Arithmetic properties and decomposability of Jacobians

Arithmetic Properties and Decomposability of

Jacobians

by

Soohyun Park

Submitted to the Department of Mathematics

in partial fulfillment of the requirements for the degree of

Master of Science in Mathematics

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

February 2018

@

Massachusetts Institute of Technology 2018. All rights reserved.

Author ...

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Department of Mathematics

January 18, 2018

CetiieSbi~gnature

redacted

Certified by.

...

Bjorn Poonen

Professor

A

Thesis Supervisor

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Accepted by....

...

Davesh Maulik

Graduate Co-chair

MA ACHS STITUT OF TECHNOLOGY to

w

MAR 122018

Arithmetic Properties and Decomposability of Jacobians

by

Soohyun Park

Submitted to the Department of Mathematics on January 18, 2018, in partial fulfillment of the

requirements for the degree of Master of Science in Mathematics

Abstract

We first give an overview of methods used to study the decomposability of Jacobians of curves over the complex numbers. This involves studying the action of a finite group on an abelian variety in general. Next, we use methods for point counting properties of curves over finite fields to study the decomposability of Jacobians over number fields and finite fields. For example, we show that the genus of curves over number fields whose Jacobians are isomorphic to a product of elliptic curves satisfying certain reduction conditions is bounded and give restrictions on curves over number fields whose Jacobians are isomorphic to a product of elliptic curves.

Thesis Supervisor: Bjorn Poonen Title: Professor

Contents

1 Introduction 7

2 Explicit decompositions of Jacobians up to isogeny 11

3 Applications to decomposability of Jacobians of curves over number

Chapter 1

Introduction

We will spend most of our time on various methods of studying decomposability of Jacobians over C or in number-theoretic settings such as number fields and finite fields. For example, here is one question we will study:

Question 1.1. Is there an upper bound on the genus of curves over a fixed number field which have Jacobians isomorphic to a product of elliptic curves as an unpolarized

abelian variety?

This is motivated by some questions raised by Ekedahl and Serre [10] for curves over C. More specifically, they asked the following questions on the decomposability of Jacobians of smooth projective curves over C into products of elliptic curves up to isogeny or isomorphism:

1. Is it true that for every g > 0 there exists a curve of genus g over C. whose

Jacobian is isogenous to a product of elliptic curves? 2. If not, is the set of such g bounded?

3. Is there a curve of genus g > 3 such that its Jacobian is isomorphic to a product

of elliptic curves as an unpolarized abelian variety?

There has been some previous work on the first and second problem (e.g. [291, [33], [41) which has been successful in producing a large number of explicit decompositions

coming from representations of the endomorphism algebra of the Jacobian of a curve induced by the action of a finite group. Most recently, they gave many new examples of completely decomposable Jacobians which extend Ekedahl and Serre's original list

[301.

Another perspective on the problem has to do with unlikely intersections in-volving (weakly) special subvarieties of Shimura varieties (e.g. [28]). One example of the tools used is analogues of positivity results from complex algebraic geometry in Arakelov theory such as slope inequalities. Motivated by the second approach, Chen, Lu, and Zuo

[81

prove a finiteness result for Jacobians of curves over number fields in the self-product case assuming the Sato-Tate conjecture. This is interesting since some of the authors working on curves over C seem to guess that the genus is unbounded in the second question posed by Ekedahl and Serre (see remark below theorem on p. 2 of [301). In our case, we will study bounds on the genus of decom-posable Jacobians over finite fields and give point counting properties of reductions of curves which have completely decomposable Jacobians.

The methods which are commonly used to find explicit decompositions of Jaco-bians up to isogeny will be described in further detail in Chapter 2. Although we will not focus on the exact methods used in recent examples of decomposable Jacobians, what we do discuss will give a general idea of what is used in these decompositions. Finally, Chapter 3 will apply point counting methods to decomposability questions over number fields and finite fields.

Chapter 2

Explicit decompositions of Jacobians

up to isogeny

As mentioned above, most of the previous work on the second question of Ekedahl and Serre was on explicit decompositions of Jacobians into elliptic curves or Jacobians of smaller dimension. Some of this has to do with representations of the endomorphism algebra of a Jacobian induced by the action of a finite group on a curve. For example, this has been used by Paulhus [291 and Rojas [331 to produce many examples of com-pletely decomposable Jacobians which extend Ekedahl and Serre's original list in [101.

