SPECIAL VALUES OF E-FUNCTIONS AS
SPECIAL VALUES OF E-FUNCTIONS AS EXPONENTIAL PERIODS PETER JOSSEN
1. Exponential motives - setting the stage Let k ⊆ Q ⊆ C be a number field. Exponential motives over k with rational coefficients form a neutral tannakian, Q-linear category Mexp (k) which contains objects H n (X, Y, f )(i) for every variety X over k, closed subvariety Y ⊆ X and regular function f : X → A1 . This category of exponential motives is constructed along the same lines as Nori’s category of mixed motives over k. There is indeed a canonical, fully faithful functor MNori (k) → Mexp (k) sending a Nori-motive H n (X, Y )(i) to the exponential motive H n (X, Y, 0)(i). The details of the construction of Mexp (k) are not important for the understanding of the rest of this text, so I refer the reader to the book in preparation [FJ] for these details. As a notational convention, we agree to drop f from the notation is f = 0, to drop the twist i from the notation if i = 0, and to drop Y from the notation is Y is the empty subvariety of X. – 1.1. The category of exponential motives comes with a panoply of realisation functors and comparison functors. The Betti-realisation is a functor RB : Mexp (k) → VecQ which associates with a motive H n (X, Y, f ) the rational vector space dual to the finite dimensional space (1.1.1)
Hnrd (X, Y, f ) = lim Hnsing (X(C), Y (C) ∪ f −1 (Sr ))Q r→∞
where Sr stands for the half plane Re(z) > r. It is an easy consequence of Verdier’s constructibility theorem or of Ehresmann’s Lemma that this limit stabilises, meaning that for r 0 the transition maps in the limit, which are induced by inclusions Sr ⊆ Sr0 for r 6 r0 , are isomorphisms. The vector space given by (1.1.1) is called rapid decay homology of (X, Y, f ), and was in this form introduced by Bloch and Esnault in [BE]. The deRhamrealisation is a functor RdR : Mexp (k) → Veck Date: May 2018. 1
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whose description I will only give on motives of the form H n (X, f ) where X is a smooth variety. We consider the usual deRham complex Ω0X → Ω1X → Ω2X → · · · of sheaves of differential forms on X, but equip it with the differential df given on local sections by the formula df (ω) = dω − df ∧ ω. The deRham realisation of the motive H n (X, f ) is the vector space n n HdR (X, f ) = HZar (X, (Ω•X , df ))
which is indeed finite dimensional.
– 1.2. The Betti and the deRham realisation can be compared, as one is accustomed from the theory of classical motives. The comparison comes in the form of a pairing (1.2.1)
n (X, f ) → C Hnrd (X, f ) × HdR
which is defined as follows. An element of Hnrd (X, f ) is given by a coherent sequence of n (X, f ) is given, at least when topological n-cycles γ = (γr )r 0 , whereas an element of HdR X is affine, by a global algebraic differential n-form ω on X. The pairing (1.2.1) is given by Z (γ, ω) 7−→ lim
r→∞ γ r
e−f ω
and is indeed well defined, as one can show using a version of Stokes’s theorem for improper integrals. Much harder to show is the following theorem, analogous deRham’s theorem. Significant steps in its proof are due to Deligne, Malgrange, Esnault, Bloch, Sabbah, Hien and Roucairol, see [HR].
Theorem 1.3. The pairing (1.2.1) induces a natural isomorphism of complex vector spaces α
M RB (M ) ⊗Q C −−− → RdR (M ) ⊗k C
for every exponential motive M , which is moreover compatible with tensor products and duality in Mexp (k).
– 1.4. In order to formulate a period conjecture for exponential motives, let us introduce a category P(k) of period structures. Its objects are triples P = (V, W, α) consisting of a finite dimensional Q-vector space V , a finite dimensional k-vector space W , and an isomorphism α
V ⊗Q C −−→ W ⊗k C of complex vector spaces. We denote by ωB : P(k) → VecQ
and
ωdR : P(k) → Veck
SPECIAL VALUES OF E-FUNCTIONS AS EXPONENTIAL PERIODS
3
the obvious forgetful functors sending P = (V, W, α) to V , respectively to W . The category P(k) is tannakian Q-linear, and ωB is a neutral fibre functor. We can associate with any object P = (V, W, α) of P(k) the following things: (1) The linear algebraic group GP ⊆ GLV representing Aut⊗ (ωB |hP i⊗ ). This is the Tannakian fundamental group of the tannakian category hP i⊗ ⊆ P(k) generated by P . (2) The GP -torsor TP over k, representing the functor Isom⊗ (ωB,k |hP i⊗ , ωdR |hP i⊗ ). Here, ωB,k : P(k) → Veck is the functor sending an object P1 = (V1 , W1 , α1 ) to V1 ⊗Q k. We call TP the period torsor of P . (3) The k-algebra AP ⊆ C generated by det(α)−1 and the coefficients of the matrix of α with respect to any Q-basis of V and k-basis of W . We call AP the period algebra of P . We call its elements periods of P . The period torsor and the period algebra are related as follows: We can regard the given isomorphism α : V ⊗Q C → W ⊗k C as a complex point α ∈ TP (C). This gives rise to an evaluation map evα : OTP → C whose image is AP . Equivalently, Spec AP is a closed k-subscheme of TP , and indeed the smallest closed k-subscheme containing α as a complex point. The algebra of regular functions OTP of TP is called algebra of formal periods.
– 1.5. The combination of the Betti realisation, the deRham realisation and the comparison isomorphism leads to a functor Rper : Mexp (k) → P(k) which we call period realisation. This functor is an exact faithful ⊗-functor between tannakian categories, by definition compatible with fibre functors. Hence if M is a motive with period realisation P , there is an inclusion of algebraic groups G P ⊆ GM where GM stands for the tanakian fundamental group of M , that is, the fundamental group of the tannakian category hM i⊗ ⊆ Mexp (k) generated by M . We call AP the period algebra of M and elements of AP periods of M . Summarily, exponential periods are complex numbers which are periods of some exponential motive, whereas classical periods are complex numbers which are periods of some classical motive.
