Introduction to Mixed Modular Motives

Transcription

1 Duke University March 24, 2017

2 Acknowledgments Antecedents and Inspiration Beilinson and Levin: elliptic polylogarithms Yuri Manin: iterated integrals of modular forms; Francis Brown: multiple modular values (and motives) The many who contributed to the mixed Tate motive story (to be named later... ), which I will review shortly. Collaborators Francis Brown (ongoing) Makoto Matsumoto (universal mixed elliptic motives)

3 The Protagonist Context M 1,1 is the moduli stack of elliptic curves. It has two manifestations of concern to us: 1 the analytic orbifold: M an 1,1 = SL 2(Z)\\h, where h dentoes the upper half plane Im τ > 0. 2 the algebraic stack over Q: M 1,1/Q = G m \\ ( A 2 Q {u3 27v 2 = 0} ), where t (u, v) = (t 4 u, t 6 v).

4 Introduction Context Each (normalized) Hecke eigen cusp form f of SL 2 (Z) of weight 2n + 2 determines a pure motive V f of weight 2n + 1 and Hodge type (2n + 1, 0), (0, 2n + 1). Its Hodge and l-adic Galois realizations occur as summands of the cuspidal cohomology group Hcusp(SL 1 2 (Z), S 2n H) = Hcusp(M 1 1,1, S 2n H) = f eigen cusp form weight 2n+2 where S 2n H is the 2nth symmetric power of the standard representation and S 2n H is the corresponding local system. The Eisenstein series G 2n+2 determines the motive Q( 2n 1) of weight 4n + 2: H 1 (M 1,1, S 2n H) = H 1 cusp(m 1,1, S 2n H) Q( 2n 1). V f

5 For distinct eigenforms f 1,..., f m, we then have the motive V := S r 1 V f1 S rm V fm with its Hodge and l-adic Galois realizations. There are expected regulator mappings Ext 1 mot(q, V (d)) Ext 1 MHS (R, V R(d)) F = V ± R Ext 1 mot(q, V (d)) H 1 f (G Q, V Ql (d)) which are conjectured by Beilinson and Bloch Kato to be isomorphisms after tensoring with R and Q l, respectively. Their conjectured dimensions are given by the order of vanishing of a convolution L-function at an appropriate negative integer.

6 The main goal of this talk is to address the question: Question: Where does one find such extensions in nature, and what might their periods be? I believe that the case m = 1, r 1 = 1 is well understood. Many of the ideas in this talk for how to approach this are due to Francis Brown.

7 Iterated Integrals Context Definition (K.-T. Chen) Suppose that M is a smooth manifold. For 1-forms ω 1,..., ω r and a piecewise smooth path γ : [0, 1] M, define ω 1... ω r = ω 1 ω 2 ω r γ r = f 1 (t 1 )f 2 (t 2 )... f r (t r ) dt 1 dt 2... dt r 0 t 1 t 2 t r 1 where γ ω j = f j (t)dt and r is the time ordered r-simplex 0 t 1 t 2 t r 1 in [0, 1] r.

8 Properties: reflect combinatorial properties of simplices 1 shuffle product: ω 1... ω r ω r+1... ω r+s = ω σ(1) ω σ(2)... ω σ(r+s) γ 2 coproduct: γ σ sh(r,s) γ γµ ω 1... ω n = n ω 1... ω j j=1 γ µ ω j+1... ω n 3 antipode: ω 1... ω r = ( 1) r γ 1 γ ω r... ω 1

9 Example: The n-logarithm: ln n (z) := k 1 z n z k n = 0 dw 1 w n 1 {}}{ dw w... dw w, z < 1 and its special value ln n (1) = ζ(n) are iterated integrals.

10 Multiple Zeta Values Definition Suppose that r 1 and that n 1,..., n r are positive integers with n r > 1. The multiple zeta number ζ(n 1,..., n r ) is defined by the convergent series ζ(n 1,..., n r ) = 1 k n 1 0<k 1 < <k r 1 k n k. r nr The integer r is called the depth, and n n r is called weight of ζ(n 1,..., n r ). These satisfy lots of combinatorial identities. They are graded by weight and filtered by depth.