Although we have been discussing decomposability of Jacobians specifically, many of the methods used by Paulhus and Rojas are connected to decompositions up to isogeny of arbitrary abelian varieties with a finite group acting on them studied earlier

by Lange and Recillas in [24]. For this reason, we will focus on giving an outline of

the proof of some of the main results in this work instead of listing specific decom-posability results, in order to give a general idea of what kinds of methods are used.

Before mentioning the specific type of decomposition we will consider, we mention a standard result on decomposing abelian varieties up to isogeny.

abelian variety over K. If B -- A is an abelian subvariety, then there exists an abelian subvariety B' -> A such that the map B x B' --* A induced by addition is

an isogeny.

As mentioned in

[51,

using this theorem as an analogue of Maschke's theorem for representations of finite groups can give us some idea of what the "smallest" abelian varieties we can get from this are. This motivates the definition of a simple abelian variety over K, which is a nonzero abelian variety without any nontrivial proper

abelian subvarieties over K.

Corollary 2.2. [5] The abelian variety A is K-isogenous to a finite product

fJ

Bi for ej > 1, where the Bi are K-simple abelian varieties, pairwise non-isogenous over

K. Moreover, the isogeny class of each Bi and the multiplicities ej are uniquely determined up to permuting the factors.

The type of decomposition that we will consider is the isotypic decomposition, which is the decomposition of an abelian variety with a group action up to isogeny into factors which are invariant under the group action without any homomorphisms between distinct factors of the Q-algebra. However, the factors may no longer be sim-ple unlike those in the decomposition we get from the Poincar6 reducibility theorem. Let A be an abelian variety with a finite group G acting on it. In [24], Lange and

Recillas work with the Q-algebra homomorphism p : Q[G] -- + Endo A End A 0 Q

induced by this action.

The main tool which is used for the isotypic decomposition is the fact that Q[G]

is a finite-dimensional semisimple Q-algebra. Suppose that W is an irreducible

Q-representation of G. Let D = EndG W and n = dimD W. Then, the action of G on W

induces a surjective Q-algebra homomorphism Q[G -+ Cw := MI(D). Taking the

product over all W yields an isomorphism Q[G] -+

HW

Cw. Let ew be the identity

element of Cw. Then, 1 = Ew ew in Q[G]. These elements ew are the centrally primitive idempotents in Q[G].

We first use this property to show the existence of such a decomposition for an abelian variety over a field k.

Theorem 2.3. (Lange and Recillas [24]) Let A be an abelian variety over a field k and G be a finite group acting on A. For each irreducible Q-representation W of G, choose m E Z>1 such that mew E Z[G]. Then, mew acts on A. Let Aw = (mew)A.

0. The definition of Aw is independent of the choice of m. 1. Aw is a G-stable abelian variety.

2. The addition map induces an isogeny

y : flAw -+ A. w

3. Let W and W' be irreducible Q-representations of G. If W 7 W', then HomG (Aw, Aw')

0.

The product

H~w

Aw in Theorem 2.3 is the isotypic decomposition of A.

Remark 2.4. Suppose that k = C. According to Carocca and Rodriguez (p. 2 of

[4]), methods of obtaining isotypic decompositions of abelian varieties with the ac-tion of a finite group G before work of Lange and Recillas [241 assumed that the irreducible rational representations of G were absolutely irreducible. In [41, Carocca and Rodriguez present examples of Jacobians where the irreducible Q-representations of G corresponding to simple components of Q[G giving rise to nontrivial isotypic

decompositions are not irreducible over C. In Theorem A.1 of [41, Carocca and

Rodriguez give examples of decompositions where isotypic decompositions give com-ponents which cannot be studied by only using representations irreducible over C. In this theorem, they use the method in [241 for obtaining an isotypic decomposition of a Jacobian with a group of order 80 acting on it.

Proof of Theorem 2.3. 1. Since the ew are in the center of A and G acts on A, we

gAw = g(mew)A

= (mew)gA

C (mew)A Aw

for every g E G.

