = ∑_{n≥1} a_{n}q^{n} has a_{2} ∈ 𝔪, where 𝔪 is now a maximal ideal above 2, then the localization at 2 is Gorenstein also. Finally, he proves that if the maximal ideal is non-Eisenstein and the associated Galois representation has mod 2 multiplicity one in the Jacobian variety J_{0}(q) then 𝕋 is Gorenstein, so by proving that 𝕋 is not Gorenstein, we show that mod 2 multiplicity 1 fails also. All of the 𝔪 that we deal with in this paper are non-Eisenstein.

We will be dealing in this paper, for the sake of simplicity, with representations attached to modular forms defined over ℤ. We will obtain these by first considering elliptic curve over ℚ, and then attaching modular forms, using the Shimura-Taniyama-Weil conjecture; in our case, we are dealing with semistable elliptic curves, so we refer to Wiles-Taylor-Wiles [17, 18] for the proof. We call the elliptic curve E, and the modular form associated to it will be denoted f.

It is proved in Matsumura [11] that if a ℤ_{2}-module is finitely generated and a complete intersection, then it is Gorenstein. Hence we have that the 𝕋_{q} above are not complete intersections. The proof of Fermat’s Last Theorem by Wiles [18] proves that certain Hecke algebras are complete intersections at various crucial places. Hence this paper gives a bound on how effective Wiles and Taylor’s methods (those used in the proof that the Hecke algebras are complete intersections [17]) can be in proving that Hecke algebras are complete intersection rings and that the associated Galois representations appear with mod 2 multiplicity one.

We follow Ribet [13], Section 5, in defining the Galois representation

ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝕋/𝔪), with tr(ρ(Frob_{l})) ≡ a_{l} mod 𝔪, l ∤ 2q.

which is well-defined up to conjugation. We consider absolutely irreducible representations only here; all of the q we deal with in this paper have such an irreducible representation associated with them. If we have that there exists an 𝔪 such that 𝕋/𝔪 = 𝔽_{2}, as we do for q ∈ 𝕊, then the image of this is S_{3}, the symmetric group on 3 letters.

The trace of this representation at r ∤ 2q is given by the splitting of r in the extension of ℚ of degree 6 cut out by ρ. If r splits completely, or if it splits into 3 primes, then the trace is even. If r splits into two primes, then the trace is odd. So we have that if r splits completely, the image of Frob_{r} is the identity in the Galois group. If r splits into 3, the image of Frobenius has order 2 in the Galois group, and if r splits into 2, then the image of Frobenius has order 3 in the Galois group.

Gross [9] proves that if the weight k satisfies either k < ℓ or k = ℓ and a_{ℓ} ≠ 1, then the Hecke algebra is Gorenstein, and indeed that this is equivalent to the representation being of mod 2 multiplicity one. In Chapter 12 (pp 487–494) he states as an open problem whether his theorems can be extended to cover this case. Here we show that it does not extend to k = ℓ = 2, for certain choices of modular forms.

Edixhoven [7] proves that if k = ℓ and the representation is ramified at ℓ, then mod ℓ multiplicity 1 holds (Theorem 9.2, part 3).

Buzzard [4] and [5] shows that if the image of Frobenius at 2 is not contained in the scalars of GL_{2}(𝕋/𝔪) then the Hecke algebra is Gorenstein; see Proposition 2.4 of that paper. This is equivalent to 2 splitting completely in the field ℚ(E[2]), where E is the elliptic curve associated to the modular form.

Dickinson shows in his paper [6] that certain Hecke algebras are complete intersections, but the hypotheses he proves this under exclude the case that we will look at; again, he has to assume that the image of Frobenius at 2 is not the sum of two identical unramified characters. We have that this image is in fact the identity, so his theorem does not affect this case.

The search plan is to first search for elliptic curves, which correspond to modular forms over ℤ with trivial character, such that 2 splits completely into 6 primes in ℚ(E[2]), and which are ordinary at 2, and have no 2-torsion over ℚ. This gives us forms which are not covered by the theorems quoted above, so we know that we can proceed with some hope of finding non-Gorenstein algebras. We then calculate the appropriate Hecke algebras over ℚ_{2}, and eliminate all of the forms which do not give the representation. This leaves us with a smaller subring of 𝕋_{q}. We then show that this subring is not Gorenstein.

