Twisted limit formula for torsion and cyclic base change

Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to 1, e.g. $G= \SL_2 (\C) \times \SL_2 (\C)$ or $\SL_3 (\C)$. Then the fundamental rank of $G$ is $2,$ and according to the conjecture made in \cite{BV}, lattices in $G$ should have 'little' --- in the very weak sense of 'subexponential in the co-volume' --- torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with the \emph{square root} of the volume. This is deduced from a general theorem that compares twisted and untwisted $L^2$-torsions in the general base-change situation. This also makes uses of a precise equivariant 'Cheeger-M\"uller Theorem' proved by the second author \cite{Lip1}.

1. Introduction 1.1. Asymptotic growth of cohomology. Let Γ be a uniform lattice in a semisimple Lie group G. Let Γ n ⊂ Γ be a decreasing sequence of normal subgroups with trivial intersection. It is known that lim n→∞ dim H j (Γ n , C) [Γ : Γ n ] converges to b (2) j (Γ), the jth L 2 -betti number of Γ. If b (2) j = 0 for some j, it follows that cohomology is abundant.
However, it is often true that b (2) j (Γ) = 0 for all j; this is the case whenever δ(G) := rank C G − rank C K = 0. What is the true rate of growth of b j (Γ n ) = dim H j (Γ n , C) when δ(G) = 0? In particular, is b j (Γ n ) non-zero for sufficiently large n?
We address this question for 'cyclic base-change.' Before stating a general result, let's give two typical examples of this situation.
Examples. 1. The real semisimple Lie group G = SL 2 (C) satisfies δ = 1. Let σ : G → G be the real involution given by complex conjugation.
Note that when the Γ n 's are congruence subgroups of an arithmetic lattice Γ, then vol(Γ σ n \SL 2 (R)) grows like vol(Γ n \G) 1 order(σ) . N.B. is a member of the Institut Universitaire de France.
In this paper we shall more generally consider the case where G is obtained form a real algebraic group by 'base change.' Let G be a connected semisimple quasi-split algebraic group defined over R. Let E be anétale R-algebra such that E/R is a cyclic Galois extension with Galois group generated by σ ∈ Aut(E/R). Concretely, E is either R s or C s . In the first case σ is of order s and acts on R s by cyclic permutation. In the second case σ is of order 2s and acts on C s by (z 1 , . . . , z s ) → (z s , z 1 , . . . , z s−1 ). The automorphism σ induces a corresponding automorphism of the group G of real points of Res E/R G. We will furthermore assume that H 1 (σ, G) = {1}; see §2.4 for comments on this condition. The following proposition is 'folklore' (see e.g. Borel-Labesse-Schwermer [7], Rohlfs-Speh [25] and Delorme [11]).
1.2. Proposition. Let Γ n ⊂ Γ be a sequence of finite index, σ-stable subgroups of G. Suppose that δ(G σ ) = 0. Then we have: We prove Proposition 1.2 for certain families {Γ n } in §4.3 but our real interest here is rather how the torsion cohomology grows.
1.3. Asymptotic growth of torsion cohomology. Let ρ : G → GL(V ) be a homomorphism of algebraic groups over R and suppose that the Γ n 's stabilize some fixed lattice O ⊂ V. The first named author and Venkatesh [4] prove that for 'strongly acyclic' [4, §4] representations ρ, there is a lower bound j log |H j (Γ n , O) tors | ≫ c(G, ρ) · [Γ : Γ n ].
for some constant c(G, ρ). In fact, they prove a limiting identity and prove that c(G, ρ) is non-zero exactly when δ(G) = 1. The numerator of the left side of (1.3.1) should be thought of as a 'torsion Euler characteristic.' The purpose of this article is to prove an analogous theorem about 'torsion Lefschetz numbers.' Suppose as above that G is obtained form a real algebraic group by 'cyclic base change,' i.e. that G is the group of real points of Res E/R G where E/R is a cyclic Galois extension of R by anétale R-algebra. Let σ be the generator of the corresponding Galois extension and let G be the group of real points of Res E/R G. In this introduction, we will furthermore assume that the Γ n 's are principal congruence of level p n for some infinite sequence of primes.
Recall that ρ : G → GL(V ) is a homomorphism of algebraic groups over R. Assume moreover that the Γ n 's are σ-stable and that V is a σ-equivariant strongly twisted acyclic representation (see §2.6) of G. It corresponds to a representationρ of Res E/R G. In particular,ρ = ρ ⊗ ρ is the corresponding representation of Res C/R G.
Analogously to (1.3.1), we prove (1.4.1) by proving a limiting identity for torsion Lefschetz numbers. For example, when σ 2 = 1, conditional on an assumption about the growth of the Betti numbers dim F2 H j (Γ σ n , O F2 ) we prove that where the superscript ± denote the ±1 eigenspaces. Assume that the maximal compact subgroup K ⊂ G is σ-stable and let X = G/K and X σ = G σ /K σ . The proof of (1.4.2) crucially uses the equivariant Cheeger-Müller theorem, proven by Bismut-Zhang [5]. This allows us to compute the left side of (1.4.2) (up to a controlled integer multiple of log 2) by studying the eigenspaces of the Laplace operators of the metrized local system V → Γ n \X together with their σ action. More precisely, the left side is nearly equal to the equivariant analytic torsion log T σ Γn\X (ρ). Using the simple twisted trace formula and results of Bouaziz [8], we prove a 'limit multiplicity formula.' 1.5. Theorem. Assume that the Γ n 's are σ-stable and that V is a σ-equivariant strongly twisted acyclic representation (see §2.6) of G. Then we have: where E = R s or C s , and r = 0 in the first case and r = rank R G(C) − rank R G(R) in the second case.
Here t (2) X σ (ρ) is the (usual) L 2 -analytic torsion of the symmetric space X σ twisted by the finite dimensional representation ρ. It is explicitly computed in [4]. Note that it is non-zero if and only if δ(G σ ) = 1.
The authors hope that the limit multiplicity formula (1.5.1) together with the twisted endoscopic comparison implicit in Section 7 will be of interest independent of torsion in cohomology. These computations complement work by Borel-Labesse-Schwermer [7] and Rohlfs-Speh [25].
The authors would like to thank Abederrazak Bouaziz, Laurent Clozel, Colette Moeglin and David Renard for helful conversations.