2. Since there are finitely many irreducible Q-representations W, there is a single

m E Z>1 such that mew E ZG for every W. Note that 1 = Ew ew. This

means that m = EW mew and composing the homomorphism A ----+

Jw

Aw

given by multiplication by mew in the W-position with the addition map

I :

fw

Aw -- A gives the multiplication by m map A -+ _{A. Since the} multiplication map is surjective, p is surjective.

If (aw)w c

Hj

Aw is in ker jt, then Ew aw = 0. Let V be an irreducible

Q-representation of G. Since mevew = 0 for V 9 W, applying mev yields

mav = 0. Thus, ker p C (JIw Aw) [m], which is finite and /y is an isogeny.

3. Choose m E Z;>1 as in part 2. Since ew is an idempotent and the multiplication by m map is surjective, we have

(mew)Aw = (mew)2 A = m(mew)A -mAw -Aw.

Suppose that f E HornG(Aw, Aw'). Take x E Aw. Since (mew)Aw = Aw

from our work above, x = (mew)y for some y E Aw. Similarly, we have that

f

(x) = (mew,)z for some z E Aw'. Combining these together, we get

f(x) = f((mew)y) = (mew)f(y)

= (mew)(mew,)z

= 0. l

Recall that factors in an isotypic decomposition are not necessarily simple. After obtaining an isotypic decomposition as in Theorem 2.3, we can sometimes decompose the isotypic components even further using the structure of Q[G].

We can also further decompose the central orthogonal idempotents ew to obtain decompositions of the isotypic components.

Proposition 2.5. (Lange and Recillas [24]) Let A be an abelian variety over a field K as in Theorem 2.3. Let n := dimD W with D := EndG W. Then there is an abelian subvariety B of Aw such that Aw is isogenous to B'.

Proof. By Schur's lemma, we have that D = EndG W is a finite-dimensional division

ring over Q. This implies that

Q

~ EndD W since W is then a finite-dimensional

left-vector space over D. Let n = dim W. The n x n matrices Ei in EndD W with 1 at the ith row and ith column and 0 elsewhere correspond to primitive orthogonal idempotents qi in

Q.

This gives rise to a decompositiion ew = q

+...

+ qn.

The matrix idempotents Ei corresponding to the qi are conjugate to each other

since Ei = PZjEjP-j1, where Pij is the permutation matrix for the transposition

(i,

j).

Moving this back to the corresponding elements of

Q,

we find that the qi are all conjugate to each other. Finally, this implies that the Bi := piA are pairwise

isogenous to each other. As in Theorem 2.3, piA is the image of mpi for some m. E Z>1 such that mpi E Z[G].

Combining this with Theorem 2.3, we have the following:

Theorem 2.6. (Lange and Recillas [24]) Let G be a finite group acting on an abelian

variety A over a field K. For each irreducible Q-representation of G in distinct isotypic components, let Dw = EndG W and nw = dimDW W. Then, there are abelian subvarieties Bw of A such that we have an isogeny

A ~ jJBn.

w

This is called the isogeny decomposition of A with respect to G.

Let TOA be the tangent space to A at the identity. Given the representation

of G on TOA induced by the action of G on A and the analytic representation (p.

138 of [241, p. 10 of [21) Endo A -+ End ToA of End0 A, we can use images of the

central orthogonal idempotents under the composition of the analytic representation and p : Q[G] -- + End0 A in order to decompose TOA into spaces which are the images of the idempotents in the representation of G on End TOA. Actually computing these subspaces involves expressing the idempotents in terms of representations of G. This will be explained in more detail in the proof of the following theorem:

Proposition 2.7. (Lange and Recillas /24]) Each isotypic projector of Q[G] is the

sum of an isotypic projector in C[G] and its Gal(Q/Q)-conjugates.

Proof. The isotypic projectors in C[G] are given by centrally primitive idempotents.

Let S be the set of centrally primitive idempotents in C[G]. There is a bijection between the subsets of S and the set of central idempotents in C[G]. This is given

by sending a subset to its sum. Since this bijection is compatible with the action

of Gal(Q/Q), the Gal(Q/Q)-stable subsets of S correspond to the Gal(Q/Q)-stable central idempotents. These are exactly the central idempotents in Q[G]. Thus, the Gal(Q/Q)-orbits in S correspond to the centrally primitive idempotents in Q[G].