So we prove:

Theorem 1 Assume that p ∈ 𝕊. Then 𝕋_{p} is not Gorenstein, and there exists a cusp form of level Γ_{0}(p) and weight 2 for which mod 2 multiplicity one does not hold.

These calculations were made possible with the hecke package, running on the Magma computer algebra system. Magma includes an environment for specialised number-theoretic calculations, and also incorporates Cremona’s database of elliptic curves, and the hecke package, written by William Stein, implements efficient algorithms to generate spaces of modular forms of arbitrary level, weight (≥ 2) and character over global and finite fields. Without this computing package this paper could not have been written.

An application of this paper’s results can be found in Section 6 of Emerton [8], where it is shown that 𝕋_{431} and 𝕋_{503} have non-invertible 𝕋-modules associated with them. The exact sequence of Gal(\overline{K}/K)-modules of Gross [9], on page 488, is split in the Gorenstein case. Emerson proves that the analogue of this short exact sequence in the non-Gorenstein case is never split. He uses this to prove results about the 𝔪-adic Tate module of J_{0}(q).

In Ribet-Stein [14] the existence of non-Gorenstein Hecke algebras is discussed in the context of the level optimisation procedure associated with Serre’s conjecture (see section 3.7.1).

I would like to thank Kevin Buzzard for suggesting this problem and for giving me much encouragement, and William Stein for helpful suggestions and technical support.

1.2 𝕋_{431}

There are two non-isogenous modular elliptic curves of conductor 431, with minimal Weierstrass equations

E_{1} : y^{2} + xy = x^{3} − 1

and

E_{2} : y^{2} + xy + y = x^{3} − x^{2} − 9x − 8

so there are two corresponding modular forms, say d_{1} and d_{2}, defined over ℤ. These curves are both ordinary at 2 and 2 splits completely in the splitting field generated by the two-torsion of these curves, so the theorems of Mazur, Gross, Edixhoven and Buzzard do not apply.

There are 10 conjugacy classes of normalised Hecke eigenforms over ℚ_{2}, which correspond to the factorisation of the characteristic polynomial for the Hecke operator T_{2} over ℚ_{2}. We give a table in Appendix A which identifies which numbering we have chosen (or, more accurately, which numbering Magma chose for me). We give the table of coefficients which disqualify the forms. The careful reader will note that this table omits the form d_{6}, which is defined over a quadratic extension of ℚ_{2}. The Sturm bound for this space of modular forms is 72 (see [15]), and the three conjugacy classes of forms d_{1}, d_{2}, d_{6} are all congruent up to this bound, so do all give the Galois representation. One can also check, using any standard computer algebra package, that the fields ℚ(E_{i}[2]) are isomorphic, which shows that the two elliptic curves have the same mod 2 Galois representation.

Form Coefficient Degree of Polynomial
3  11  3
4  11  3
5  13  4
7  13  4
8  347  6
9  5  6
10  307  6

This means that the 32-dimensional space spanned by these forms is completely disjoint from the 4-dimensional space spanned by the forms which give the representation, so we can write the Hecke algebra 𝕋_{431} as 𝕋_{431} = 𝕋 ⊕ (ℤ_{2})^{32}. We now concentrate on the ℤ_{2}-module 𝕋.

The basis matrix obtained by hecke can be refined to show that 𝕋 is generated by

𝕋 = ℤ_{2}⟨(1, 1, 1) (0, 2, 0), (0, 0, α + 1), (0, 0, 2)⟩ ⊂ ℤ_{2} × ℤ_{2} × ℤ_{2}[α],

where α satisfies a quadratic polynomial over ℤ_{2}.

This has unique maximal ideal

𝔪 = ⟨(2, 0, 0), (0, 2, 0), (0, 0, α + 1), (0, 0, 2)⟩.

Now we show that (𝕋)_{𝔪} is Cohen-Macaulay and that there is a parameter ideal — one which contains a power of the maximal ideal — which is not irreducible. This shows that it is not Gorenstein, using Matsumura [11], Theorem 18.1. In this case, the ring is Cohen-Macaulay, as it has Krull dimension one and has no nonzero nilpotent elements, from [11], Section 17, page 139.