Conjecture 1.6. Let M be an exponential motive, and write P = (V, W, α) for its period realisation. Then (1) The inclusion GP ⊆ GM is an equality. In other words, the period realisation functor Rper : hM i⊗ → hP i⊗ is full and essentially surjective.
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(2) The point α ∈ TP (C) is dense, so Spec AP = TP . Equivalently, the evaluation map OTP → C is injective.
– 1.7. Little is known about this conjecture, even for classical motives. In fact, most positive examples in which we can show the conjecture to hold involve genuine exponential motives. For instance, the classical Lindemann-Weierstrass theorem can be seen as an illustration of the conjecture. The period conjecture is often formulated as the equality of numbers dim GM = trdegk AP which indeed follows from 1.6, but is slightly less precise. Part (1) of the conjecture has been called formal period conjecture and can sometimes be deduced from quite limited information about GP and GM by invoking results from the theory of linear algebraic groups. At the heart of part (2) of the conjecture are transcendence statements which are infamously difficult to prove.
– 1.8. It might come as a surprise that for any given exponential motive M the motivic fundamental group GM of M is, generally speaking, not very hard to compute. Write V for the Betti-realisation of M , so GM is a linear algebraic subgroup of GLV . Realisations of M in interesting categories (Hodge structures, `-adic realisations, etc.) give rise to interesting and presumably large subgroups of GM , whereas endomorphisms of M or of tensor constructions of M cut out constraints for GM . If all goes well, there is but one possibility left for GM . One particularly interesting and important realisation functor for exponential motives is the perverse realisation Rperv : Mexp (k) → Perv0 where Perv0 is the following tannakian category. Its objects are perverse sheaves of Qvector spaces F on A1k , with the property RΓF = 0. It’s not difficult to show that such perverse sheaves are precisely those of the form F [1] for some constructible sheaf F satisfying H 0 (A1 , F ) = 0 and H 1 (A1 , F ) = 0. Morphisms in Perv0 are morphisms of perverse sheaves, and the tensor product is given by additive convolution F ∗ G = Rsum∗ (pr∗1 F ⊗ pr∗2 G) with the obvious maps pr1 , pr2 , sum : A2k → A1k . A fibre functor is the nearby-fibre-atinfinity functor which we can define as Ψ∞ (F [1]) = colimF (Sr ) r→∞
for Sr = {z ∈ C | Re(z) > r}. It is a nontrivial matter to show that Ψ∞ is a fibre functor, especially its compatibility with associativity and commutativity constraints is tricky. The
SPECIAL VALUES OF E-FUNCTIONS AS EXPONENTIAL PERIODS
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perverse realisation of a motive M of the form M = H n (X, Y, f ) is given by Rperv (M ) = Hn ((X × A1 , Y × A1 ∪ Γf )/A1 )[1] where Γf ⊆ X × A1 stands for the graph of f , and Hn ((−)/A1 ) is relative cohomology over A1 which could as well be expressed as a higher direct image. It is clear from the definitions that there is a canonical and natural isomorphism Ψ∞ (Rperv (M )) ∼ = RB (M ) of vector spaces, compatible with tensor products and duality. Therefore, if F [1] is the perverse realisation of an exponential motive M , then there is a canonical inclusion GF [1] ⊆ GM where GF [1] and GM are the tannakian fundamental groups of F [1] and M respectively.
Theorem 1.9 (Gabber, Katz). Let f ∈ k[x] be an irreducible polynomial of degree d+1 > 0, viewed as a regular function on A1 = Spec k[x]. Set M = H 1 (A1 , f ) and write F for the perverse realisation of M and set V = Ψ∞ F , so V ' Qd . Suppose that f 0 has no multiple roots, and that for any four not necessarily distinct roots α1 , α2 , α3 , α4 of f 0 the implication f (α1 ) + f (α1 ) = f (α3 ) + f (α4 )
=⇒
{α1 , α2 } = {α3 , α4 }
holds. Then, the tannakian fundamental group GF ⊆ GLV of F contains SLV . Moreover, equality GF = GLV if and only if X b := f (α) α
is nonzero, where the sum ranges over all zeroes α of f 0 . The constructible sheaf F is indeed F = (f∗ Q)/Q. Its singularities are at the branch points of f , that is, at the zeroes of the discriminant ∆(z) of f (x) − z. The quantity b is the only singularity of the one dimensional object det F . Example 1.10. Fix an element z ∈ k, and consider the motive M = H 1 (A1 , f ) where f is the polynomial f (x) = 31 x3 − zx seen as a regular function on A1 = Spec k[x]. The rapid decay homology H1rd (X, f ) is the homology of the pair (C, f −1 (Sr )) for large r. It is a Q-vector space of dimension 2 with a basis γ1 , γ2 given by cycles as illustrated in Figure 1. Notice that in this particular example the basis γ1 , γ2 is independent of z, which à priori it need not be. The story for f (x) = zx3 − x would be quite different!
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Figure 1. A basis of H1rd (X, f ) 1 (A1 , f ) is given by the global differential forms dx A basis of the deRham cohomology HdR and xdx. The period pairing (1.2.1) as well as the corresponding comparison isomorphism of Theorem 1.3 is given in terms of these bases by the matrix P (z), with coefficients Z 1 3 (1.10.1) Pij (z) = e− 3 x +zx xi−1 dx γj
for 1 6 i, j 6 2. The integrals (1.10.1) are well known: They define the value at z ∈ k of the complex Airy functions. We have ! ! P11 (z) P12 (z) Ai1 (z) Ai2 (z) P (z) = = P21 (z) P22 (z) Bi1 (z) Bi2 (z) where Ai1 and Ai2 are linearly independent solutions of the Airy equation u00 (z) − zu(z) = 0 and Bij (z) = Ai0j (z). The discrinimant of f (x) − t as a polynomial in x equals ∆(z, t) = 13 (4z 3 − 9t2 ) √ so its zeroes as a function of t are at ± 32 z. Hence the perverse realisation of the motive M is a perverse sheaf F [1], where the constructible sheaf F is given by a local system of rank √ 2 on A1 \ {± 23 z}. Unless z = 0, the tannakian fundamental group of the object F [1] of Perv0 is SL2 . As for the motive M itself, one can show that its determinant equals Q(1), hence GM = GL2 except when z = 0.
