Iwasawa theory for the symmetric square of a modular form

David Loeffler; Sarah Livia Zerbes

doi:10.1515/crelle-2016-0064

Article

Iwasawa theory for the symmetric square of a modular form

David Loeffler and Sarah Livia Zerbes

Published/Copyright: December 14, 2016

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2019 Issue 752

Abstract

We construct a compatible family of global cohomology classes (an Euler system) for the symmetric square of a modular form, and apply this to bounding Selmer groups of the symmetric square Galois representation and its twists.

To the memory of Rudolf Zerbes (1944–2015)

Funding statement: The authors’ research was supported by the following grants: Royal Society University Research Fellowship (Loeffler); ERC Consolidator Grant “Euler Systems and the Birch–Swinnerton-Dyer conjecture” (Zerbes).

A Kolyvagin systems for direct sums

In this section we prove a simple modification of a crucial lemma from [17], in order to permit us to work with Euler systems for reducible Galois representations (and their associated Kolyvagin systems). Our aim is to show a precise form of the following statement:

“A Kolyvagin system for a direct sum T1⊕T2 that happens to take values in T1 is almost as good as a Kolyvagin system for T1”.

A.1 Setup

In this subsection we will use the following notations:

p is a prime,
R is a complete Noetherian local ring, with finite residue field 𝐤 of characteristic p,
𝔪 is the maximal ideal of R.

We are interested in Kolyvagin systems for modules of the form T1⊕T2, where Ti are R⁢[G𝐐]-modules, both free of finite rank over R. We let Σ be a finite set of places of 𝐐 containing p, ∞, and all the primes at which either T1 or T2 is ramified; and we choose a Selmer structure^[3]ℱ for T1 with Σ⁢(ℱ)=Σ.

For t≥1, we define the following set of primes (cf. [17, Definition 3.1.6]): 𝒫t♯ is the set of all primes ℓ∉Σ such that

ℓ=1mod𝔪k∩𝐙,
T1/(𝔪k⁢T1+(Frobℓ-1)⁢T1) is free of rank 1 over R/𝔪k,
Frobℓ-1 acts bijectively on T2.

It is clear that 𝒫1♯⊇𝒫2♯⊇𝒫3♯ and so on; and if 𝔪k=0, then 𝒫t♯=𝒫k♯ for t≥k. Finally, we let 𝒫 be a set of primes disjoint from Σ.

If T is any 𝐙p⁢[G𝐐]-module, we shall write T∨=Hom⁡(T,μp∞).

A.2 Hypotheses

We shall impose a set of hypotheses (H.0♯)–(H.6♯) on the collection (T1,T2,ℱ,𝒫). These are slight adaptations of the hypotheses (H.0)–(H.6) of [17, Section 3.5].

The Ti are free R-modules.
T1/𝔪⁢T1 is an absolutely irreducible 𝐤-representation.
There is a τ∈G𝐐 such that τ=1 on μp∞, T1/(τ-1)⁢T1 is free of rank 1 over R, and τ-1 is bijective on T2.
If Ω=𝐐⁢(T1,T2,μp∞) is the smallest extension of 𝐐 acting trivially on T1, T2, and μp∞, then H1⁢(Ω/𝐐,T1/𝔪⁢T1)=H1⁢(Ω/𝐐,T1∨⁢[𝔪])=0.
Either of the following holds:
1. Hom𝐅p⁢[G𝐐]⁡(T1/𝔪⁢T1,T1∨⁢[𝔪])=0,
2. p≥5.
We have 𝒫1♯⊇𝒫⊇𝒫t♯ for some t.
For every ℓ∈Σ, the local condition ℱ at ℓ is Cartesian on the category of quotients of R in the sense of [17, Definition 1.1.4].

Note that hypotheses (H.1♯), (H.4♯), (H.5♯) and (H.6♯) are identical to Mazur and Rubin’s hypotheses (H.i) for T=T1; it is only (H.2♯) and (H.3♯) which are different. Note, also, that if T2={0} then all our hypotheses are identical to their non-sharpened versions.

A.3 Choosing useful primes

In this subsection, we suppose that (H.0♯)–(H.5♯) are satisfied, and that the coefficient ring R is Artinian and principal.