The simple twisted trace formula
Let G be a connected semisimple quasi-split algebraic group defined over R. Let E be anétale R-algebra such that E/R is a cyclic Galois extension with Galois group generated by σ ∈ Aut(E/R). The automorphism σ induces a corresponding automorphism of the group G of real points of Res E/R G. We furthermore choose a Cartan involution θ of G that commutes with σ and denote by K the group of fixed points of θ in G. Here we follow Labesse-Waldspurger [18].
2.1. Twisted spaces. We associate to these data the twisted space The left action of G on G, turns G into a left principal homogeneous G-space equipped with a G-equivariant map Ad : G → Aut(G) given by We also have a right action of G on G by This allows to define an action by conjugation of G on G and yields a notion of G-conjugacy class in G. Note that taking δ = 1 ⋊ σ we have: We similarly define the twisted space K = K ⋊ σ.

Twisted representations.
A representation of G, in a vector space V , is the data for every δ ∈ G of a invertible linear map π(δ) ∈ GL(V ) and of a representation of G in V : such that for x, y ∈ G and δ ∈ G, π(xδy) = π(x) π(δ)π(y).
Therefore π(δ) intertwines π and π • Ad(δ). Note that π determines π; we will say that π is the restriction of π to G. Conversely π is determined by the data of π and of an operator A which intertwines π and π • σ: and whose p-th power is the identity, where p is the order of σ. We reconstruct π by setting Say that π is essential if π is irreducible. If π is unitary and essential, Schur's lemma implies that π determines A up to a p-th root of unity.
There is a natural notion of equivalence between representations of G -see e.g. [18, §2.3]. This is the obvious one; beware however that even if π is essential the class of π does not determine the class of π since the intertwiner A is only determined up to a root of unity.
We have a corresponding notion of a (g, K)-module.
If π is unitary and f ∈ C ∞ c ( G) we set It follows from [18,Lemma 2.3.2] that π(f ) is of trace class. Moreover: trace π(f ) = 0 unless π is essential. In the following, we denote by Π( G) the set of irreducible unitary representations π of G (considered up to equivalence) that can be extended to some (twisted) representation π of G. Note that the extension is not unique.
2.3. Twisted trace formula (in the cocompact case). Let Γ be a cocompact lattice of G that is σ-stable. Associated to Γ is the (right) regular representation R Γ of G on L 2 (Γ\G) where the restriction R Γ of R Γ is the usual regular representation in L 2 (Γ\G) and Given δ ∈ G we denote by G δ its centralizer in G (for the (twisted) action by conjugation of G on G). Corresponding to Γ is a (non-empty) discrete twisted subspace Γ ⊂ G. Given δ ∈ Γ we denote by {δ} its Γ-conjugacy class (where here again Γ acts by (twisted) conjugation on Γ).
Let f ∈ C ∞ c ( G). The twisted trace formula is obtained by computing the trace of R Γ (f ) in two different ways. It takes the following form (the LHS is the spectral side and the RHS is the geometric side): Here π is some extension of π to a twisted representation of G and where m( π ′ ) is the multiplicity of π ′ in R Γ and λ( π ′ , π) ∈ C × is the scalar s.t. for all δ ∈ G, we have π ′ (δ) = λ( π ′ , π) π(δ). 1 Note that λ( π ′ , π) is in fact a p-th root of unity.