Before giving a specific application of isotypic decompositions, we show that giving tangent spaces

Proposition 2.8. Suppose that k = C. Then, To(Aw) = ew(ToA).

Proof. For any

j

E Z[G], we have that To(jA) = j(ToA). Setting

j

= mew, we get

To(Aw) = ew(TA). D

Next, we give an application of isotypic decompositions of abelian varieties to Jacobians. More specifically, we give a sufficient condition for all the isotypic compo-nents of a Jacobian to be nonzero.

Theorem 2.9. (Lange and Recillas [24]) Let 7r : X -+ Y = X/G be a Galois cov-ering of smooth projective curves over C. If gy 2, all irreducible Q-representations

of G correspond to components which appear in the isotypic decomposition of J(X). Remark 2.10. Note that 7 is affine since it is finite and Ox is quasicoherent since it is coherent. By Exercise 3.4.1 of

114],

H'(X, Ox) = H'(Y,7rOx). The main

idea is to relate H'(Y, 7rOx) to components of the isotypic decomposition of the Jacobian J(X). This is done by considering 7r.Ox as a Q[G]-module which breaks

into components from the decomposition of

Q[G

into simple algebras. Then, we use

a version of Riemann-Roch for vector bundles to show that the dimension of these components is positive when gy > 2. To control the degree term of this formula, we use a generalization of Serre duality. Finally, we show that we can decompose J(X) in parallel to H1(X, 7, Ox).

We first relate TrOx to the regular representation of G.

Lemma 2.11. (Lange and Recillas [24]) Let X be a smooth projective curve with a faithful action of a finite group G. Denote by Y = X/G the quotient curve and by

7r : X -+ Y the covering map. For each irreducible Q-representation W of G, let

Fw = ew7rOx. Then, Tw is a locally free Oy-module of positive rank, and

7r,Ox = Fw.

Proof. Since 7r is a finite flat morphism of degree n, 7rOx is locally free of rank

n =

jGI.

The action of G on X induces one on r,Ox since r.,Ox(W) = Ox(7rM(U))

for open U C Y and 7r-1 (W) c X. Then, we can consider 7r.Ox as a (Q[G 9, Oy)

-module. From the decomposition of Q[G] into simple Q-algebras Qw (see Theorem

2.3), we get a decomposition of ir.Ox into a direct sum of (Qw 0Q Oy)-algebras Fw.

Note that a subsheaf F of a locally free sheaf

g

over a smooth projective curve is locally free. We have that Oy,, is a DVR since Y is a smooth projective curve. Our claim follows from the fact that a submodule of a free module over a PID is free since a sheaf W on Y is locally free if and only if l, is a free Oy,-module for every y E Y.

Applying this to .Fw C 7r.Ox, we find that the Fw are locally free.

To show that one of the Fw is equal to Oy, we take a look at the action of G on

rOx again. Let r7 E X be the generic point. On (7rOx),, G = Gal(C(X)/C(Y)) acts

as the regular representation. Note that the Fw are the images of the action of the central orthogonal idempotents on 7rOx. Since the images of these idempotents give the projections of representations onto the isotypic components, the Fw correspond to isotypic components of the regular representation. From the trivial representation, we find that one of the Fw is equal to Oy. The (Fw),7 are of positive dimension

over Ox,n since their dimensions are equal to those of the isotypic components of the regular representation of G. Thus, the Bw are of positive rank.

Now we can start the proof of Theorem 2.9.

Proof of Theorem 2.9. Since H1(X, Ox) 2 H'(Y, 7rOx) and cohomology commutes

with direct sums, the decomposition from Lemma 2.11 gives

H'(X, Ox) H1 (Y, Tr, Ox)

where the W are the nontrivial irreducible Q-representations of G. What we would like to do is show that dim H1(Y, Fw) > 0. Since the dimension of the first term is gy and gy > 2, it must have positive dimension. To look at the remaining terms, we use a version of Riemann-Roch for vector bundles (see p. 231 of [15] as an application of Hirzebruch-Riemann-Roch) to get

h'(Y, Fw) = h (Y, Fw) - deg Fw + (gy - 1)rank Fw.

Here H0(X, Ox) 2 H0(Y, 7rOx) by definition since ir is surjective. We also have

that ho(X, Ox) = ho(Y, Oy) = 1 since global functions on curves are constants.

