EULER SYSTEMS DAVID LOEFFLER / SARAH ZERBES These are the notes from my talk at the Harris conference at MSRI in December 2014. 1. A conjecture of Perrin-Riou Let M be a motive over Q which is not the trivial motive. Denote by L(M, s) the L-function of M . Assumption 1.1. L(M, s) has analytic continuation to C and satisfies a functional equation. Denote by Mp the p-adic realisation of M , so Mp is a finite-dimensional Qp -vector space with a continuous action of GQ = Gal(Q/Q). Denote by Hf1 (Q, Mp ) the Bloch-Kato Selmer group of Mp , which is a subgroup of H 1 (Q, Mp ), cut out by local conditions • the unramified local condition at ` 6= p; • the Hf1 -local condition at p. The Bloch-Kato conjecture predicts the size of Hf1 (Q, Mp∗ (1)) in terms of the L-function of M : Conjecture 1.2. (Bloch-Kato) ords=0 L(M, s) = dim Hf1 (Q, Mp∗ (1)) − dim H 0 (Q, Mp∗ (1)), together with an explicit formula for the leading term up to a p-adic unit. Definition. The motive M is effective if all Hodge numbers (= steps in the Hodge filtration of MdR ) are ≥ 0. Example. The trivial motive M = Q and the motive M (f ) attached to a modular form of weight ≥ 2 are effective, but M = Q(1) is not effective. Fact 1.3. If M is effective and M 6= Q, then • the Hf1 =local condition at p is the relaxed local condition unless the local L-factor vanishes; • ords=0 L(M, s) = dim(MBetti )c=1 , where c denotes complex conjugation ⇒ the Bloch-Kato conjecture predicts that dim Hf1 (Q, Mp∗ (1)) = dim(MBetti )c=1 =: d+ . Euler systems are a tool for proving special cases of the Bloch-Kato conjecture for effective motives: Conjecture 1.4. (Perrin-Riou [PR98]) If M is effective, there exists a non-trivial system of elements (zm )m≥1 , + zm ∈ d ^ H 1 Q(µm )+ , Mp∗ (1) , such that Q(µ )+ coresQ(µm` + (zm` ) m) ( zm = P` (σ`−1 )zm if `|m or Mp∗ (1) is ramified at ` otherwise, where P` (X) = det(1 − Xσ`−1 |Mp ) and σ`−1 denotes the arithmetic Frobenius at `. Definition. Such a system of elements (zm )m≥1 is called a rank d+ Euler system for Mp . Remark. If M is defined over a number field K, then an Euler system for M should have classes over all the abelian extensions of K. 1 Theorem 1.5. (Kolyvagin, Rubin [Rub00], Perrin-Riou [PR98]) If such a rank d+ Euler system exists and z1 6= 0, then (under some technical hypotheses) the Selmer group Hf1 (Q, Mp∗ (1)) is d+ -dimensional, as prediced by the Bloch-Kato conjecture. Remark. In proving this theorem, it is very important that the Hf1 -local condition at p is the relaxed one. Examples. (rank d+ = 1 Euler systems) • M = Q (cyclotomic units) • M = M (f ), where f is a modular form of weight ≥ 2 (Kato’s Euler system, see [Kat04]) • M = Q over an imaginary quadratic field (eliptic units) Remark. There exist other Euler systems attached to self-dual motives (i.e. M ∼ = M ∗ (1))) satisfying a sign condition, but these don’t give access to non-central L-values. Problem. There are no non-trivial examples for d+ > 1! (Unless one assumes the ‘prectic conjecture’ of Nekovar-Scholl, which gves access to certian settings related to totally real fields.) 