The definition of trace π(f ) depends on a choice of a Haar measure dx on G. On the geometric side the volumes vol(Γ δ \G δ ) depend on choices of Haar measures on the groups G δ . We will make precise choices later on. For the moment we just note that the measure dẋ on the quotient G δ \G is normalized by:
Equivalently, (g 1 , . . . , g p ) ⋊ σ is conjugated to σ in G by some element in G. We furthermore note that H 1 (C/R, SL n (C)) = H 1 (C/R, Sp n (C)) = {1}, see e.g. [26,Chap. X]. Therefore, condition (2.4.1) holds if G is a product of factors SL n or Sp n or of factors whose group of real points is isomorphic to a complex Lie group viewed as a real Lie group.

2.5.
Under assumption (2.4.1), the map necessarily has trivial image. In other words: if δ represents a class in H 1 (σ, Γ) then δ is conjugate to σ by some element of G. In particular, we have vol(Γ δ \G δ ) = vol(Γ σ \G σ ) and We may therefore write the geometric side of the twisted trace formula as: 2.6. Finite dimensional representations of G. Note that the complexification of G may be identified with the complex points of Res E/R G, i.e. G(C) p . Every complex finite dimensional σ-stable irreducible representation ( ρ, F ) of G can therefore be realized in a space F = F ⊗p 0 where (ρ 0 , F 0 ) is an irreducible complex linear representation of G(C). The action of G is defined by the tensor product action ρ ⊗p 0 if E = R p and by ⊗ p/2 i=1 (ρ 0 ⊗ρ 0 ), whereρ 0 is obtained by composing the complex conjugation in G(C) by ρ 0 , if E = C p/2 . In both cases, we choose the action of σ on F = F ⊗p 0 to be the cyclic permutation A : Let g be the Lie algebra of G. Say that ( ρ, F ) is strongly twisted acyclic if there is a positive constant η depending only on F such that: for every irreducible unitary (g, K)-module V for which is satisfied. Here Λ F , resp. Λ V , is the scalar by which the Casimir acts on F , resp. V . Write ν for the highest weight of F 0 . The following lemma can be proven analogously to [4, Lemma 4.1].

Lemma.
Suppose that ν is not preserved by the Cartan involution θ then ρ is strongly twisted acyclic.

Lefschetz number and twisted analytic torsion
Let G, σ and Γ be as in §2.3.1 and let (ρ, F ) be a complex finite dimensional σ-stable irreducible representation of G. We denote by g be the Lie algebra of G.
3.1. Twisted (g(C), K)-cohomology and Lefschetz number. We can define an action of σ on each cohomology group H i (Γ\X, F ) and thus define a Lefschetz number If V is a (g, K)-module, we have a natural action of σ on the space of (g, K)-cochains C • (g(C), K, V ) which induces an action on the quotient H • (g(C), K, V ). We denote by the trace of the corresponding operator. We then define the Lefschetz number of V by If F is a finite dimensional representation of G then F ⊗ V is still a (g, K)-module; we denote by Lef(σ, F, V ) its Lefschetz number.
The function L ρ is called the Lefschetz function for σ and (ρ, F ).