11 Let ω 0 = dw w and ω 1 = dw 1 w. The iterated integral 1 0 ω ɛ 1... ω ɛr converges if and only if ɛ 1 = 1 and ɛ r = 0. Kontsevich formula ζ(n 1,..., n r ) = 1 ω 1 0 n 1 1 n 2 1 n r 1 {}}{{}}{{}}{ ω 0... ω 0 ω 1 ω 0... ω 0... ω 1 ω 0... ω 0 Divergent iterated integrals can be regularized provided that one chooses non-zero tangent vectors v 0 at 0 and v 1 at 1. The regularization of this iterated integral is also an MZV when v 0 and v 1 are ± / w.

12 Regularizing iterated integrals: trivial coefficients Suppose that X is a complex curve, P, Q X and ω 1,..., ω r are holomorphic 1-forms on X := X {P, Q} with at worst logarithmic singularities at P, Q. (Ie., each ω j is a differential of the 3rd kind.) To regularize the possibly divergent Q P ω 1... ω r choose non-zero tangent vectors v T P X and w T Q X. Choose holomorphic coordinates s and t on X centered at P, Q, respectively, where v = / s and w = / t. Q P v γ w

13 Set f (s, t) = Q(t) P(s) ω 1... ω r. Then the theory of ODEs with regular singular points implies f (s, t) = r r a j,k (s, t)(log s) j (log t) k j=0 k=0 where each a j,k (s, t) is holomorphic near (0, 0). Define w v ω 1... ω r := a 0 (0). It depends only on v and w and not on the choice of t and s.

14 Regularization is compatible with properties (shuffle product, antipode, coproduct) of iterated integrals. Examples Take z = C, P = 0, Q = 1, v = / w and w = / w. 1 Since 1 w v t dw/w = log t, w v dw/w = 0. Similarly, dw/(1 w) = 0. 2 Since 1 0 ω 1ω 0 = ζ(2), the shuffle product formula 0 = w v w ω 0 ω 1 = gives w v ω 0ω 1 = ζ(2). v w v ω 0 ω 1 + w v ω 1 ω 0.

15 1 Manin considered iterated integrals of classical modular forms. 2 Brown considered regularized iterated integrals of classical modular forms, where the base point is the tangent vector / q at the origin of the q-disk. Such regularized iterated integrals will be regarded as periods of a suitable completion of π 1 (M 1,1, / q) = SL 2 (Z). Later we will explain why ± / q are the most (only?) natural choices of base point.

16 Suppose that f is a modular form of SL 2 (Z) of weight 2n + 2. The corresponding cohomology class in H 1 (SL 2 (Z), S 2n H) is represented by the SL 2 (Z)-invariant 1-form ω f (b) := f (τ)(b τa) 2n dτ. where a, b is a basis of the standard representation H of SL 2 (Z). Now suppose that f j, j = 1,..., r are modular forms where f j has weight 2n j + 2. One considers the iterated integral ω f1 (b 1 )ω f2 (b 2 )... ω fr (b r ) which takes values in S 2n 1H S 2nr H.

17 Multiple modular values are numbers obtained by evaluating ω f1 (b 1 )ω f2 (b 2 )... ω fr (b r ) along the imaginary axis, which is a path in M 1,1 from / q to / q, composed with a projection defined over Q. S 2n 1 H S 2nr H C Remark: Regularized iterated integrals of Eisenstein series include all MZVs (RMH), but are more general (Brown: they include periods cusp forms). So multizeta values form a subring of mixed modular values.

18 Tate Motives Context The Tate motives over Q are Q(n), where n Z. Examples: 1 Q(0) = H 0 (point); 2 Q(1) = H 1 (G m ) 3 Q( 1) = H 1 (G m ) = H 2 (P 1 ) 4 Q(n) = Q(1) n ; Hodge type ( n, n). Realizations of Q( 1) Betti: H 1 (C ; Q) de Rham: H 1 DR (G m/q) = Q dw w l-adic: H 1 «et(g m /Q ) = Q l on which G Q acts by the inverse of the l-adic cyclotomic character.

