## Constructing elements in Shafarevich–Tate groups of modular motives (2003)

Venue: | in Number theory and algebraic geometry, London Mathematical Society Lecture Notes, Volume 303 |

Citations: | 14 - 1 self |

### BibTeX

@INPROCEEDINGS{Dummigan03constructingelements,

author = {Neil Dummigan and William Stein and Mark Watkins},

title = {Constructing elements in Shafarevich–Tate groups of modular motives},

booktitle = {in Number theory and algebraic geometry, London Mathematical Society Lecture Notes, Volume 303},

year = {2003},

pages = {91--118},

publisher = {University Press}

}

### OpenURL

### Abstract

We study Shafarevich–Tate groups of motives attached to modular forms on Γ0(N) of weight> 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich–Tate groups, and give 16 examples in which a strong form of the Beilinson–Bloch conjecture would imply the existence of such elements. We also use modular symbols and observations about Tamagawa numbers to compute nontrivial conjectural lower bounds on the orders of the Shafarevich–Tate groups of modular motives of low level and weight ≤ 12. Our methods build upon the idea of visibility due to Cremona and Mazur, but in the context of motives rather than abelian varieties. 1

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Citation Context ..., Tλ(j))tors − ordλ(Pp(p −j )) = length ( H 0 (Qp, Aλ(j))/H 0 ( Qp, Vλ(j) Ip /Tλ(j) Ip)) . We omit the definition of ordλ(cp(j)) for λ | p, which requires one to assume Fontaine’s de Rham conjecture (=-=[Fo1]-=-, Appendix A6), and depends on the choices of TdR and TB, locally at λ. (We shall mainly be concerned with the q-part of the Bloch–Kato conjecture, where q is a prime of good reduction. For such prime... |

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Citation Context ... H1 (Qq, A ′[q](k/2)) is in the image of H1 f (Qq, T ′ q(k/2)), by construction, and therefore is in the image of h1 (D′(k/2)/qD′(k/2)). By the fullness and exactness of the Fontaine–Lafaille functor =-=[FL]-=- (see [BK], Theorem 4.3), D ′(k/2)/qD′(k/2) is isomorphic to D(k/2)/qD(k/2). It follows that the class resq(c) ∈ H1 (Qq, A[q](k/2)) is in the image of h1 (D(k/2)/qD(k/2)) by the vertical map in the ex... |

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Citation Context ... (Ip, A[q](k/2))), by inflation-restriction. The order of this group is the same as the order of the group H0 (Qp, A[q](k/2)) (this is [W], Lemma 1), which we claim is trivial. By the work of Carayol =-=[Ca1]-=-, the level N is the conductor of Vq(k/2), so p | N implies that Vq(k/2) is ramified at p, hence dim H0 (Ip, Vq(k/2)) = 0 or 1. As above, we see that dim H0 (Ip, Vq(k/2)) = dim H0 (Ip, A[q](k/2)), so ... |

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Citation Context ...(Ip, A ′ q(k/2)) of the restriction of c is zero, hence that the restriction of c to H1 (Ip, A ′[q](k/2)) ≃ H1 (Ip, A[q](k/2)) is zero. Hence the restriction of γ to H1 (Ip, Aq(k/2)) is also zero. By =-=[Fl1]-=-, line 3 of p. 125, H1 f (Qp, Aq(k/2)) is equal to (not just contained in) the kernel of the map from H1 (Qp, Aq(k/2)) to H1 (Ip, Aq(k/2)), so we have shown that resp(γ) ∈ H1 f (Qp, Aq(k/2)). Case 2, ... |

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Citation Context ...k ≥ 2 for Γ0(N), with coefficients in an algebraic number field E, which is necessarily totally real. Let λ be any finite prime of E, and let ℓ denote its residue characteristic. A theorem of Deligne =-=[De1]-=- implies the existence of a two-dimensional vector space Vλ over Eλ, and a continuous representation such that ρλ: Gal(Q/Q) → Aut(Vλ), 1. ρλ is unramified at p for all primes p not dividing ℓN, and 2.... |

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Citation Context ...tor of the algebraic number L(f, k/2)/vol∞, where vol∞ is a certain canonical period. In fact, we show how this divisibility may be deduced from the vanishing of L(g, k/2) using recent work of Vatsal =-=[V]-=-. The point is, the congruence between f and g leads to a congruence between suitable “algebraic parts” of the special values L(f, k/2) and L(g, k/2). In slightly more detail, a multiplicity one resul... |

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Citation Context ...ch conjecture [B, Be] generalises the part about the order of vanishing at the central point, identifying it with the rank of a certain Chow group. This paper is a partial generalisation of [CM1] and =-=[AS]-=- from abelian varieties over Q associated to modular forms of weight 2 to the motives attached to modular forms of higher weight. It also does for congruences between modular forms of equal weight wha... |