Twisted heat kernels. Let
be the heat kernel for L 2 -forms of degree i with values in the bundle associated to (ρ, F ). Note that we have a natural action of σ on ∧ i (g/k) * ⊗ F ; we denote by A σ the corresponding linear operator and let . Eventually we shall apply the twisted trace formula to h ρ,i,σ t . The heat kernel H ρ,i t is not compactly supported. However, it follows from [3, Proposition 2.4] that it belongs to all Harish-Chandra Schwartz spaces C q ⊗End(∧ i (g/k) * ⊗ F ), q > 0. This is enough to ensure absolute convergence of both sides of the twisted trace formula.
3.3. Lemma. Let π be an essential admissible representation of G and let V be its associated (g, K)-module. We have: Proof. It follows from the K × K equivariance of H ρ,i t , and Kuga's Lemma that relative to the splitting we have: We furthermore note that this decomposition is σ-invariant since K is σ-stable. We conclude that we have: Here π is the restriction of π to G and A is the intertwining operator between π and π • σ that determines π. Now let {ξ n } n∈N and {e j } j=1,...,m be orthonormal bases of V and ∧ i (g/k) * ⊗ F , respectively. Then we have: The lemma follows.
K×K the heat kernel for L 2 -functions on X, the following proposition follows from [23, Proposition 5.3] and the definition of strong twisted acyclicity.

Proposition.
Assume that ( ρ, F ) is strongly twisted acyclic. Then there exist positive constants η and C such that for every x ∈ G, t ∈ (0, +∞) and i ∈ {0, . . . , dim X}, one has: it defines a function in C q ( G), for all q > 0.
3.5. Twisted analytic torsion. The twisted analytic torsion T σ Γ\X (ρ) is then defined by Note that if ( ρ, F ) is strongly twisted acyclic each trace(σ | H i (Γ\X, F )) is trivial. From now on we will assume that ( ρ, F ) is strongly twisted acyclic. In particular, we have: Remark. We should explain the notation Lef ′ . Given a group G and a G-vector space V , we denote by det[1 − V ] the virtual G-representation (that is to say, element of K 0 of the category of G-representations) defined by the alternating of exterior powers. This is multiplicative in an evident sense: This explains our notation Lef ′ for the twisted (g(C), K)-torsion.
For future reference we note that we have: We also note that Labesse's proof of the existence of L ρ can be immediately modified to get a function L ′ ρ ∈ C ∞ c ( G) such that for every essential admissible representation ( π, V ) of G one has Lef ′ (σ, F, V ) = trace π(L ′ ρ ).

3.7.
It follows from Lemma 3.3 that the spectral side of the twisted trace formula evaluated in k ρ,σ where V π is the (g, K)-module associated to the extension π.

L 2 -Lefschetz number, L 2 -torsion and limit formulas
Let G, σ and Γ be as in §2.3.1 and let (ρ, F ) be as in the preceding sections. Let f ∈ C ∞ c ( G).

Proposition.
Let {Γ n } be a sequence of finite index σ-stable subgroups of Γ. Assume that for any δ ∈ Γ with δ / ∈ Z 1 (σ, Γ) we have: Proof. It follows from (2.5.1) that we have: Since f is compactly supported, the last sum above is finite: choosing R > 0 so that the support of f is contained in we may restrict the sum on the right side of the above equation to δ that are contained in B R . It follows from Lemma 4.2 that the corresponding sum if finite.

4.4.
Let {Γ n } be a normal chain with ∩ n Γ n = {1}. Then r Γn → ∞ as n → ∞. The following lemma implies that the hypotheses of Proposition 4.3 are satisfied for {Γ n }.