19 Mixed Tate Motives Mixed Tate motives (MTMs) over Q are successive extensions of Tate motives. They have Betti, de Rham and l-adic realizations. The l-adic realization is G Q -module; Betti and de Rham define a mixed Hodge structure. Example: H 1 (G m, {1, x}), x Q 0 H 1 (G m ) H 1 (G m, {1, x}) H0 ({1, x}) 0 Q(1) Q(0) This is unramified outside {primes p : ν p (x) 0}.

20 Tannakian Categories The axioms of a neutral tannakian category over a char 0 field F are what you obtain if you axiomatize the properties of the category of finite dimensional representation of a group over F. Essential features: 1 F-linear abelian category (internal Hom sets are vector spaces over F) 2 associative and commutative tensor product 3 trivial object and duals 4 compatibilities 5 faithful functor to Vec F (fiber functor)

21 Tannaka duality Context Tannaka duality: Every neutral tannakian category C over F is equivalent to the category of representations of an affine group over F. It depends on the choice of a fiber functor ω and is denoted π 1 (C, ω). All are canonically isomorphic up to inner automorphisms. Examples: 1 The category of semi-simple Tate motives over Q is tannakian. Fiber functors Betti or de Rham. The fundamental group is G m/q. 2 The category MHS of Q-mixed Hodge structures.

22 Mixed Tate Motives Theorem (Borel, Voevodsky, Levine, Deligne Goncharov) There is a Q-linear tannakian category of MTMs unramified over O K,S and whose ext groups Ext j MTM(O K,S )(Q, Q(n)) vanish when j > 1 and satisfy: Ext 1 MTM(O K,S ) (Q, Q(n)) = K 2n 1 (O K,S ) Q. This implies that there is a natural isomorphism π 1 (MTM(Z), ω DR ) = G m K where K is a prounipotent group whose Lie algebra is the free Lie algebra k = L(σ 3, σ 5, σ 7, σ 9,... )

23 Digression: unipotent completion Suppose that Γ is a discrete group and that F is a field of characteristic zero. The unipotent completion Γ un /F of Γ over F is the fundamental group of the category of unipotent representations Γ U(F) of Γ defined over F. It is a prounipotent F-group. There is a canonical homomorphism Γ Γ un (F). Universal mapping property Given a representation representation ρ : Γ U(F) of Γ, where U is unipotent over F, there is a unique homomorphism U such that ρ is the composite Γ un /F Γ Γ un (F) U(F).

24 Example: unipotent completion of a free group If Γ is the free group Γ = u, v and ad bc 0, then θ : FΓ F X, Y is a (complete) Hopf algebra isomorphism when where U, V L(X, Y ) and θ(u) = exp U and θ(v) = exp V U ax + by and V cx + dy mod J 2. Theta induces an isomorphism Γ un (F) exp L(X, Y ).

25 Theorem (K.-T. Chen) If M is a smooth manifold and x M, then the coordinate ring O(π1 un (M, x)) is isomorphic to the set of closed iterated integrals on M. Example: M = P 1 {0, 1, } The homomorphism Θ : Cπ 1 (P 1 {0, 1, }, x) C e 0, e 1 γ 1 + ω 0 e 0 + ω 1 e 1 + γ γ + γ γ ω 0 ω 0 e ω 0 ω 1 e 0 e 1 γ ω 1 ω 0 e 1 e 0 + ω 1 ω 1 e produces an isomorphism O(π un 1 (P1 {0, 1, }, x)) with the Hopf algebra of iterated integrals ω ɛ1... ω ɛr, where ɛ j {0, 1}. γ

26 Question: Where does one find extensions of mixed Tate motives in nature? Following older work of Deligne (1987), Deligne and Goncharov proved: Theorem (Deligne Goncharov, 2005) The coordinate ring of the unipotent completion of π 1 (P 1 {0, 1, }, v) is an ind-object of MTM(Z), where v = / w T 1 P 1.