2. A new Euler system Theorem 2.1. (Lei-LZ [LLZ14], Kings-LZ [KLZ14]) Let f, g be modular forms of weights k+2, k 0 +2 ≥ 2, and let 0 ≤ j ≤ min{k, k 0 }. Then there exists a rank 1 Euler system for M = M (f ) ⊗ M (g)(1 + j), related to the p-adic L-value Lp (f, g, 1 + j). The existence of this Euler system does not fit the setting of Perrin-Riou’s conjecture: • d+ (M ) = 2, but M is not effective: its Hodge numbers are −1 − j, , k 0 − j, k − j, k + k 0 + 1 − j; • ords=0 L(M, s) = ords=1+j L(f, g, s) = 1; • the Hf1 -local condition at p is not the relaxed one; • the Euler system classes take values in the Bloch-Kato Selmer group. We therefore need a generalisation of Perrin-Riou’s conjecture for motives which are not effective. We first generalize the notion of effective: Definition. Let r ≥ 0. Then M is r-critical if the Archimedean Γ-factor L∞ (M, s) has a pole at s = 0 of order r, and L∞ (M ∗ (1), 0) 6= ∞. Remarks. (1) 0-critical is precisely Deligne’s definition of critical; (2) if M is r-critical, then ords=0 L(M, s) ≥ r (and this lower bound shold be sharp); (3) M is r-critical if and only if d+ − r Hodge numbers are < 0 ⇒ M is effective if and only if it is d+ -critical. Conjecture 2.2. If M is r-critical, there exists a non-trivial rank r Euler system (zm )m≥1 , zm ∈ r ^ Hf1 Q(µm )+ , Mp∗ (1) . Note. The Euler system in Theorem 2.1 is an example of an Euler system for a 1-critical motive. Remarks. • Conjecture 2.2 reduces to Perrin-Riou’s conjecture when r = d+ . (Note that in this case the Hf1 -local condition is relaxed, so the condition that the Euler system take values in Hf1 Q(µm )+ , Mp∗ (1) is pretty much automatic.) • A rank 0 Euler system should be thought of as a p-adic L-function. I will elaborate on this later. Theorem 2.3. (LZ) If M is 1-critical and (zm )m≥1 is a rank 1 Euler system with z1 6= 0, then (under some technical hypotheses) Hf1 (Q, Mp∗ (1)) is 1-dimensional, as predicted by the Bloch-Kato conjecture. Remark. For proving this theorem, we need to adapt the Euler system machine to take into account the non-relaxed local condition at p. Here are some examples of 1-critical motives: 2 (1) M = M (f ) ⊗ M (g)(1 + j), where f, g are modular forms of weights k + 2, k 0 + 2 ≥ 2 and 0 ≤ j ≤ min{k, k 0 }; (2) M = MAsai (f )(1 + j), where f is a quadratic Hilbert modular form of weights (k + 2, k 0 + 2) and 0 ≤ j ≤ min{k, k 0 }; (3) M = MSpin (F )(1 + j), where F is a genus 2 Siegel modular form of weights (k + 3, k 0 + 3), k ≥ k 0 ≥ 0 and 0 ≤ j ≤ k 0 ; (4) M (1), where M ⊂ h2 (Sh(U (2, 1))) is a rank 3 motive over an imaginary quadratic field. Remark. There are lots more examples: if M is any motive with distinct Hodge numbers, then some twist of it will be 1-critical. For the examples above, Conjecture 2.2 predicts the existence of a rank 1 Euler system: case (1) is Antonio Lei Theorem 2.1, and cases (2), (3) and (4) are joint work in progress with Francesco Lemma . Chris Skinner In other words, we can construct some exmaples of Euler systems for 1-critical motives. However, the interesting cases are those twists of the motives which are 0-critical. We can get at those using p-adic deformation. 3. Euler systems in p-adic families Definition. Let A be a complete local Zp -algebra, and let X = Spf(A). A family of motivic Galois representations is a finite free A-module V with an A-linear action of GQ such that for a Zariski dense set Xcl ⊂ X, we have for some motive Mx . Vx ∼ = Mx,p The key example is that of the cyclotomic deformation of a motivic Galois representation: Example. Let A = Λ(Z× p ) and V = Mp ⊗ A, where GQ acts on A via multiplication by the canonical × ,→ A . Note that V specializes to Mp (j) for all j. character GQ → Z× p In order to vary a rank r Euler system in a p-adic family, we need to make an auxiliary choice called an r-refinement. Definition. An r-refinement of V is an A-direct summand W ⊂ V which is GQp -stable together with a Zariski-dense set of points Xcl,W ⊂ Xcl , such that ∀x ∈ Xcl,W , • Mx is r-critical, • Wx has