4.7.
Corollary. Let {Γ n } be a normal chain of finite index σ-stable subgroups of Γ with ∩ n Γ n = {1}. Then we have: Proof. Apply Proposition 4.3 (and Lemma 4.5) to the Lefschetz function L ρ .
where t (2)σ X (ρ) -which depends only on the symmetric space X, the involution σ, and the finite dimensional representation ρ -is defined by Note that k ρ,σ t is not compactly supported and that we have to prove that the RHS of (4.8.2) is indeed well defined. Recall however that k ρ,σ t belongs to C q ( G). Lemma 4.2 therefore implies that the series converges absolutely and locally uniformly. This implies that the integral of this series along a (compact) fundamental domain D for the action of Γ on G is absolutely convergent. Restricting the sum to the δ's that belong to the (twisted) Γ-conjugacy class of σ we conclude in particular that, for every positive t, the integral is absolutely convergent. We postpone the proof of the fact that (4.8.2) is indeed well defined until sections 6 and 7 where we will explicitly compute t (2)σ X (ρ). In the course of the computations we will also prove the following lemma. 4.9. Lemma. There exist constants C, c > 0 such that Granted this we conclude this section by the proof of the following 'limit multiplicity theorem'.

Theorem.
Assume that ( ρ, F ) is strongly twisted acyclic. Let {Γ n } be a sequence of finite index σ-stable subgroups of Γ. Assume that there exists a constant A s.t. for every δ ∈ Γ with δ / ∈ Z 1 (σ, Γ) the sequence remains uniformly bounded by A and converges to 0 as n tends to infinity. 2 Then Proof. Since k ρ,σ t ∈ C q ( G), for all q > 0, we still have: Note that at this point it is not clear that the sum on the right (absolutely) converges. This is however indeed the case: it first follows from (4.1.1) that if δ / ∈ Z 1 (σ, Γ) and x ∈ G we have r(xδx −1 ) ≥ 2r Γ /p. Now recall from [4, Lemma 3.8] or [23, Proposition 3.1 and (3.14)] that we have, for t ∈ (0, 1], 2 Note that, in the untwisted case, the condition is equivalent to the BS-convergence of the compact quotients Γn\X towards the symmetric space X, see [1]. (Here c ′ depends on r Γ .) From this and Lemma 4.2, it follows that the geometric side of the trace formula evaluated in k ρ,σ t indeed absolutely converges. Moreover, it follows from (4.10.1) together with our uniform boundedness assumption that we have: is holomorphic is s in a half-plane containing 0, so Now it follows from Proposition 3.4 that there exists some positive η such that |k ρ,σ The above sum is absolutely convergent and independent of t and n, implying that where the implicit constant does not depend on n. Using Lemma 4.9, we conclude that both are absolutely convergent uniformly in n. We are therefore reduced to proving that uniformly in t when t belongs to a compact interval of [0, +∞). But this follows from the proof of Proposition 4.3 and the fact that for every δ ∈ Γ the sequence |{γ ∈ Γ n \Γ : remains uniformly bounded by A and converges to 0 as n tends to infinity.
Remark. Though reminiscent of a natural condition in the non-twisted case, we do not know how to check the condition on c Γn (δ) stated in Theorem 4.10 for any non-arithmetic Γ. However, we show in the next section that it holds for many sequences of congruence subgroups of arithmetic groups.

5.
Bounding the growth of c Γn (δ) for congruence subgroups of arithmetic groups We begin with a lemma to be used heavily in the proof of Proposition 5.2, which bounds the growth of c Γp (δ).