27 Periods of MTM Context Every mixed Tate motive V has a Betti realization V B and a de Rham realization V DR. These are rational vector spaces and are isomorphic after tensoring with C: comp : V DR Q C V B Q C The periods of V are the complex numbers φ(comp(v)), where v V DR and φ Hom Q (V B, Q). The periods of O(π un 1 (P1 {0, 1, }, v) are precisely the multiple zeta values γ ω ɛ 1... ω ɛr. Question: Are the periods of all objects of MTM(Z) multiple zeta values?

28 Brown s Theorem Context Theorem (Brown 2012) The action of π 1 (MTM(Z), ω DR ) on O(π un 1 (P1 {0, 1, }, v)) is faithful. Corollary 1 The periods of every object of MTM(Z) are multiple zeta values. 2 The category MTM(Z) is the smallest tannakian subcategory of MHS Q that contains O(π un 1 (P1 {0, 1, }, v)).

29 Loose Ends... Context While many of the main questions about mixed Tate motives and multiple zeta values have be resolved, there are important problems which remain. These include: 1 Understanding periods MTM(O K,S ) for other rings of S-integers, especially the case of Z[µ N, 1/N]. Cf. work of Goncharov, Deligne on periods of π1 un(g m µ N, v) and values of polylogarithms at roots of unity. 2 The relationship between classical modular forms and multiple zeta values: work of Gangl Kaneko Zagier and the Broadhurst Kreimer Conjecture. Why should elliptic modular forms control multiple zeta values? More about that later.

30 Context Beyond Mixed Tate Motives Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient Where to next? mixed Tate motives: G m /Z mixed elliptic motives: E/K mixed modular motives: M 1,1 /Z motives associated to the universal elliptic curve: E M 1,1 Issues: We do not currently have a tannakian category of mixed motives with the correct exts in the elliptic or modular cases. The problem is that Beilinson Soulé vanishing is not known except in the mixed Tate case.

31 Geography Context Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient E 0 E / q / w identity section q = 0 / q

32 Mixed Elliptic Motives Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient To each elliptic curve E/K, one can associate the pure motive H 1 (E) of weight 1. The category of elliptic motives associated to E should be a tannakian category whose simple objects are quotients of the Tate twists S m H 1 (E)(r) of symmetric powers of H 1 (E). Working hypothesis: The coordinate ring O(π un 1 (E, w)) of E := E {id E } with basepoint a non-zero vector in T id E should be an object of this category. Could it generate it? Example: The smooth locus of the nodal cubic E 0 is G m. So E 0 = P1 {0, 1, } and the mixed elliptic motives associated to E 0 should be MTM(Z). Further, one can show that π un 1 (E / q, w) is in MTM(Z).

33 Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient It is useful to consider all elliptic curves at the same time. To this end we consider the universal elliptic curve π : E M 1,1 and the local system H = R 1 π Q. Elliptic motives in this situation are motivic local systems V over M 1,1 whose weight graded quotients W r V/W r 1 V are direct sums of the the simple local systems S m H(d). 1 Here W is the weight filtration 0 W r 1 V W r V V. 1 Here one can take a motivic local system to be a compatible set of l-adic lisse sheaves and an admissible variation of MHS.

34 Universal Mixed Elliptic Motives Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient To rigidify the possible extensions, we insist that the fiber of such a V over / q be an object of MTM(Z). These local systems are called universal mixed elliptic motives. Similarly one can define universal mixed elliptic motives over M 1,2. Theorem The local system over M 1,2 whose fiber over (E, id E, x) is the Lie algebra of π1 un(e, x) is a (pro) universal mixed elliptic motive. Local systems V over M 1,1 correspond to representations of on their fiber V := V / q. π 1 (M 1,1, / q) = SL 2 (Z)

35 Relative Unipotent Completion Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient The standard representation of SL 2 (Q) is H := H 1 (E / q ). The category of finite dimensional representations of SL 2 (Z) that admit a filtration 0 = V 0 V 1 V N = V with the property that each graded quotient V r /V r 1 is isomorphic to a direct sum of copies of S m H is tannakian over Q. Definition The relative (unipotent) completion G rel of SL 2 (Z) is the tannakian fundamental group of this category.