Substituting this into

H0

(X, Ox) - H (Y, 7r,Ox) = H (Y, Oy)

e

®

H (Y, .Fw), w

we find that ho(Y, .Fw) = 0. This means that

h'(Y, Fw) = - deg.Fw + (gy - 1) rank Fw. (2.1)

We now show that h1(Y, F,) > 0. By Grothendieck duality (see Definition 4.18 on p. 243 of [261), we have that (7rOx)v - -rwxly, where wx/y is the relative dualizing sheaf. Since 7r is finite, we have that 7rwx/y is nef by Corollary 3.2 of [39]. Dualizing the decomposition from Lemma 2.11, we have

(7rOx)v Oy E ®Fv

w

Since every quotient bundle of a nef vector bundle has a nonnegative degree (see Theorem 6.4.15 of [251), this implies that - deg.Fw = deg F, > 0. Substituting this

back into equation 2.1, we find that h1(Y, Fw) > 0 if gy > 2 since rankFw > 0.

So far, we have shown that the decomposition H1(Y, ir.Ox) has positive-dimensional

iso-typic decomposition of J(X), it suffices to show that the isoiso-typic decomposition of

J(X) is parallel to the decomposition of H'(Y, 7rOx). Let ToJ(X) be the tangent

space to the Jacobian J(X) at the identity and W be an isotypic component of the regular representation of G. Since

ToJ(X)v _ H0

(J(X), lj(X)) = H0(X, Wx) = Hl(X, Ox)v

by Serre duality, TOJ(X) ' H'(X, Ox) = H1(Y, 7rOx). Applying the central

or-thogonal idempotent ew corresponding to W on both sides gives To(J(X)w) on the left by Proposition 2.8 and H'(Y, Fw) on the right. Since the previous paragraph showed that H'(Y, Fw) is nonzero, we also have that J(X)w is nonzero. Thus, all isotypic components of the regular representation must appear in the isotypic

decom-position. E

Using properties of Galois covers and Prym varieties along with tools used earlier in their paper [241, Lange and Recillas obtain more specific results on factors such as Jacobians of smaller dimension which appear in decompositions of Jacobians up to isogeny. Finally, they give examples of decompositions of Jacobians with group action by a given group, such as the symmetric group.

Chapter 3

Applications to decomposability of

Jacobians of curves over number

fields and finite fields

Assuming the Sato-Tate conjecture and building on the work of Kukulies [22], Chen, Lu, and Zuo [81 prove that the genus of smooth projective curves over number fields of bounded degree whose Jacobians are isogenous to a self-product a single elliptic curve is bounded (see Theorem 1.2 of [81). Using the Grothendieck-Lefschetz trace formula, we can turn this decomposability question into a point counting problem. This can be used to find bounds on the genus of decomposable Jacobians under some reduction conditions and give point counting properties of curves with decomposable Jacobians. We first recall the trace formula.

Theorem 3.1. (Grothendieck-Lefschetz trace formula) (see p. 85 of [38]) Let Fq be

a finite field. Writing q = pr for a prime p, let o be the r th power of the absolute Frobenius map for X and pi for the induced map on H Qi) for a prime I ,L p. Let X be a smooth proper variety of X over Fq of dimension n. Then, we have

2n

#X(Fq) = E(-1)iTr W.

Remark 3.2. This is an 6tale cohomology analogue of the Lefschetz fixed point

theo-rem.

We can now state the results of this chapter.

Theorem 3.3.

1. (a) Given q = pin, there is a bound G(q) such that for all elliptic curves E over

Fq and smooth projective curves C of genus g over Fq such that J(C) is isogenous to a power of E, we have that g G(q).

(b) Given positive integers h, d, and m, there exists a constant G = G(h, d, m) such that if K is a number field of degree at most m and E is an elliptic

curve over K of Faltings height at most h, and C is a genus g curve over K such that C and J(C) have good reduction at the same primes and C has an isogeny of degree at most d from J(C) to E9, then g G.

(c) Let K be a number field with a real place and OK be its ring of integers. Let C be a curve over K such that J(C) is isogenous to E9 for some elliptic

curve E over K. Then, there are infinitely many primes p C 9

K such that C is maximal or minimal after a field extension of degree < 3.