Proposition A.3.1 (cf. [17, Proposition 3.6.1]).

Let

c1,c2∈H1⁢(𝐐,T1),c3,c4∈H1⁢(𝐐,T1∨)

be non-zero elements. For every k≥1 there is a positive-density set of primes S⊆Pk♯ such that for all ℓ∈S, the localisations (cj)ℓ are all non-zero.

Proof.

We may assume without loss of generality that 𝔪k=0. We simply imitate the arguments of op.cit. for T=T1, but with the field F replaced by the slightly larger field

F♯=𝐐⁢(T1,T2,μpk).

If τ∈G𝐐 is as in (H.2♯), then any prime ℓ whose Frobenius in Gal⁡(F♯/𝐐) is conjugate to τ acts invertibly on T2, so the set of primes S constructed in op.cit. is automatically contained in 𝒫k♯. ∎

Remark A.3.2.

In fact, a slight refinement of this statement is true: if R=R~/I for some larger ring R~, and Ti=T~i⊗R, then we may arrange that S is contained in the set 𝒫~k♯ defined with the T~i in place of the Ti. Note that 𝒫~k♯ will in general be smaller than 𝒫k♯, even in the case T2={0}. This minor detail appears to have been overlooked in [17], and this slight strengthening of [17, Proposition 3.6.1] is actually needed for some of the proofs in op.cit., in particular for those dealing with the module KS¯⁢(T) of generalised Kolyvagin systems.

A.4 Bounding the Selmer group

With Proposition A.3.1 in place of Mazur–Rubin’s Proposition 3.6.1, we obtain generalisations of all of the theorems of [17] to this setting. For example, we have the following generalisation of [17, Theorem 5.2.2 (and Remark 5.2.3)]:

Corollary A.4.1.

Suppose R is the ring of integers of a finite extension of Qp, and (T1,T2,F,P) satisfy (H.0♯)–(H.6♯). Let

𝜿∈KS¯⁢(T,ℱ,𝒫)≔lim←k⁡(lim→j⁡KS⁢(T/𝔪k⁢T,ℱ,𝒫∩𝒫j♯)).

Then

lengthR⁡(Hℱ∨1⁢(𝐐,T∨))≤max⁡{j:κ1∈𝔪j⁢H1⁢(𝐐,T)}.

The same also applies to [17, Theorem 5.3.10] (bounding a Selmer group over the cyclotomic 𝐙p-extension of 𝐐) and to [13, Theorem 12.3.5] (allowing a “Greenberg-style” local condition at p).

Acknowledgements

It is a pleasure to thank John Coates and Andrew Wiles for their interest in our work and helpful discussions; and Samit Dasgupta for explaining many aspects of [4] to us. We would also like to thank Henri Darmon for valuable comments on an earlier draft of this paper, and the anonymous referee for his suggestions on how to improve the exposition of the results.

References

[1] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift. Vol. I, Progr. Math. 86, Birkhäuser, Boston (1990), 333–400. 10.1007/978-0-8176-4574-8_9Search in Google Scholar

[2] J. Coates and C.-G. Schmidt, Iwasawa theory for the symmetric square of an elliptic curve, J. reine angew. Math. 375/376 (1987), 104–156. 10.1515/crll.1987.375-376.104Search in Google Scholar

[3] A. Dabrowski and D. Delbourgo, S-adic L-functions attached to the symmetric square of a newform, Proc. Lond. Math. Soc. (3) 74 (1997), no. 3, 559–611. 10.1112/S0024611597000191Search in Google Scholar

[4] S. Dasgupta, Factorization of p-adic Rankin L-series, Invent. Math. 205 (2016), no. 1, 221–268. 10.1007/s00222-015-0634-4Search in Google Scholar

[5] T. Dokchitser, Computing special values of motivic L-functions, Exp. Math. 13 (2004), no. 2, 137–149. 10.1080/10586458.2004.10504528Search in Google Scholar

[6] M. Flach, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math. 109 (1992), no. 2, 307–327. 10.1007/BF01232029Search in Google Scholar