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Citation Context ...arrangement of [BK], (5.15.1). The Bloch–Kato conjecture has been reformulated and generalised by Fontaine and Perrin-Riou, who work with general E, though that is not really the point of their work. =-=[Fo2]-=-, Section 11 sketches how to deduce the original conjecture from theirs, in the case E = Q. Lemma 4.1 vol∞/a ± = c(2πi) k/2a ± Ω±, with c ∈ E and ordλ(c) = 0 for λ ∤ Nk!. Proof We note that vol∞ is eq... |

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10 | Adjoint motives of modular forms and the Tamagawa number conjecture, preprint
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Citation Context ...R in the Betti and de Rham realisations VB and VdR of Mf. We do this in such a way that TB and TdR ⊗OE OE[1/Nk!] agree respectively with the OE-lattice Mf,B and the OE[1/Nk!]-lattice Mf,dR defined in =-=[DFG1]-=- using cohomology, with nonconstant coefficients, of modular curves. (See especially [DFG1], Sections 2.2 and 5.4, and the paragraph preceding Lemma 2.3.) For any finite prime λ of OE, define the Oλ m... |

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Crystalline cohomology and GL(2
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Citation Context ...ar motives cohomology of X1(N) with nonconstant coefficients. This would be the case if δ ± f and δ± g generate the same one-dimensional subspace upon reduction modulo q. But this is a consequence of =-=[FJ]-=-, Theorem 2.1(1) (for which we need the irreducibility of A[q]). □ Remark 5.2 The signs in the functional equations of L(f, s) and L(g, s) are equal. They are determined by the eigenvalue of the Atkin... |

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Citation Context ... ∆± f ∈ SΓ1(N)(k, OE,q). We are of [V], 1.7, the cohomology class ω ± f now dealing with cohomology over X1(N) rather than M(N), which is why we insist that q ∤ ϕ(N). It follows from the last line of =-=[St]-=-, Section 4.2 that, up to some small factorials which do not matter locally at q, Φ ± f ([∞] − [0]) = ∑k−2 rf(j)X j Y k−2−j . j=0, j≡(k/2)−1 (mod 2) Since ω ± f = Ω± f δ± f , we see that ∆ ± f ([∞] − ... |

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Citation Context ...H1 (Ip, A[q](k/2)) is zero. Then resp(c) comes from H1 (Dp/Ip, H0 (Ip, A[q](k/2))), by inflation-restriction. The order of this group is the same as the order of the group H0 (Qp, A[q](k/2)) (this is =-=[W]-=-, Lemma 1), which we claim is trivial. By the work of Carayol [Ca1], the level N is the conductor of Vq(k/2), so p | N implies that Vq(k/2) is ramified at p, hence dim H0 (Ip, Vq(k/2)) = 0 or 1. As ab... |

7 | The Bloch-Kato conjecture for adjoint motives of modular forms
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Citation Context ...± . Then the proof is completed by noting that, locally away from primes dividing Nk!, the index of TdR in its dual is equal to the index of TB in its dual, both being equal to the ideal denoted η in =-=[DFG2]-=-. □ Remark 4.2 Note that the “quantities” a ± Ω± and vol∞/a ± are independent of the choice of δ ± f . Lemma 4.3 Let p ∤ N be a prime and j an integer. Then the fractional ideal cp(j) is supported at ... |

7 |
Heights of Heegner cycles and derivatives of L-series, Invent
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Citation Context ... would be nice if likewise one could explicitly produce algebraic cycles predicted by the Beilinson–Bloch conjecture in the above examples. Since L ′(g, k/2) = 0, Heegner cycles have height zero (see =-=[Z]-=-, Corollary 0.3.2), so ought to be trivial in CH k/2 0 (Mg) ⊗ Q.Neil Dummigan, William Stein and Mark Watkins 107 7.2 How the computation was performed We give a brief summary of how the computation ... |

5 |
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Citation Context ... 0 (Ip, A[q](j)) = dimEq H 0 (Ip, Vq(j)), since this ensures that H0 (Ip, Aq(j)) = Vq(j) Ip /Tq(j) Ip , and therefore that H0 (Qp, Aq(j)) = H0 (Qp, Vq(j) Ip /Tq(j) Ip ). Suppose that Condition (b) of =-=[L]-=-, Proposition 2.3 is not satisfied. Then there exists a character χ: Gal(Q/Q) → O × q of q-power order such that the p-part of the conductor of Vq ⊗ χ is strictly smaller than that of Vq. Let fχ denot... |

4 | Symmetric square L-functions and Shafarevich-Tate groups
- Dummigan
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Citation Context ...f of Lemma 4.6, Tq is the Oq[Gal(Qq/Qq)]-module associated to the filtered module TdR ⊗ Oq.Neil Dummigan, William Stein and Mark Watkins 103 Since also q > k, we may now prove, in the same manner as =-=[Du1]-=-, Proposition 9.2, that resq(γ) ∈ H1 f (Qq, Aq(k/2)). For the convenience of the reader, we give some details. In [BK], Lemma 4.4, a cohomological functor {hi}i≥0 is constructed on the Fontaine–Lafail... |