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2. Some families of exponential motives on the affine line Let us consider more generally families exponential motives, depending on a formal parameter z, of the form M = H 1 (A, f ) with f (x) =
d+1 1 d+1 x
− zg(x)
for some integer d > 1 and polynomial g(x) of degree 6 d. Set ζ = e2πi/(d+1) . As in example 1.10, a basis γ1 , . . . , γd of the rapid decay homology is given by paths (γj,r )r 0 in the complex plane, where γj,r runs from ζ j r to r. A basis of the deRham cohomology is given by the global forms dx, xdx, . . . , xd−1 dx. Just as in example 1.10, the period pairing (1.2.1) as well as the corresponding comparison isomorphism of Theorem 1.3 is given in terms of these bases by the d-by-d matrix P (z), with coefficients Z 1 d+1 +zg(x) i−1 x − x dx e d+1 Pij (z) = γj
for 1 6 i, j 6 d. – 2.1. In order to examine the periods Pij of M , which we view now as entire functions in the complex variable z, we introduce Z 1 d+1 − x +zg(x) Ej (z, h) = e d+1 h(x)dx γj
for h(x) ∈ k[x]. We have Pij (z) = Ej (z, xi−1 ). Besides the obvious k-linearity in h, the following relations hold: (2.1.1)
Ej (z, h0 ) = Ej (z, xd h) − zEj (z, g 0 h)
(2.1.2)
∂ Ej (z, h) = Ej (z, gh) ∂z
Using these, it is elementary to show that P (z) is a fundamental matrix of solutions of a matricial differential equation u0 = Lu for some d-by-d matrix L with coefficients in k[z ±1 ]. The fact that the Pij (z) are entire functions shows that the only singularity of this equation is at infinity, and by using Fuchs’s criterion (i.e. because some of the Pij (z) grow exponentially fast) one can see that infinity is an irregular singularity lest g = 0.
– 2.2. We can expand the entire functions Pij (z) as Taylor series around 0. To compute the Taylor coefficients, the relations (2.1.1) and (2.1.1) help. We use them in the form (2.2.1)
Ej (0, h0 ) = Ej (0, xd h)
(2.2.2)
∂n Ej (z, h) = Ej (z, g n h) ∂z n
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and notice that because of relation (2.2.1), Ej (0, g n h) can be written as a k-linear combination of Ej (0, 1), Ej (0, x), . . . , Ej (0, xd−1 ), with coefficients Aikn which are independent of j. The Taylor expansion of Pij (z) = Ej (z, xi−1 ) reads Pij (z) =
∞ X 1 Ej (0, g n xi−1 )z n n!
n=0
d ∞ X 1 X Aikn Ej (0, xk−1 )z n = n! n=0
=
k=1
d X ∞ X k=1 n=0
=
d X
1 Aikn z n Pkj (0) n!
Aik (z)Pkj (0)
k=1
hence, in summary, we find the relation P (z) = A(z)P (0) where P (0) is a constant d-by-d matrix whose coefficients are independent of f . A direct computation reveals Z 1 d+1 ζ ij − 1 x i Pij (0) = e d+1 xi−1 dx = Γ d+1 d/(d+1) (d + 1) γj where ζ = e2πi/(d+1) . In particular, all coefficients Pij (0) of P (0) belong to the field √ 1 2 d Q(ζ, Γ( d+1 ), Γ( d+1 ), . . . , Γ( d+1 )). The determinant of P equals cπ d for some nonzero rational c which we could, but do not care to determine. The power series Aij (z) =
∞ X 1 Aikn z n n!
n=0
define entire functions, have coefficients in k, and arranged in a matrix satisfy the same differential equation u0 = Lu as do the Pij .
Example 2.3. If we carry out the computations in 2.2 for the example of f (x) = 31 x3 − zx, we find z3 z6 z9 z 12 z 15 + 4 + 4 · 7 + 4 · 7 · 10 + 4 · 7 · 10 · 13 + ··· 3! 6! 9! 12! 15! z4 z7 z 10 z 13 z 16 A12 (z) = z + 2 + 2 · 5 + 2 · 5 · 8 + 2 · 5 · 8 · 11 + 2 · 5 · 8 · 11 · 14 + ··· 4! 7! 10! 13! 16!
A11 (z) = 1 +
as well as A21 (z) = A011 (z) and A22 (z) = A012 (z). It follows from Abel’s theorem that the Wronskian A11 (z) A12 (z) W (z) = det A21 (z) A22 (z)
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is constant, equal to 1. The equality P (z) = A(z)P (0) amounts to the classical power series expansion of the Airy functions.
– 2.4. In general, the determinant of A(z) is not constant. It is given by X f (α) det(A(z)) = eb(z) b(z) = f 0 (α)=0
where the sum runs over all zeroes α of f 0 (x) = xd − zg 0 (x), while regarding z as a fixed complex parameter. As a function of z, the quantity b(z) is a polynomial with coefficients in k, satisfying b(0) = 0.