2. Given positive integers m, n > 2, let C(n, m) be the smooth projective curve

corresponding to x" + ym = 1. Consider C(n, m) as a curve over Q(i). If J(C(n, in)) is isogenous to the power of an elliptic curve E over Q(i) with CM

by an order in Q(i), then there are infinitely many primes p of K where C mod p is a maximal curve. At each these of these primes where C mod p is also non-hyperelliptic, each twist of C mod p has a finite extension of the residue field F, at which it is maximal.

Remark 3.4 (Corollary of Part 1(a)). If C is a smooth projective curve of genus g over

a number field K such that J(C) is isogenous to a power of an elliptic curve and C has good reduction at a prime p of norm < N, then g G(K, N) for some constant

G(K, N) depending on K and N.

Proof. 1. (a) Let p C OK _{be a prime where E has good reduction and let Fq = F,.}

By Theorem 3.1, #E(Fqn) =1 - (a" + 3f") + q", where a and 3 are the

roots of the characteristic polynomial of the action of Frobq on Ht (Cq , Qi).

Since J(C) is isogenous to E" over Fq, we have that

g

H't(J(C)qQi) = H (E ,,qQi)

in a way compatible with the action of the absolute Galois group.

Com-bining this with the fact that Ht(J(C)p,,Qi) 2

Hi(Cy,,Qi),

we find

that

#C(Fqn)

= 1 - g(a" + n") + q"n. Since 3 = Z, this means that

#C(Fqn)

= 1 - 2g Re a" + q". Consider the powers 1,a, . . ., a'. By the

pigeonhole principle, there are two numbers 0 < k, I < 7 such that ak and a' have arguments which are within 27r/8 = 7r/4 of each other. Dividing these powers, we find an n < 7 such that -7r/4 < arg an < 7r/4. Since

#C(Fqn) > 0 and

Ianj

= qfn/2 , we have 2g Re < -1 + q" 1 + q"l g <I+qn - 2Rean 1 + q"n V/2qn/2

This gives us a bound for a fixed prime.

(b) By the first sentence of the proof of Proposition 4.4 of [81, there are only

finitely many possibilities for (K, E). Thus, we may assume that K and

E are fixed. Choose a prime p of K where E has good reduction. If J(C)

is isogenous to E", then J(C) has good reduction at p. This implies that

of C mod p, we obtain a bound on g.

(c) By [111, there are infinitely many primes p of K such that E mod p is supersingular. For any such p lying above a rational prime p ;> 5, the Hasse-Weil bound is attained after an extension of degree ; 3 and Table 1 below Lemma 6.1 of [19J.

2. In

[19],

Karemaker and Pries classified supersingular abelian varieties over finite fields according to the maximality or minimality of point counts after a finite extension. They put the supersingular abelian varieties A over finite fields Fq in the following categories:

" If each of the Fq-twists of A has a finite extension of Fq where it attains

the Hasse-Weil upper bound, then A is

fully

maximal.

" If none of the Fq-twists of A have this property, then A is

fully

minimal.

"

If some (but not all) of the Fq-twists of

A

attain the Hasse-Weil upper bound over some finite extension, then A is mixed.

Let K = Q(i) and C =

C(n,

m). By Deuring's criterion (see Theorem 1.1 of

[31),

an elliptic curve

E

with CM over

K

and good reduction at a prime p C OK

lying over a rational prime p has supersingular reduction at p if and only if p is the unique prime lying over p. For example, this includes the inert primes

p _ 3 (mod 4). Suppose that gcd(p, n) = gcd(p, m) = 1. By Theorem 5 of [351, the reduction of C mod a prime p lying over an inert prime p is maximal

over F, = F 2 if and only if we also have that p = -1 (mod n) and p -1

(mod m). Combining this with the condition p = 3 (mod 4), there are infinitely many inert primes p where C mod p is maximal over F, = F,2 by Dirichlet's theorem on primes in arithmetic progressions. The last statement follows from

Since a Jacobian has an irreducible principal polarization, it it isogenous to a power of a single elliptic curve E- if it is isomorphic to some product of elliptic curves

El x ... x Eg (see Lemma 2.2 of [23]). This gives the following corollary:

Corollary 3.5. Part 1 of Theorem 3.3 with J(C) ~ E1 x ... x E. substituted for

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