Now we notice that 𝔦 = ⟨(2, 2, α + 1)⟩ is a parameter ideal, as it contains 𝔪^{2}. We observe that the ideals 𝔦_{1} = ⟨(2, 0, 0), (0, 2, α+1)⟩ and 𝔦_{2} = ⟨(0, 2, 0), (2, 0, α+ 1)⟩ have intersection exactly 𝔦. Hence 𝔦 is a reducible parameter ideal, hence (𝕋_{431} )_{𝔪} is not Gorenstein. Therefore 𝕋 is not Gorenstein, and hence the full Hecke algebra 𝕋_{431} is not Gorenstein.

Hence there exists a cusp form of level 431 for which mod 2 multiplicity one does not hold.

1.3 𝕋_{503}

There are three non-isogenous modular elliptic curves of conductor 503, with minimal Weierstrass equations:

F_{1} : y^{2} + xy + y = x^{3} − 32x − 71
F_{2} : y^{2} + xy = x^{3} − x^{2} + 2x − 1
F_{3} : y^{2} + xy = x^{3} − 210x − 1189

and there are three corresponding modular forms — say f_{1}, f_{2} and f_{3} — defined over ℤ.

We check that these elliptic curves are all ordinary, and that 2 splits completely into 6 distinct primes in the splitting field of the cubics. Hence, as before, we are not bound by the theorems quoted above, and so therefore we have Gal(ℚ(F_{i}[2])/ℚ) ≅ S_{3}.

We now use the hecke package, on the Magma system, to compute the space of modular forms over ℚ_{2}. We find there are 10 conjugacy classes of normalised Hecke eigenforms over ℚ_{2} , called f_{1}, f_{2}, …, f_{10}, which correspond to the irreducible factors of the characteristic polynomial of T_{2} over ℤ_{2}.

Four of these - f_{1}, f_{2}, f_{3} and f_{6} - are defined over ℤ_{2}. It can be seen that, up to the Sturm bound, which is 84 (see [15]), they are all congruent mod 2, so give the same irreducible representation.

We now check that the other six forms do not give the representation. The forms are numbered according to the table in Appendix A, which is in fact the ordering imposed on them by the Magma operating system.

If a form f_{j} gives the representation, then its i^{th} coefficient will reduce modulo the maximal ideal, which will be something above 2, to give the trace of the image of Frobenius, or equivalently the i^{th} coefficient of one of the four forms above; without loss of generality we use f_{1}.

We simply provide a table showing which coefficients can be inspected to show that the congruence fails:

Form Coefficient Degree of polynomial
4  23  3
5  79  10
7  23  3
8  283  6
9  59  6
10  79  10

As before, the 38-dimensional space spanned by these forms is completely seperate from the 4-dimensional space spanned by those forms which do give the representation, so we can write, as above, 𝕋_{503} = 𝕋 ⊕ (ℤ_{2})^{38}.

So we consider the remaining module 𝕋:

𝕋 = ⟨(1, 1, 1, 1), (0, 2, 0, 0), (0, 0, 2, 2), (0, 0, 0, 4)⟩.

The maximal ideal 𝔪 is

𝔪 = ⟨(2, 0, 0, 0), (0, 2, 0, 0), (0, 0, 2, 2), (0, 0, 0, 4)⟩.

We see that (𝕋)_{𝔪} is Cohen-Macaulay, as it has Krull dimension 1 and no nilpotent elements. 𝔦 = ⟨(2, 2, 2, 2)⟩ is a parameter ideal, as we have 𝔪^{2} ⊆ 𝔦.

Now the ideals 𝔦_{1} = ⟨(0, 2, 2, 2), (2, 0, 0, 0)⟩ and 𝔦_{2} = ⟨(2, 0, 2, 2), (0, 2, 0, 0)⟩ have 𝔦 as their intersection, hence 𝔦 is reducible, and therefore 𝕋 is not Gorenstein and 𝕋_{503} is not Gorenstein.

Hence there exists a cusp form of level 503 for which mod 2 multiplicity one fails.