3. Consequences of the Siegel-Shidlovskii theorem We are interested in the transcendence of the coefficients of the matrix P (1), since these coefficients together with det(P (1))−1 , generate the period algebra of the motive H 1 (A1 , f ), 1 xd+1 − g(x). with f (x) = d+1 – 3.1. Let s > 0 be a positive rational number. An arithmetic Gevrey series of order −s is an entire function given by a power series E(z) =
∞ X
an z n
n=0
where the coefficients an are algebraic numbers, satisfying the following estimates for some real constant C > 0. (1) For an algebraic number a, denote by kak the maximum norm of the complex conjugates of a. Then, k(n!)s an k < C n holds for all n. (2) For every n > 1, there exists an integer N 6 C n such that N a0 , N a1 , . . . , N an are all algebraic integers. Condition (1) guarantees the convergence of the power series on the whole complex plane, so any formal series satisfying (1) and (2) is an arithmetic Gevrey series. We are interested in holonomic Gevrey series, that is, Gevrey series which are annihilated by some nontrivial differential operator D ∈ Q(z)[∂]. Typical examples of holonomic arithmetic Gevrey series include polynomials, the exponential function, the Bessel functions of the second kind Jm for integral m, or the function ∞ X zn E(z) = n · n! n=1
which is a variant of the exponential integral function. To verify condition (2) for this series requires the use of the prime number theorem.
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Theorem 3.2 (Siegel-Shidlovskii, André, Beukers). Let d > 0 be an integer, and let L be a d-by-d matrix with coefficients in Q(z). Let T (z) ∈ Q[z] be a common denominator of the coefficients of L. Let E be a d-by-m matrix satisfying E 0 = LE and whose coefficients Eij (z) are arithmetic Gevrey series. Let z0 be an algebraic number satisfying z0 T (z0 ) 6= 0. Then, every Q-polynomial relation between the complex numbers Eij (z0 ) is obtained by specialisation at z = z0 from a Q(z)-polynomial relation between the functions Eij (z). In particular, the equality trdegQ(z) Q(z, E11 (z), . . . , Edm (z)) = trdegQ Q(E11 (z0 ), . . . , Edm (z0 )) holds. References are [A] (there is a summary by the same author in J. Th. des nombres de Bordeaux, 2003) and [B]. – 3.3. We come back to families exponential motives of the form M = H 1 (A, f ) with 1 f (x) = d+1 xd+1 − zg(x). We have seen that the periods of M , with respect to appropriate bases, can be written as P (z) = A(z)P (0) where A(z) is a fundamental matrix solution of a matricial differential equation, consisting of arithmetic Gevrey series of order < 0. Conditions (1) and (2) in 3.1 are not hard to verify using the recurrence relations satisfied by the coefficients Aikn . The algebraic relations between the functions Aij (z) are governed by the differential Galois group of the corresponding linear differential equation u0 = Lu. Granted we understand this group, we can apply the Siegel-Shidlovskii theorem to estimate the transcendence degree of the field generated by the complex numbers Aij (1). We have alread computed P (0) and seen that the field extension of k generated by the coefficients of P (0) is the field 1 d K := k(ζ, Γ( d+1 ), . . . , Γ( d+1 )).
We can use the well-known fact that the transcendence degree over Q of this field K is at most (conjecturally equal to) 1 + ϕ(d + 1)/2 to estimate the transcendence degree of the period field obtained by adjoining to k the coefficients of P (1) = A(1)P (0). Here, ϕ stands for the Euler-totient function.
Example 3.4. Consider once again the case f (x) = 13 x3 −zx. The differential Galois group of the Airy equation is SL2 , hence all algebraic relations between the functions Aij (z) are consequences of the determinant relation A11 A22 − A12 A21 = 1
SPECIAL VALUES OF E-FUNCTIONS AS EXPONENTIAL PERIODS
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which is ultimately a consequence of the fact that f is odd. The Siegel-Shidlovskii theorem implies thus trdegQ Q(A11 (1), A12 (1), A12 (1), A22 (1)) = 3 and shows more precisely that all Q-algebraic relations between the numbers Aij (1) are generated by A11 (1)A22 (1) − A12 (1)A21 (1) = 1. The field K is in this situation K = Q(ζ, Γ( 31 ), Γ( 23 )) = Q(ζ, Γ( 31 ), 2πi) and has transcendence degree 2. The transcendence degree of the period field obtained by adjoining to Q the coefficients of Pij (1) is thus at least ?
4 = trdegQ Q(P11 (1), P12 (1), P12 (1), P22 (1)) > 3 − 2 + 1 = 2 where the "+1" comes from the fact that the determinant of P (1) and 2πi are algebraically dependent. 1 xd+1 − zg(x) be any generic either odd or even polynomial Example 3.5. Let f (x) = d+1 of degree d + 1 > 6. If f is even, suppose f (0) = 0. The motivic fundamental group of M = H 1 (A, f ) is SLd , and the differential Galois group of the corresponding differential equation u0 = Lu is SLd as well. We can estimate the transcendence degree of the field generated by the periods Pij (1) by
ϕ(d + 1) −1+1 2 where the "+1" comes again from the fact that det P (1) and 2πi are algebraically dependent. Denote by GP the tannakian fundamental group of the period realisation of M |z=1 . The group GP is a subgroup of GLd and is not contained in SLd . It is of dimension at least > d2 − ϕ(d+1) . A little algebraic group theory (thank you Jean-Benoît) reveals that 2 for d > 5 only GP = GLd is possible (there is no proper subgroup of codimension < d − 1 in SLd , and ϕ(d + 1)/2 + 1 < d − 1). Thus, part (1) of the period conjecture holds for such √ motives. If f is not odd or even, then the determinant of P (1) takes the form c · eb · π for √ some rational number c and algebraic b, and it is not known whether c · eb · π is irrational or not (though its expected to be a transcendental number), hence its not clear that GP is not contained in SLd . ?
dim SLd = d2 − 1 = trdegQ Q(P11 (1), . . . , Pdd (1)) > d2 −
References [A] Y. André, Séries Gevrey de type arithmétique parts I and II, Annals of math. 151, p. 705–756, (2000) [B] F. Beukers, A refined version of the Siegel-Shidlovskii theorem, Annals of math. 163, p.369–379, (2006) [BE] S. Bloch and H. Esnault, Homology for irregular connections, J. Th. des Nombres de Bordeaux. 16, p. 65–78 (2004) [HR] M. Hien and C. Roucairol, Integral representations for solutions of exponential Gauss-Manin systems., Bull. Soc. Math. France. 136, no. 4 (2008) 505–532. [FJ] J. Fresán and P. Jossen, Exponential Motives, Book in preparation.