36 Properties of G rel Context Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient 1 There is a Q-DR manifestation GDR rel of Grel and a canonical isomorphism GDR rel Q C = G rel Q C. So one can speak of periods of G rel. All MMVs are periods. 2 The coordinate ring O(G rel ) and the Lie algebra g rel have natural (ind and pro, resp.) MHS. 3 There is a natural action of G Q on G rel Q Q l. It is unramified at all p l (Mochizuki Tamagawa) and crystalline at l (Olsson). (This is why we choose the base point to be ± / q.) 4 It is a split extension 1 U rel G rel SL 2 1 where U rel is prounipotent. So G rel = SL(H) U rel. (Levi)

37 Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient 5 The Lie algebra u rel of U rel is a free pronilpotent Lie algebra isomorphic to L(H 1 (u rel )). 6 There is a canonical SL(H)-invariant isomorphism of MHS H 1 (u rel ) = n 2 ( S 2n H(2n + 1) f eignform weight 2n+2 V f S 2n H(2n + 1) where H = Q(0) Q( 1). It is also G Q -equivariant after tensoring with Q l. The weight graded quotients of u rel are therefore products of V (d), where V := S r 1 V f1 S rm V fm. )

38 Mixed Modular Motives Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient There should be a tannakian category of mixed motives, unramified over Z whose simple objects are V (d), where V = S r 1 V f1 S rm V fm where f 1,..., f m are distinct eigenforms. We are far from having a construction of this category from Voevodsky motives. Brown s end run motivated by his result for MTM(Z): Definition (Brown) Define the category MMM = MMM(Z) of mixed modular motives unramified over Z to be the full tannakian subcategory of MHS Q generated by O(G rel ). Their periods include all multiple modular values.

39 Mixed Modular Motives: wrinkles Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient There are regulators Ext 1 MMM (Q, V (d)) Ext1 MHS (R, V (d))f and Ext 1 MMM (Q, V (d)) H1 f (G Q, V Ql (d)). One would hope that these are isomorphisms after tensoring with R and Q l, respectively. However, Brown recently observed that the Hodge regulator cannot be an isomorphism when r r m > 1 and d is small. Still, one hopes that these are isomorphisms once d is large enough.

40 The Eisenstein Quotient Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient The inclusion MTM(Z) MHS induces a surjection π 1 (MHS) π 1 (MTM). Theorem The fundamental group of the category of universal mixed elliptic motives is π 1 (MTM) G eis where G eis is the maximal quotient of G rel on which π 1 (MTM) acts. It is called the Eisenstein quotient of G rel. Its DR realization is canonically split: GDR eis = SL 2 UDR eis. The coordinate ring O(U eis ) is contained in (and should equal) the set of linear combinations of regularized iterated integrals of Eisenstein series whose periods are MZVs.

41 Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient Representations of π 1 (MTM(Z) G eis ) are universal mixed elliptic motives. They correspond to motivic local systems V over M 1,1 whose weight graded quotients are direct sums of S m H(d) and whose fiber over / q is in MTM(Z). Examples of universal mixed elliptic motives 1 the elliptic polylogs of Beilinson Levin 2 the local system over M 1,2 whose fiber over (E, id E, x) is the Lie algebra of π1 un(e, x). Its Q de Rham realization is the elliptic KZB connection studied by Calaque Etingof Enriquez and Levin Racinet.