1.4 Abelian Intersections

William Stein suggested another way [16] to check that these rings were not Gorenstein, and used it to show that 𝕋_{2071} was not Gorenstein, and that mod 2 multiplicity one failed for J_{0}(2071).

Let {E_{i}} be the set of optimal elliptic curves (also called strong Weil curves) of conductor q. We check that the newforms they generate are congruent mod 2. If mod 2 multiplicity one is true, then J_{0}(q)[𝔪] = E_{i}[2] for all of the i. In particular, E_{i} ⋂ E_{j} should contain E_{i}[2]. We find that, for q ∈ {431, 503}, each elliptic curve which gives the representation is isolated in its isogeny class, so it is optimal. We may conclude that mod 2 multiplicity one and the Gorenstein property fail if, using the Magma command “IntersectionGroup”, we find that E_{i} ⋂ E_{j} = {0}, for any E_{i} ≠ E_{j}, which both give the Galois representation.

This command is more fully explained in [2]; there is also a formal proof of the theory which it relies upon in this reference.

1.5 𝕋_{2089}

There are five isomorphism classes of elliptic curves with conductor 2089, of which four — labelled 2089A, 2089C, 2089D, 2089E in Cremona’s tables — give the representation. We also check that each of these is a strong Weil curve, in the language of the previous section. The one which does not, 2089B, has rational 2-torsion, so its associated Galois representation is reducible.

Using the results of the previous section, we find that the intersection of any pair of these as abelian varieties is {0}. Hence we do not have mod 2 multiplicity one and therefore the Gorenstein property fails to hold here.

1.6 Further Studies

This paper answers the question raised in [4] and in [9], in that it exhibits specific examples of non-Gorenstein Hecke algebras. However, this then raises the natural question of proving exactly which Hecke algebras are not Gorenstein, without explicitly computing them, and proving theorems like those of Buzzard, Edixhoven, Gross and Mazur for these algebras. The natural next step is to ask the following:

Question 2 Are there infinitely many prime or squarefree q ∈ ℤ such that 𝕋_{q} is not Gorenstein, and hence where mod 2 multiplicity one fails?

William Stein suggested that the recent paper of Emerton [8] could be used to provide a very quick way to show that certain Hecke algebras of prime level were not Gorenstein. Theorem 0.5 of [8] shows that the Gorenstein property for 𝕋_{q} is equivalent to another module, X, being an invertible 𝕋_{q}-module, where X is the free ℤ-module of divisors supported on the set of singular points of the curve X_{0}(q) in characteristic q. Now X can be calculated very quickly using the Magma software. This allows some more computations to be undertaken, and the set of Hecke algebras 𝕋_{q}, with q ∈ 𝕊_{2} = {4481, 4799, 11321, 18097} can be shown not to be Gorenstein.

However, this still does not give an infinite number of non-Gorenstein examples. The method used here relies on finding elliptic curves of prime or squarefree conductor, and it is not known if there are infinitely many elliptic curves of prime conductor.

It should be noted that just because an elliptic curve E has the property that its conductor N is squarefree and ℚ(E[2]) has Galois group S_{3}, this does not mean automatically that the Hecke algebra of level N is not Gorenstein. If we look at the unique elliptic curves of conductor 1439 and 1607, which both have this property, we find that the algebras 𝕋_{1439} and 𝕋_{1607} are in fact Gorenstein, as the only part of them which corresponds to a modular form giving the mod 2 representation is isomorphic to ℤ_{2}, which is of course Gorenstein, as it has parameter ideal generated by ⟨2⟩.

Appendix A

Modular forms defined over ℤ_{2}

To aid the reader in identifying which congugacy classes of modular forms are denoted by d_{i} and f_{j} in this article, we display a table of the factorisation of the characteristic polynomial of the Hecke operator T_{3} at level 431, and the characteristic polynomial of T_{11} at level 503, identifying factors of the polynomials with conjugacy classes. These are the first characteristic polynomials where all the forms are distinct.

The polynomial factors corresponding to the forms d_{6} to d_{10} and f_{6} to f_{10} are not defined over ℤ. The factors are given here to an accuracy of 2^{40}. This gives enough accuracy to factor the polynomials into irreducible factors over ℤ_{2}.