1. Exponential motives - setting the stage Let k ⊆ Q ⊆ C be a number field. Exponential motives over k with rational coefficients form a neutral tannakian, Q-linear category Mexp (k) which contains objects H n (X, Y, f )(i) for every variety X over k, closed subvariety Y ⊆ X and regular function f : X → A1 . This category of exponential motives is constructed along the same lines as Nori’s category of mixed motives over k. There is indeed a canonical, fully faithful functor MNori (k) → Mexp (k) sending a Nori-motive H n (X, Y )(i) to the exponential motive H n (X, Y, 0)(i). The details of the construction of Mexp (k) are not important for the understanding of the rest of this text, so I refer the reader to the book in preparation [FJ] for these details. As a notational convention, we agree to drop f from the notation is f = 0, to drop the twist i from the notation if i = 0, and to drop Y from the notation is Y is the empty subvariety of X. – 1.1. The category of exponential motives comes with a panoply of realisation functors and comparison functors. The Betti-realisation is a functor RB : Mexp (k) → VecQ which associates with a motive H n (X, Y, f ) the rational vector space dual to the finite dimensional space (1.1.1)
Hnrd (X, Y, f ) = lim Hnsing (X(C), Y (C) ∪ f −1 (Sr ))Q r→∞
where Sr stands for the half plane Re(z) > r. It is an easy consequence of Verdier’s constructibility theorem or of Ehresmann’s Lemma that this limit stabilises, meaning that for r 0 the transition maps in the limit, which are induced by inclusions Sr ⊆ Sr0 for r 6 r0 , are isomorphisms. The vector space given by (1.1.1) is called rapid decay homology of (X, Y, f ), and was in this form introduced by Bloch and Esnault in [BE]. The deRhamrealisation is a functor RdR : Mexp (k) → Veck Date: May 2018. 1
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PETER JOSSEN
whose description I will only give on motives of the form H n (X, f ) where X is a smooth variety. We consider the usual deRham complex Ω0X → Ω1X → Ω2X → · · · of sheaves of differential forms on X, but equip it with the differential df given on local sections by the formula df (ω) = dω − df ∧ ω. The deRham realisation of the motive H n (X, f ) is the vector space n n HdR (X, f ) = HZar (X, (Ω•X , df ))
which is indeed finite dimensional.
– 1.2. The Betti and the deRham realisation can be compared, as one is accustomed from the theory of classical motives. The comparison comes in the form of a pairing (1.2.1)
n (X, f ) → C Hnrd (X, f ) × HdR
which is defined as follows. An element of Hnrd (X, f ) is given by a coherent sequence of n (X, f ) is given, at least when topological n-cycles γ = (γr )r 0 , whereas an element of HdR X is affine, by a global algebraic differential n-form ω on X. The pairing (1.2.1) is given by Z (γ, ω) 7−→ lim
r→∞ γ r
e−f ω
and is indeed well defined, as one can show using a version of Stokes’s theorem for improper integrals. Much harder to show is the following theorem, analogous deRham’s theorem. Significant steps in its proof are due to Deligne, Malgrange, Esnault, Bloch, Sabbah, Hien and Roucairol, see [HR].
Theorem 1.3. The pairing (1.2.1) induces a natural isomorphism of complex vector spaces α
M RB (M ) ⊗Q C −−− → RdR (M ) ⊗k C
for every exponential motive M , which is moreover compatible with tensor products and duality in Mexp (k).
– 1.4. In order to formulate a period conjecture for exponential motives, let us introduce a category P(k) of period structures. Its objects are triples P = (V, W, α) consisting of a finite dimensional Q-vector space V , a finite dimensional k-vector space W , and an isomorphism α
V ⊗Q C −−→ W ⊗k C of complex vector spaces. We denote by ωB : P(k) → VecQ
and
ωdR : P(k) → Veck
SPECIAL VALUES OF E-FUNCTIONS AS EXPONENTIAL PERIODS
3
the obvious forgetful functors sending P = (V, W, α) to V , respectively to W . The category P(k) is tannakian Q-linear, and ωB is a neutral fibre functor. We can associate with any object P = (V, W, α) of P(k) the following things: (1) The linear algebraic group GP ⊆ GLV representing Aut⊗ (ωB |hP i⊗ ). This is the Tannakian fundamental group of the tannakian category hP i⊗ ⊆ P(k) generated by P . (2) The GP -torsor TP over k, representing the functor Isom⊗ (ωB,k |hP i⊗ , ωdR |hP i⊗ ). Here, ωB,k : P(k) → Veck is the functor sending an object P1 = (V1 , W1 , α1 ) to V1 ⊗Q k. We call TP the period torsor of P . (3) The k-algebra AP ⊆ C generated by det(α)−1 and the coefficients of the matrix of α with respect to any Q-basis of V and k-basis of W . We call AP the period algebra of P . We call its elements periods of P . The period torsor and the period algebra are related as follows: We can regard the given isomorphism α : V ⊗Q C → W ⊗k C as a complex point α ∈ TP (C). This gives rise to an evaluation map evα : OTP → C whose image is AP . Equivalently, Spec AP is a closed k-subscheme of TP , and indeed the smallest closed k-subscheme containing α as a complex point. The algebra of regular functions OTP of TP is called algebra of formal periods.
– 1.5. The combination of the Betti realisation, the deRham realisation and the comparison isomorphism leads to a functor Rper : Mexp (k) → P(k) which we call period realisation. This functor is an exact faithful ⊗-functor between tannakian categories, by definition compatible with fibre functors. Hence if M is a motive with period realisation P , there is an inclusion of algebraic groups G P ⊆ GM where GM stands for the tanakian fundamental group of M , that is, the fundamental group of the tannakian category hM i⊗ ⊆ Mexp (k) generated by M . We call AP the period algebra of M and elements of AP periods of M . Summarily, exponential periods are complex numbers which are periods of some exponential motive, whereas classical periods are complex numbers which are periods of some classical motive.