42 Pollack Relations Context Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient 1 The Lie algebra u eis is not free. This reflects the fact (due to Brown) that the periods of iterated integrals of Eisenstein series contain periods of cusp forms. 2 We know that each normalized eigenform gives a countable set of minimal relations (up to the actions of SL 2 and G Q ) between the Eisenstein generators. 3 Standard conjectures about the size of Ext 1 mot(q, V f (d)) imply that there are no more relations.

43 Derivations Context Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient Set p = π un 1 (E / q, w). It is isomorphic to the rank 2 free Lie algebra L(H). The monodromy action induces a homomorphism k g eis Der L(H) where k = L(σ 3, σ 5, σ 7, σ 9,... ). The generator e 2n of u eis corresponding to G 2n goes to a derivation ɛ 2n of L(H); the generator σ 2m 1 determines a derivation ζ 2m 1 that corresponds to G 2m. Relations between the generators of k g eis give relations between the ɛ 2n s and ζ 2m 1 s.

44 Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient Pollack (2009) found such relations in degree 2, and (mod depth 3 ) found relations of all degrees. All of his relations are known to lift to u eis and they account for all known relations. Example: quadratic relations c a [ɛ 2a+2, ɛ 2b+2 ] = 0 a+b=n if and only if there is a cusp form f of SL 2 (Z) of weight 2n + 2 with r + f (a, b) = c a a 2a b 2n 2a He also found remarkable congruences between several of these relations. One example is given on the following slides.

45 Pollack Congruences Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient The quartic relation corresponding to the normalized cusp form of weight 12 is a multiple of 0 = ω12, ω4 10, ω4 8, [ɛ 4, ω8,2 3 ] [ɛ 4, ω6,4 3 ] [ɛ 6, ω6,2 3 ] [ɛ 6, ω4,4 3 ] [ɛ 8, ω4,2 3 ] [ɛ 10, ω2,2 3 ] [ɛ 4, [ɛ 4, [ɛ 6, ɛ 4 ]]]

46 Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient This is congruent mod 691 to 318 times the relation corresponding to 691G 12 : 691[ζ 3, ɛ 12 ] = 210 ω 4 12, [ɛ 4, ω 3 8,2 ] [ɛ 4, ω 3 6,4 ] [ɛ 6, ω 3 6,2 ] [ɛ 8, ω 3 4,2 ] [ɛ 10, ω 3 2,2 ] [ɛ 4, [ɛ 4, [ɛ 6, ɛ 4 ]]]. He compute another in weight 16. These and Brown s computations of the periods of twice iterated integral of Eisenstein series suggest that there is an action of the Hecke algebra on (say) the cohomology groups H ( π 1 (MTM(Z) G eis, S 2n H(d)) ) = Ext MEM (Q, S 2n H(d)), where MEM denotes the category of universal mixed elliptic motives over M 1,1.

47 Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient Future Projects and Possible Applications (with Brown) 1 Investigate analogues (two kinds) of Beilinson s conjecture in MMM by studying extensions in O(G rel ). 2 The problem of constructing Hecke operators in this non-abelian setting. 3 Pollack relations and their manifestations in higher genus; their relationship to the cohomology of outer automorphism groups of free groups. 4 Mixed modular motives and universal elliptic motives of level N > 1; their relationship to MTM ( O(µ N, 1/N) ) and its missing periods. 5 The Broadhurst Kreimer conjecture. 6 Prove that Pollack s congreunces generalize to all weights.

48 References Context Mixed Elliptic Motives Mixed Modular Motives The Eisenstein Quotient 1 F. Brown: Multiple modular values for SL 2 (Z), preprint, [arxiv: ]. 2 R. Hain: The Hodge-de Rham Theory of Modular Groups, in Recent Advances in Hodge Theory edited by Matt Kerr and Gregory Pearlstein, LMS Lecture Notes 427, R. Hain, M. Matsumoto: Universal mixed elliptic motives, preprint, [arxiv: ]. 4 Y. Manin: Iterated integrals of modular forms and noncommutative modular symbols, Algebraic geometry and number theory, , Progr. Math., 253, A. Pollack: Relations between derivations arising from modular forms, undergraduate thesis, Duke University, 2009.