Factors over ℚ_{2} of the characteristic polynomial of the Hecke operator T_{3} at level 431. Form Polynomial Factor

1  y − 3
2  y − 1
3  y^{3} − y^{2} − 8y + 11
4  y^{3} + y^{2} − 4y − 3
5  y^{4} + 3y^{3} − 4y − 1
6  y^{2} − 310081147870y − 531740541567
7  y^{4} + 93865878899y^{3} + 492306262748y^{2} + 199271497556y + 72038598859
8  {y^{6} − 117460789714y^{5} + 140896353361y^{4} + 502127623066y^{3} − \atop 137206343406y^{2} + 285362431848y − 82394257695}
9  {y^{6} + 155104204721y^{5} − 151862234690y^{4} + 319033115208y^{3} − \atop 516207445344y^{2} + 246308839360y − 541570543317}
10  {y^{6} + 178571853963y^{5} − 50945999226y^{4} − 1711616104y^{3} + 384183165500y^{2} − \atop 45698556868y − 286094811073}

Factors over ℚ_{2} of the characteristic polynomial of the Hecke operator T_{11} at level 503. Form Polynomial Factor

1  y − 5
2  y − 3
3  y − 1
4  y^{3} − 11y^{2} + 36y − 35
5  y^{10}+3y^{9}−41y^{8}−80y^{7}+515y^{6}+537y^{5}−1961y^{4}−1781y^{3}+1973y^{2}+1920y+311
6  y + 162038350031
7  y^{3} − 98811965589y^{2} + 192166733404y + 472858366683
8  {y^{6}+60178080367y^{5}−389390875114y^{4}+388661373764y^{3}−296676097968y^{2}− \atop 416851965620y − 463470069569}
9  {y^{6} + 409592703841y^{5} − 441384946190y^{4} − 168712944024y^{3} + \atop 150920999068y^{2} − 25323310436y − 280567076689}
10  y^{10} − 532997168633y^{9} + 519304293775y^{8} + 487680728516y^{7} − ; 347836282969y^{6} − 253687675639y^{5} + 471031837483y^{4} + 31516090307y^{3} − ; 364948386179y^{2} + 420792975716y + 308442028279

Bibliography

[1] A. Agashe and W. A. Stein, Appendix to Some computations with Hecke rings and deformation rings, by Joan-C. Lario and René Schoof, to appear.

[2] A. Agashe and W. A. Stein, Visibility of Shafarevich-Tate groups of abelian varieties: Evidence for the Birch and Swinnerton-Dyer conjecture, to appear.

[3] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp. 24 (1997), no 3–4, 235–265, http://www.math.usyd.edu.au:8080/u/magma.

[4] K. Buzzard, On level lowering for mod 2 representations, Math. Res. Lett., vol. 7 no. 1 (2000), pp 95–110.

[5] K. Buzzard, Appendix to Lectures on Serre’s Conjectures, IAS/ Park City Mathematics Institute Lecture Series, to appear.

[6] , M. Dickinson, On the modularity of certain 2-adic Galois representations, to appear in Duke Math. Journal.

[7] B. Edixhoven, The weight in Serre’s conjectures on modular forms, Invent Math. 109 (1992), pp563–594.

[8] M. Emerton, Supersingular Elliptic Curves and Weight Two Modular Forms, to appear.

[9] B. Gross, A Tameness Criterion for Galois Representations associated to Modular Forms (mod p), Duke Math. Journal vol. 61 no. 2 (1990), pp 445– 517.

[10] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer, New York (1993).

[11] H. Matsumura, Commutative Ring Theory, CUP, 1989.

[12] B. Mazur, Modular Curves and the Eisenstein Ideal, IHES Publ. Math. 47 (1990), pp. 33–186.

[13] K. Ribet, On modular representations of Gal(\overline{ℚ}/ℚ) arising from modular forms, Invent. Math. 100 (1990), pp 431–476.

[14] K. Ribet and W. Stein, Lectures on Serre’s Conjectures, IAS/ Park City Mathematics Institute Lecture Series, to appear.

[15] W. A. Stein, Generating the Hecke algebra as a ℤ-module, preprint, 1999.

[16] W. A. Stein, Personal communication, March 2000.

[17] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995) pp 553–572.

[18] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), pp 443–551.