Conjecture 1.6. Let M be an exponential motive, and write P = (V, W, α) for its period realisation. Then (1) The inclusion GP ⊆ GM is an equality. In other words, the period realisation functor Rper : hM i⊗ → hP i⊗ is full and essentially surjective.
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PETER JOSSEN
(2) The point α ∈ TP (C) is dense, so Spec AP = TP . Equivalently, the evaluation map OTP → C is injective.
– 1.7. Little is known about this conjecture, even for classical motives. In fact, most positive examples in which we can show the conjecture to hold involve genuine exponential motives. For instance, the classical Lindemann-Weierstrass theorem can be seen as an illustration of the conjecture. The period conjecture is often formulated as the equality of numbers dim GM = trdegk AP which indeed follows from 1.6, but is slightly less precise. Part (1) of the conjecture has been called formal period conjecture and can sometimes be deduced from quite limited information about GP and GM by invoking results from the theory of linear algebraic groups. At the heart of part (2) of the conjecture are transcendence statements which are infamously difficult to prove.
– 1.8. It might come as a surprise that for any given exponential motive M the motivic fundamental group GM of M is, generally speaking, not very hard to compute. Write V for the Betti-realisation of M , so GM is a linear algebraic subgroup of GLV . Realisations of M in interesting categories (Hodge structures, `-adic realisations, etc.) give rise to interesting and presumably large subgroups of GM , whereas endomorphisms of M or of tensor constructions of M cut out constraints for GM . If all goes well, there is but one possibility left for GM . One particularly interesting and important realisation functor for exponential motives is the perverse realisation Rperv : Mexp (k) → Perv0 where Perv0 is the following tannakian category. Its objects are perverse sheaves of Qvector spaces F on A1k , with the property RΓF = 0. It’s not difficult to show that such perverse sheaves are precisely those of the form F [1] for some constructible sheaf F satisfying H 0 (A1 , F ) = 0 and H 1 (A1 , F ) = 0. Morphisms in Perv0 are morphisms of perverse sheaves, and the tensor product is given by additive convolution F ∗ G = Rsum∗ (pr∗1 F ⊗ pr∗2 G) with the obvious maps pr1 , pr2 , sum : A2k → A1k . A fibre functor is the nearby-fibre-atinfinity functor which we can define as Ψ∞ (F [1]) = colimF (Sr ) r→∞
for Sr = {z ∈ C | Re(z) > r}. It is a nontrivial matter to show that Ψ∞ is a fibre functor, especially its compatibility with associativity and commutativity constraints is tricky. The
SPECIAL VALUES OF E-FUNCTIONS AS EXPONENTIAL PERIODS
5
perverse realisation of a motive M of the form M = H n (X, Y, f ) is given by Rperv (M ) = Hn ((X × A1 , Y × A1 ∪ Γf )/A1 )[1] where Γf ⊆ X × A1 stands for the graph of f , and Hn ((−)/A1 ) is relative cohomology over A1 which could as well be expressed as a higher direct image. It is clear from the definitions that there is a canonical and natural isomorphism Ψ∞ (Rperv (M )) ∼ = RB (M ) of vector spaces, compatible with tensor products and duality. Therefore, if F [1] is the perverse realisation of an exponential motive M , then there is a canonical inclusion GF [1] ⊆ GM where GF [1] and GM are the tannakian fundamental groups of F [1] and M respectively.
Theorem 1.9 (Gabber, Katz). Let f ∈ k[x] be an irreducible polynomial of degree d+1 > 0, viewed as a regular function on A1 = Spec k[x]. Set M = H 1 (A1 , f ) and write F for the perverse realisation of M and set V = Ψ∞ F , so V ' Qd . Suppose that f 0 has no multiple roots, and that for any four not necessarily distinct roots α1 , α2 , α3 , α4 of f 0 the implication f (α1 ) + f (α1 ) = f (α3 ) + f (α4 )
=⇒
{α1 , α2 } = {α3 , α4 }
holds. Then, the tannakian fundamental group GF ⊆ GLV of F contains SLV . Moreover, equality GF = GLV if and only if X b := f (α) α
is nonzero, where the sum ranges over all zeroes α of f 0 . The constructible sheaf F is indeed F = (f∗ Q)/Q. Its singularities are at the branch points of f , that is, at the zeroes of the discriminant ∆(z) of f (x) − z. The quantity b is the only singularity of the one dimensional object det F . Example 1.10. Fix an element z ∈ k, and consider the motive M = H 1 (A1 , f ) where f is the polynomial f (x) = 31 x3 − zx seen as a regular function on A1 = Spec k[x]. The rapid decay homology H1rd (X, f ) is the homology of the pair (C, f −1 (Sr )) for large r. It is a Q-vector space of dimension 2 with a basis γ1 , γ2 given by cycles as illustrated in Figure 1. Notice that in this particular example the basis γ1 , γ2 is independent of z, which à priori it need not be. The story for f (x) = zx3 − x would be quite different!
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Figure 1. A basis of H1rd (X, f ) 1 (A1 , f ) is given by the global differential forms dx A basis of the deRham cohomology HdR and xdx. The period pairing (1.2.1) as well as the corresponding comparison isomorphism of Theorem 1.3 is given in terms of these bases by the matrix P (z), with coefficients Z 1 3 (1.10.1) Pij (z) = e− 3 x +zx xi−1 dx γj
for 1 6 i, j 6 2. The integrals (1.10.1) are well known: They define the value at z ∈ k of the complex Airy functions. We have ! ! P11 (z) P12 (z) Ai1 (z) Ai2 (z) P (z) = = P21 (z) P22 (z) Bi1 (z) Bi2 (z) where Ai1 and Ai2 are linearly independent solutions of the Airy equation u00 (z) − zu(z) = 0 and Bij (z) = Ai0j (z). The discrinimant of f (x) − t as a polynomial in x equals ∆(z, t) = 13 (4z 3 − 9t2 ) √ so its zeroes as a function of t are at ± 32 z. Hence the perverse realisation of the motive M is a perverse sheaf F [1], where the constructible sheaf F is given by a local system of rank √ 2 on A1 \ {± 23 z}. Unless z = 0, the tannakian fundamental group of the object F [1] of Perv0 is SL2 . As for the motive M itself, one can show that its determinant equals Q(1), hence GM = GL2 except when z = 0.