[7] T. Fukaya and K. Kato, A formulation of conjectures on p-adic zeta functions in noncommutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society. Vol. XII, Amer. Math. Soc. Transl. Ser. 2 219, American Mathematical Society, Providence (2006), 1–85. 10.1090/trans2/219/01Search in Google Scholar

[8] S. Gelbart and H. Jacquet, A relation between automorphic representations of GL⁢(2) and GL⁢(3), Ann. Sci. Éc. Norm. Supér. (4) 11 (1978), no. 4, 471–542. 10.24033/asens.1355Search in Google Scholar

[9] R. Greenberg, Iwasawa theory for p-adic representations, Algebraic number theory, Adv. Stud. Pure Math. 17, Academic Press, Boston (1989), 97–137. 10.2969/aspm/01710097Search in Google Scholar

[10] H. Hida, p-adic L-functions for base change lifts of GL2 to GL3, Automorphic forms, shimura varieties, and L-functions. Vol. II (Ann Arbor 1988), Perspect. Math. 11, Academic Press, Boston (1990), 93–142. Search in Google Scholar

[11] H. Jacquet and J. A. Shalika, A non-vanishing theorem for zeta functions of GLn, Invent. Math. 38 (1976), no. 1, 1–16. 10.1007/BF01390166Search in Google Scholar

[12] K. Kato, P-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applications arithmétiques. III, Astérisque 295, Société Mathématique de France, Paris (2004), 117–290. Search in Google Scholar

[13] G. Kings, D. Loeffler and S. L. Zerbes, Rankin–Eisenstein classes and explicit reciprocity laws, preprint (2015), http://arxiv.org/abs/1503.02888; to appear in Camb. J. Math. 10.4310/CJM.2017.v5.n1.a1Search in Google Scholar

[14] G. Kings, D. Loeffler and S. L. Zerbes, Rankin–Eisenstein classes for modular forms, preprint (2015), http://arxiv.org/abs/1501.03289. 10.1353/ajm.2020.0002Search in Google Scholar

[15] A. Lei, D. Loeffler and S. L. Zerbes, Euler systems for Rankin–Selberg convolutions of modular forms, Ann. of Math. (2) 180 (2014), no. 2, 653–771. 10.4007/annals.2014.180.2.6Search in Google Scholar

[16] A. Lei, D. Loeffler and S. L. Zerbes, Euler systems for modular forms over imaginary quadratic fields, Compos. Math. 151 (2015), 1585–1625. 10.1112/S0010437X14008033Search in Google Scholar

[17] B. Mazur and K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (2004), no. 799, 1–96. 10.1090/memo/0799Search in Google Scholar

[18] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25–68. 10.1007/BF01239508Search in Google Scholar

[19] K. Rubin, Euler systems, Ann. of Math. Stud. 147, Princeton University Press, Princeton 2000. 10.1515/9781400865208Search in Google Scholar

[20] C.-G. Schmidt, p-adic measures attached to automorphic representations of GL⁢(3), Invent. Math. 92 (1988), no. 3, 597–631. 10.1007/BF01393749Search in Google Scholar

[21] P. Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), no. 2, 181–205. 10.1007/BF01214195Search in Google Scholar

[22] G. Shimura, On the holomorphy of certain Dirichlet series, Proc. Lond. Math. Soc. (3) 31 (1975), no. 1, 79–98. 10.1007/978-1-4612-2076-3_19Search in Google Scholar

[23] J. Sturm, Special values of zeta functions, and Eisenstein series of half integral weight, Amer. J. Math. 102 (1980), no. 2, 219–240. 10.2307/2374237Search in Google Scholar

[24] P. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, Modular functions of one variable III (Antwerp 1972), Lecture Notes in Math. 350, Springer, Berlin (1973), 1–55. 10.1007/978-3-540-37802-0_1Search in Google Scholar

[25] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. 10.2307/2118560Search in Google Scholar

[26] R. Venerucci, Exceptional zero formulae and a conjecture of Perrin–Riou, Invent. Math. 203 (2016), no. 3, 923–972. 10.1007/s00222-015-0606-8Search in Google Scholar

[27] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. 10.2307/2118559Search in Google Scholar

Received: 2016-02-02

Revised: 2016-10-20

Published Online: 2016-12-14

Published in Print: 2019-07-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/crelle-2016-0064