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3 |
Congruences of modular forms and
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Citation Context ...rom abelian varieties over Q associated to modular forms of weight 2 to the motives attached to modular forms of higher weight. It also does for congruences between modular forms of equal weight what =-=[Du2]-=- did for congruences between modular forms of different weights. We consider the situation where two newforms f and g, both of even weight k > 2 and level N, are congruent modulo a maximal ideal q of ... |

2 |
Visible Evidence for the Birch and SwinnertonDyer Conjecture for Modular Abelian Varieties of Rank Zero.” Appendix to A. Agashe and
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(Show Context)
Citation Context ...tor, whose p-torsion is Galois-isomorphic to that of the first one, and which has positive rank. The rational points on the second elliptic curve produce classes in the common H1 (Q, E[p]). They show =-=[CM2]-=- that these lie in the Shafarevich– Tate group of the first curve, so rational points on one curve explain elements of the Shafarevich–Tate group of the other curve. 9192 Shafarevich–Tate groups of m... |

2 |
On the Degree of Modular Parametrisations.” In Séminaire de Théorie des Nombres, Paris 1991-92, edited by
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(Show Context)
Citation Context ...ities” a ± Ω± and vol∞/a ± are independent of the choice of δ ± f . Lemma 4.3 Let p ∤ N be a prime and j an integer. Then the fractional ideal cp(j) is supported at most on divisors of p. Proof As on =-=[Fl2]-=-, p. 30, for odd l ̸= p, ordλ(cp(j)) is the length of the finite Oλ-module H0 (Qp, H1 (Ip, Tλ(j))tors), where Ip is an inertia group at p. But Tλ(j) is a trivial Ip-module, so H1 (Ip, Tλ(j)) is torsio... |

2 |
p-adic Abel-Jacobi maps and p-adic heights. The arithmetic and geometry of algebraic cycles
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(Show Context)
Citation Context ... is equal to the rank of the Mordell-Weil group E(Q).) Via the q-adic Abel–Jacobi map, CH k/2 0 (Mg)(Q) maps to H1 (Q, V ′ q(k/2)), and its image is contained in the subspace H1 ′ f (Q, V q(k/2)), by =-=[Ne]-=-, 3.1 and 3.2. If, as expected, the q-adic Abel–Jacobi map is injective, we get (assuming also the Beilinson–Bloch conjecture) a subspace of H1 f ′ (Q, V q(k/2)) of dimension equal to the order of van... |

2 |
Cuspidal modular symbols are transportable
- Stein, Verrill
(Show Context)
Citation Context ...l.Neil Dummigan, William Stein and Mark Watkins 105 Note that in §7 “modular symbol” has a different meaning from in §5, being related to homology rather than cohomology. For precise definitions see =-=[SV]-=-. 7.1 Table 1: visible X Table 1 lists sixteen pairs of newforms f and g (of equal weights and levels) g deg g f deg f possible q 127k4A 1 127k4C 17 43 159k4B 1 159k4E 16 5, 23 365k4A 1 365k4E 18 29 3... |

1 |
Visualizing elements in the Shafarevich
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(Show Context)
Citation Context ...rational points, and also gives an interpretation of the leading term in the Taylor expansion in terms of various quantities, including the order of the Shafarevich–Tate group of E. Cremona and Mazur =-=[CM1]-=- look, among all strong Weil elliptic curves over Q of conductor N ≤ 5500, at those with nontrivial Shafarevich–Tate group (according to the Birch and Swinnerton-Dyer conjecture). Suppose that the Sha... |

1 |
On the non-vanishing of Lf(s) at the center of the critical strip, preprint
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(Show Context)
Citation Context ...vanish to order at least two at s = k/2. Note that Maeda’s conjecture implies that there are no examples of g of level one with positive sign in their functional equation such that L(g, k/2) = 0 (see =-=[CF]-=-). 6 Constructing elements of the Shafarevich– Tate group Let f, g and q be as in the first paragraph of the previous section. In the previous section we showed how the congruence between f and g rela... |

1 |
Conjecture “epsilon” for weight k
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(Show Context)
Citation Context ...p ̸≡ −1 (mod q) if p2 | N, Case 3 is excluded, so A[q](j) is unramified at p and ordp(N) = 1. (Here we are using Carayol’s result that N is the prime-to-q part of the conductor of Vq [Ca1].) But then =-=[JL]-=-, Theorem 1 (which uses the condition q > k) implies the existence of a newform of weight k, trivial character and level dividing N/p, congruent to g modulo q, for Fourier coefficients of index coprim... |