all Hodge-Tate weights ≥ 1, and Vx /Wx has all Hodge-Tate weights ≤ 0 (Panchishkin condition). Conjecture 3.1. Let V be a family of motivic Galois representations, and let W ⊂ V be an r-refinement. Then there exists a non-trivial rank r Euler system (zm )m≥1 , zm ∈ r ^ H 1 Q(µm )+ , V such that • locp (zm ) ∈ im H 1 (Q(µm )+ , W ) → H 1 (Q(µm )+ , V ) , • ∀x ∈ Xcl,W , the specialisation of the Euler system at x agrees with the Euler system from Conjecture 2.2. Remark. ∀x ∈ Xcl,W (generically), one has Hf1 Q(µm )+ , Vx = im H 1 Q(µm )+ , Wx → H 1 Q(µm )+ , Vx , Vr 1 so zm,x ∈ Hf as required. 3 Example: Rankin-Selberg convolutions Let f , g be Hida families. Then there exists a 3-parameter of GQ -representations × ˆ (g)∗ ⊗Λ(Z ˆ V (f )∗ ⊗V p) (two Hida parameters and one cyclotomic parameter) interpolating Mp (fk )∗ ⊗Mp (gk0 )∗ (1+j) for varying k, k 0 , j. We have the following refinements: 2-refinement: 1-refinement: 0-refinements: W2 = {0}, × ˆ + V (g)∗ ⊗Λ(Z ˆ W1 = F + V (f )∗ ⊗F p ), + ∗ˆ ∗ˆ × W0a = F V (f ) ⊗V (g) ⊗Λ(Zp ), × ˆ + V (g)∗ ⊗Λ(Z ˆ W0b = V (f )∗ ⊗F p ), Xcl,W2 = {k, k 0 ≥ 0, j ≤ −1} Xcl,W1 = {0 ≤ j ≤ min{k, k 0 }} Xcl,W0a = {k 0 + 1 ≤ j ≤ k} Xcl,W0b = {k + 1 ≤ j ≤ k 0 } Here, F + V (f ) denotes the rank 1 submodule as constructed by Wiles in [Wil88]. Theorem 3.2. (Kings-LZ, [KLZ14]) (1) Our Euler systems from Theorem 2.1 for M (fk )∗ ⊗ M (gk0 )∗ (1 + j) interpolate along W1 ; (2) (Explicit ( reciprocity law) There exist rank-lowering operators corresponding to the inclusions W0a W1 ,→ mapping the rank 1 Euler system to Hida’s two 2-variable p-adic L-functions; W0b (3) if a rank 2 Euler system exist, then it maps to our Euler system under the rank-lowering operator. Remarks. • The rank-lowering operators are multi-variable analogues of Perrin-Riou’s regulator map, as constructed in [LZ14]. • (KLZ, in progress) The Hida families can be replaced by Coleman families. In this case, the subrepresentations are replaced by sub-(ϕ, Γ)-modules over the Roba ring. In general, we expect there to be a hierachy of Euler systems, governed by inclusion of refinements, and related to each other via rank-lowering operators: if W 0 and W are r0 -, resp. r-refinements of V , then Conjecture 3.1 predicts the existence of rank r0 , resp. rank r Euler systems. If W 0 ⊂ W , then these Euler systems should be related to each other via a rank-lowering operator r ^ 0 H 1 + Q(µm ) , V - r ^ H 1 Q(µm )+ , V . References [Kat04] Kazuya Kato, P -adic Hodge theory and values of zeta functions of modular forms, Ast´ erisque 295 (2004), ix, 117–290, Cohomologies p-adiques et applications arithm´ etiques. III. MR 2104361 [KLZ14] Guido Kings, David Loeffler, and Sarah Zerbes, Rankin-Selberg Euler systems and p-adic interpolation, preprint, 2014. [LLZ14] Antonio Lei, David Loeffler, and Sarah Livia Zerbes, Euler systems for Rankin–Selberg convolutions of modular forms, Ann. of Math. 180 (2014), no. 2, 653–771. [LZ14] David Loeffler and Sarah Livia Zerbes, Iwasawa theory and p-adic L-functions for Z2p -extensions, Int. J. Number Theory (to appear) (2014). [PR98] Bernadette Perrin-Riou, Syst` emes d’Euler p-adiques et th´ eorie d’Iwasawa, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 5, 1231–1307. MR 1662231 (99m:11124) [Rub00] Karl Rubin, Euler systems, Annals of Mathematics Studies, vol. 147, Princeton University Press, 2000. MR 1749177 [Wil88] A. Wiles, On ordinary λ-adic representations associated to modular forms, Invent. Math. 94 (1988), no. 3, 529–573. MR 969243 4

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