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2. Some families of exponential motives on the affine line Let us consider more generally families exponential motives, depending on a formal parameter z, of the form M = H 1 (A, f ) with f (x) =
d+1 1 d+1 x
− zg(x)
for some integer d > 1 and polynomial g(x) of degree 6 d. Set ζ = e2πi/(d+1) . As in example 1.10, a basis γ1 , . . . , γd of the rapid decay homology is given by paths (γj,r )r 0 in the complex plane, where γj,r runs from ζ j r to r. A basis of the deRham cohomology is given by the global forms dx, xdx, . . . , xd−1 dx. Just as in example 1.10, the period pairing (1.2.1) as well as the corresponding comparison isomorphism of Theorem 1.3 is given in terms of these bases by the d-by-d matrix P (z), with coefficients Z 1 d+1 +zg(x) i−1 x − x dx e d+1 Pij (z) = γj
for 1 6 i, j 6 d. – 2.1. In order to examine the periods Pij of M , which we view now as entire functions in the complex variable z, we introduce Z 1 d+1 − x +zg(x) Ej (z, h) = e d+1 h(x)dx γj
for h(x) ∈ k[x]. We have Pij (z) = Ej (z, xi−1 ). Besides the obvious k-linearity in h, the following relations hold: (2.1.1)
Ej (z, h0 ) = Ej (z, xd h) − zEj (z, g 0 h)
(2.1.2)
∂ Ej (z, h) = Ej (z, gh) ∂z
Using these, it is elementary to show that P (z) is a fundamental matrix of solutions of a matricial differential equation u0 = Lu for some d-by-d matrix L with coefficients in k[z ±1 ]. The fact that the Pij (z) are entire functions shows that the only singularity of this equation is at infinity, and by using Fuchs’s criterion (i.e. because some of the Pij (z) grow exponentially fast) one can see that infinity is an irregular singularity lest g = 0.
– 2.2. We can expand the entire functions Pij (z) as Taylor series around 0. To compute the Taylor coefficients, the relations (2.1.1) and (2.1.1) help. We use them in the form (2.2.1)
Ej (0, h0 ) = Ej (0, xd h)
(2.2.2)
∂n Ej (z, h) = Ej (z, g n h) ∂z n
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PETER JOSSEN
and notice that because of relation (2.2.1), Ej (0, g n h) can be written as a k-linear combination of Ej (0, 1), Ej (0, x), . . . , Ej (0, xd−1 ), with coefficients Aikn which are independent of j. The Taylor expansion of Pij (z) = Ej (z, xi−1 ) reads Pij (z) =
∞ X 1 Ej (0, g n xi−1 )z n n!
n=0
d ∞ X 1 X Aikn Ej (0, xk−1 )z n = n! n=0
=
k=1
d X ∞ X k=1 n=0
=
d X
1 Aikn z n Pkj (0) n!
Aik (z)Pkj (0)
k=1
hence, in summary, we find the relation P (z) = A(z)P (0) where P (0) is a constant d-by-d matrix whose coefficients are independent of f . A direct computation reveals Z 1 d+1 ζ ij − 1 x i Pij (0) = e d+1 xi−1 dx = Γ d+1 d/(d+1) (d + 1) γj where ζ = e2πi/(d+1) . In particular, all coefficients Pij (0) of P (0) belong to the field √ 1 2 d Q(ζ, Γ( d+1 ), Γ( d+1 ), . . . , Γ( d+1 )). The determinant of P equals cπ d for some nonzero rational c which we could, but do not care to determine. The power series Aij (z) =
∞ X 1 Aikn z n n!
n=0
define entire functions, have coefficients in k, and arranged in a matrix satisfy the same differential equation u0 = Lu as do the Pij .
Example 2.3. If we carry out the computations in 2.2 for the example of f (x) = 31 x3 − zx, we find z3 z6 z9 z 12 z 15 + 4 + 4 · 7 + 4 · 7 · 10 + 4 · 7 · 10 · 13 + ··· 3! 6! 9! 12! 15! z4 z7 z 10 z 13 z 16 A12 (z) = z + 2 + 2 · 5 + 2 · 5 · 8 + 2 · 5 · 8 · 11 + 2 · 5 · 8 · 11 · 14 + ··· 4! 7! 10! 13! 16!
A11 (z) = 1 +
as well as A21 (z) = A011 (z) and A22 (z) = A012 (z). It follows from Abel’s theorem that the Wronskian A11 (z) A12 (z) W (z) = det A21 (z) A22 (z)
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is constant, equal to 1. The equality P (z) = A(z)P (0) amounts to the classical power series expansion of the Airy functions.
– 2.4. In general, the determinant of A(z) is not constant. It is given by X f (α) det(A(z)) = eb(z) b(z) = f 0 (α)=0
where the sum runs over all zeroes α of f 0 (x) = xd − zg 0 (x), while regarding z as a fixed complex parameter. As a function of z, the quantity b(z) is a polynomial with coefficients in k, satisfying b(0) = 0.