Lemma (Rational points of inner forms).
Let P be any connected, affine algebraic group over a finite field k.
If P ′ over k is any inner form of P, then |P(k)| = |P ′ (k)|.
Proof. We first reduce to the smooth case. Denote by P red the underlying reduced scheme of P. We have an isomorphism (P red × k P red ) red ∼ → (P × k P) red , see [12, Chap. I, Cor. 5.1.8]. Since the field k is perfect, we moreover have (P red × k P red ) red ∼ → P red × k P red so that the group law P × P → P induces a group law on P red . Hence, P red is a closed subgroup k-scheme of P. Since every reduced finite type scheme over a perfect field is smooth over a dense open subscheme, the standard homogeneity argument implies the group P red is smooth. But P(k) = P red (k), P ′ red (k) = P ′ (k), and P red is an inner form of P ′ red . We may therefore assume that P and P ′ are smooth.
Let P and P ′ have respective unipotent radicals U, U ′ and respective reductive quotients R, R ′ . Because H 1 (k, U) = H 1 (k, U ′ ) = 0, there are exact sequences of finite groups Because U and U ′ are forms, their dimensions are equal, say to d. Furthermore, all unipotent groups over the perfect field k are split unipotent. Therefore, applying the vanishing of H 1 to filtrations of U, U ′ by subgroups whose successive quotients are G a , we find that |U(k)| = |k| d = |U ′ (k)|. Moreover, because P, P ′ are inner forms, so are R, R ′ . But H 1 (k, Inn(R)) = 0 since Inn(R) is connected. Therefore, R ∼ = R ′ over k. In particular, The result follows.
Let G/Z[1/N ] be a semisimple group. For ease of exposition, we also assume that G is simply connected. Let Let p be a rational prime. Let for some constant C depending only on G and H. Proof. We abuse notation and write δ = δ ⋊ σ, so that now δ belongs to Γ. We then identify H(Fp) 1 ; this follows, for example, from the identity valid for arbitrarily endomorphisms A i of a finite dimensional vector space V. So in this case, . We turn to the more interesting case where O E /p = k ′ is a finite field extension of k = F p . The quantity c Γp (δ) can be expressed explicitly as a sum over twisted conjugacy classes: The groups Z H,y⋊σ and Z H,Norm(y) are inner forms [2, §1 Lemma 1.1]. By Lemma 5.1, Let M H be the maximum of |π 0 (Z H,Norm(y) )(k)| and C H,σ the maximum of |H 1 (σ, π 0 (Z H,y⋊σ )(k))| for all y ∈ I. Combining (5.2.2) and (5. 2.4) gives uniformly for all δ and all p. • To ease notation, let γ = Norm(δ). By Lang's theorem, If γ acts trivially on H\G, then γ ∈ g∈G(k) gH Q g −1 , a normal subgroup of G Q contained in H Q , implying that γ = 1 by hypothesis. Therefore, Fix(γ|H\G) is a proper subvariety of H\G for every γ = 1. Since H\G is irreducible, Fix(γ|H\G) must have strictly positive codimension in H\G. It follows that

Lemma.
Let G be an affine algebraic group over an arbitrary field F. The number of components of Z G,x , where x ranges over all elements of G(F ), is uniformly bounded.
Proof. G is a closed subvariety of SL n ⊂ End n for some fixed n. Suppose G is the simultaneous vanishing locus of polynomials f 1 , . . . , f m on the vector space End n . Let f i have degree d i , the maximum degree of all the monomials in its support. Note that Z G,x = Z End n ,x ∩ V (f 1 , . . . , f m ).
Clearly, Z G,x is Zariski open and dense in its projective completion . . . , f n ) obtained by adding a hyperplane at infinity to End n . Therefore, the number of connected components of Z G,x equals the number of connected components of Z G,x . Thus, where the final inequality follows by Bézout's theorem. This upper bound is independent of x.
Remark. The proof of Proposition 5.2 is an adaptation of Shintani's arguments [27] proving the existence of a "base change transfer" representations of GL n (k) Galois-fixed representations of GL n (k ′ ).
On the one hand, he sidesteps all component and endoscopy issues by working with G = GL n , H = parabolic subgroup. On the other hand, he proves an exact trace identity between matching principal series representations, an analogue of the fundamental lemma for GL n (k). We plan to pursue this analogue of the fundamental lemma for more general finite groups of Lie-type in a subsequent paper.
In the next sections we compute the L 2 -Lefschetz numbers and twisted L 2 -torsion and in particular prove Lemma 4.9. We distinguish two cases: we first deal with the case where E = R p (the product case) and then deal with case where E = C. The general case easily reduces to these two cases.

Computations on a product
Here we suppose that E = R p . Then G is the p-fold product of G(R) and σ cyclically permutes the factors of G. We will abusively denote by G σ the group G(R). Let (ρ 0 , F 0 ) be an irreducible complex linear representation of G(C). We denote by ( ρ, F ) the corresponding complex finite dimensional σ-stable irreducible representation of G. Recall that F = F ⊗p 0 , that G acts by the tensor product representation ρ ⊗p 0 and that σ acts by the cyclic permutation A : We finally let X and X σ be the symmetric spaces corresponding to G and G σ respectively, so X = (X σ ) p .

Heat kernels of a product. The heat kernels H ρ,j
This implies that pt (e). Here H ρ0,a pt is an untwisted heat kernel on X σ . Lemma 4.9 therefore follows from standard estimates (see e.g. [4]). Moreover, computations of the L 2 -Lefschetz number and of the twisted L 2 -torsion immediately follow from the above explicit computation.
6.2. Theorem (L 2 -Lefschetz number of a product). We have: Here X σ u is the compact dual of X σ whose metric is normalized such that multiplication by i becomes an isometry The computation then reduces to the untwisted case for which we refer to [24].
The computation of the twisted L 2 -torsion similarly reduces to the untwisted case: 6.3. Theorem (Twisted L 2 -torsion of a product). We have:

Computations in the case E = C
Throughout this section, E = C. Then G = G(C) is the group of complex points, σ : G → G is the real involution given by the complex conjugation and G σ = G(R). Recall that we fix a choice of Cartan involution θ of G that commutes with σ. 7.1. Irreducible σ-stable tempered representations of G. Choose θ-stable representatives h 0 1 , . . . , h 0 s of the G(R)-conjugacy classes of Cartan subalgebras in the Lie algebra g 0 of G(R). For each j ∈ {1, . . . , s} we write h 0 j = t j ⊕ a j for the decomposition of h 0 j w.r.t. θ, i.e. a j is the split part of h 0 j and t j is the compact part of h 0 j . We denote by h j the complexification of h 0 j ; note that a j ⊕ it j and t j ⊕ ia j are resp. the split and compact part of h j . We now fix some j. To ease notations we will omit the j index. Choose a Borel subgroup B of G = G(C) containing the torus H which corresponds to h j . Let A and T be resp. the split and compact tori corresponding to a ⊕ it and t ⊕ ia. Write µ for the differential of a character of T and λ for the differential of a character of A. Note that µ is σ-stable if and only if µ is zero on ia.
Associated to (µ, λ) is a representation π µ,λ = ind G B (µ ⊗ λ ⊗ 1). 7.2. Proposition (Delorme [11]). Every irreducible σ-stable tempered representation of G is equivalent to some π µ,λ as above (for some j) where µ is zero on ia j and λ is zero on t j and has pure imaginary image.
Note that if λ is zero on t j we may think of λ as a real linear form a → C. We denote by I µ,λ the underlying (g, K)-module. It is σ-stable and Delorme [11, §5.3] define a particular extension to a (g, K)-module, but we won't follow his convention here (see Convention I below). 7.3. Computations of the Lefschetz numbers. If G(R) has no discrete series Delorme [11,Proposition 7] proves that for any admissible (g, K)-module and any finite dimensional representation ( ρ, F ) of G, we have: Even if G(R) has discrete series Delorme's proof -see also [25,Lemma 4