3. Consequences of the Siegel-Shidlovskii theorem We are interested in the transcendence of the coefficients of the matrix P (1), since these coefficients together with det(P (1))−1 , generate the period algebra of the motive H 1 (A1 , f ), 1 xd+1 − g(x). with f (x) = d+1 – 3.1. Let s > 0 be a positive rational number. An arithmetic Gevrey series of order −s is an entire function given by a power series E(z) =
∞ X
an z n
n=0
where the coefficients an are algebraic numbers, satisfying the following estimates for some real constant C > 0. (1) For an algebraic number a, denote by kak the maximum norm of the complex conjugates of a. Then, k(n!)s an k < C n holds for all n. (2) For every n > 1, there exists an integer N 6 C n such that N a0 , N a1 , . . . , N an are all algebraic integers. Condition (1) guarantees the convergence of the power series on the whole complex plane, so any formal series satisfying (1) and (2) is an arithmetic Gevrey series. We are interested in holonomic Gevrey series, that is, Gevrey series which are annihilated by some nontrivial differential operator D ∈ Q(z)[∂]. Typical examples of holonomic arithmetic Gevrey series include polynomials, the exponential function, the Bessel functions of the second kind Jm for integral m, or the function ∞ X zn E(z) = n · n! n=1
which is a variant of the exponential integral function. To verify condition (2) for this series requires the use of the prime number theorem.
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Theorem 3.2 (Siegel-Shidlovskii, André, Beukers). Let d > 0 be an integer, and let L be a d-by-d matrix with coefficients in Q(z). Let T (z) ∈ Q[z] be a common denominator of the coefficients of L. Let E be a d-by-m matrix satisfying E 0 = LE and whose coefficients Eij (z) are arithmetic Gevrey series. Let z0 be an algebraic number satisfying z0 T (z0 ) 6= 0. Then, every Q-polynomial relation between the complex numbers Eij (z0 ) is obtained by specialisation at z = z0 from a Q(z)-polynomial relation between the functions Eij (z). In particular, the equality trdegQ(z) Q(z, E11 (z), . . . , Edm (z)) = trdegQ Q(E11 (z0 ), . . . , Edm (z0 )) holds. References are [A] (there is a summary by the same author in J. Th. des nombres de Bordeaux, 2003) and [B]. – 3.3. We come back to families exponential motives of the form M = H 1 (A, f ) with 1 f (x) = d+1 xd+1 − zg(x). We have seen that the periods of M , with respect to appropriate bases, can be written as P (z) = A(z)P (0) where A(z) is a fundamental matrix solution of a matricial differential equation, consisting of arithmetic Gevrey series of order < 0. Conditions (1) and (2) in 3.1 are not hard to verify using the recurrence relations satisfied by the coefficients Aikn . The algebraic relations between the functions Aij (z) are governed by the differential Galois group of the corresponding linear differential equation u0 = Lu. Granted we understand this group, we can apply the Siegel-Shidlovskii theorem to estimate the transcendence degree of the field generated by the complex numbers Aij (1). We have alread computed P (0) and seen that the field extension of k generated by the coefficients of P (0) is the field 1 d K := k(ζ, Γ( d+1 ), . . . , Γ( d+1 )).
We can use the well-known fact that the transcendence degree over Q of this field K is at most (conjecturally equal to) 1 + ϕ(d + 1)/2 to estimate the transcendence degree of the period field obtained by adjoining to k the coefficients of P (1) = A(1)P (0). Here, ϕ stands for the Euler-totient function.
Example 3.4. Consider once again the case f (x) = 13 x3 −zx. The differential Galois group of the Airy equation is SL2 , hence all algebraic relations between the functions Aij (z) are consequences of the determinant relation A11 A22 − A12 A21 = 1
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which is ultimately a consequence of the fact that f is odd. The Siegel-Shidlovskii theorem implies thus trdegQ Q(A11 (1), A12 (1), A12 (1), A22 (1)) = 3 and shows more precisely that all Q-algebraic relations between the numbers Aij (1) are generated by A11 (1)A22 (1) − A12 (1)A21 (1) = 1. The field K is in this situation K = Q(ζ, Γ( 31 ), Γ( 23 )) = Q(ζ, Γ( 31 ), 2πi) and has transcendence degree 2. The transcendence degree of the period field obtained by adjoining to Q the coefficients of Pij (1) is thus at least ?
4 = trdegQ Q(P11 (1), P12 (1), P12 (1), P22 (1)) > 3 − 2 + 1 = 2 where the "+1" comes from the fact that the determinant of P (1) and 2πi are algebraically dependent. 1 xd+1 − zg(x) be any generic either odd or even polynomial Example 3.5. Let f (x) = d+1 of degree d + 1 > 6. If f is even, suppose f (0) = 0. The motivic fundamental group of M = H 1 (A, f ) is SLd , and the differential Galois group of the corresponding differential equation u0 = Lu is SLd as well. We can estimate the transcendence degree of the field generated by the periods Pij (1) by
ϕ(d + 1) −1+1 2 where the "+1" comes again from the fact that det P (1) and 2πi are algebraically dependent. Denote by GP the tannakian fundamental group of the period realisation of M |z=1 . The group GP is a subgroup of GLd and is not contained in SLd . It is of dimension at least > d2 − ϕ(d+1) . A little algebraic group theory (thank you Jean-Benoît) reveals that 2 for d > 5 only GP = GLd is possible (there is no proper subgroup of codimension < d − 1 in SLd , and ϕ(d + 1)/2 + 1 < d − 1). Thus, part (1) of the period conjecture holds for such √ motives. If f is not odd or even, then the determinant of P (1) takes the form c · eb · π for √ some rational number c and algebraic b, and it is not known whether c · eb · π is irrational or not (though its expected to be a transcendental number), hence its not clear that GP is not contained in SLd . ?
dim SLd = d2 − 1 = trdegQ Q(P11 (1), . . . , Pdd (1)) > d2 −
References [A] Y. André, Séries Gevrey de type arithmétique parts I and II, Annals of math. 151, p. 705–756, (2000) [B] F. Beukers, A refined version of the Siegel-Shidlovskii theorem, Annals of math. 163, p.369–379, (2006) [BE] S. Bloch and H. Esnault, Homology for irregular connections, J. Th. des Nombres de Bordeaux. 16, p. 65–78 (2004) [HR] M. Hien and C. Roucairol, Integral representations for solutions of exponential Gauss-Manin systems., Bull. Soc. Math. France. 136, no. 4 (2008) 505–532. [FJ] J. Fresán and P. Jossen, Exponential Motives, Book in preparation.