.2.3] -shows that
Lef(σ, F, I µ,λ ) = 0 unless h 0 = t is a compact Cartan subalgebra (so that it is the split part of h). In the latter case λ = 0 (recall that I µ,λ is assumed to be σ-stable); we will simply denote by I µ the (g, K)-module I µ,0 . The following propositiondue to Delorme [11,Th. 2] 4 -computes the Lefschetz numbers in the remaining cases.
7.4. Proposition. We have: Here W is the Weyl group of (g, h) and the sign depends on the chosen extension of I µ to a (g, K)-module.
Convention I. In the following we will always assume that the extension of a σ-discrete I µ to a (g, K)-module is s. 7.5. Computations of the twisted (g, K)-torsion. Consider an arbitrary irreducible σ-stable tempered representation of G associated to some j and some (µ, λ) as in Proposition 7.2. Let P be the parabolic subgroup of G whose Levi subgroup M = 0 MA P is the centralizer in G of a. We have B ⊂ P(C) and we may write π µ,λ as the induced representation B∩ 0 M(C) (µ |t ⊗ 0) is a tempered (σ-discrete) representation of 0 M(C) and we think of λ -seen as real linear form a → C -as (the differential of) a character of A P (C).
Convention II. In the following we fix the extension of I µ,λ to a (g, K)-module to be the one associated to the interwining operator A G = ind G(C) P(C) (A M ⊗ 1) where A M is chosen according to Convention I. Let K M = K ∩ 0 M(C). Since σ stabilizes 0 M(C), µ, etc. . . it follows from Frobenius reciprocity and (3.6.2) that we have: It follows from (3.6.3) that -as a K M -module -we have: Now two simple lemmas: Proof. For any δ ∈ K M , For any X ∈ a * , In particular, (2) if dim a σ > 1, then det(t · 1 − δ|a * ) vanishes to order at least 2 at t = 1, whence Proof. Finite dimensionality is immediate since τ is admissible. Let ζ be a finite dimensional subrepresentation of τ Since K M is compact, taking K M -invariants is an exact functor from the category of finite dimensional K Mmodules to the category of finite dimensional σ-modules. Virtually trivial K M -modules therefore map to virtually trivial σ-modules. Thus, [V ⊗ ζ] KM is virtually trivial. In particular, In particular we conclude that (7.5.1) is zero unless dim a ≤ 1. In the following we compute (7.5.1) in the two remaining cases. 7.8. Computation of (7.5.1) when dim a = 1. Assume that dim a = 1. It then follows from Lemmas 7.6 and 7.7 that We can therefore compute For each w ∈ W we let ν w be the restriction of w(ρ + ν) to t. Let [W KM \W ] be the set of w ∈ W U such that ν w is dominant as a weight on t (with respect to the roots of t on k M ), i.e.: This is therefore a set of coset representatives for W KM in W . 7.9. Proposition. Assume dim a = 1. Then we have: Proof. We shall apply Proposition 7.4 to the twisted space associated to 0 M(C) to compute (7.8.1). To do so directly, we would need to decompose the virtual representation into irreducibles. This can be done by hand by reducing to the cases where G is simple of type SO(p, p) with p odd, or of type SL(3). Instead we note that Proposition 7.4 implies that only the essential σ-stable subrepresentations of (7.9.1) contribute to the final expression in (7.8.1). We may therefore reduce to considering the virtual representation Here we have realized the K M -representations a and n as representations in a 0 ⊗ a σ 0 and n 0 ⊗ n σ 0 , resp. Now it follows from Proposition 7.4 that Lef(σ, F, I µ,λ ) = 0 unless µ = 2µ 0 where µ 0 − ρ is the highest weight of a finite dimensional representation of 0 M(R). Next we use that if θ 0 denotes the discrete series of 0 M(R) with infinitesimal character µ 0 and H 0 a finite dimensional representation of 0 M(R) of highest weight ν then the (untwisted) Euler-Poincaré characteristic We can extend χ(·, θ 0 ) to any virtual representation. Applying the above to H 0 = det ′ [1 − a * 0 ⊕ n * 0 ] ⊗ F 0 , it follows from Proposition 7.4 and (7.8.1) that we have: . We are therefore reduced to the untwisted case. And the proposition finally follows from the computations made in [4, §5.6].
Remark. 1. The proof above and the transfer of infinitesimal characters under base change shows that (twisted heat kernel for F )(2t) 2. Note that this base change calculation includes, as a special case, that of a product G = G ′ × G ′ . But we've worked out separately that Lef ′ (σ, F 0 ⊗ F 0 , π ⊗ π) = 2det ′ (F 0 , π). There is no contradiction here: if either side is non zero then dim a = 1, but in that special case dim a = dim t since the real group is in fact a complex group. 7.10. Computation of (7.5.1) when dim a = 0. We now assume that dim a = 0 and follow an observation made by Mueller and Pfaff [23]. In that case M = G, K M = K and π µ,λ = π µ,0 is σ-discrete. Now dim g(C)/k equals 2d, where d is the dimension of the symmetric space associated to G(R). Note that -as K-modules -we have It follows that as K-representations we have:

Proof of Lemma 4.9.
If φ is any smooth compactly supported function on G, Bouaziz [8] shows that (7.12.1) where φ G ∈ C ∞ c (G(R)) is the transfer of φ. Now we can use the Plancherel theorem of Herb and Wolf [14] (as in [25,Proposition 4.2.14]) and get We can group the terms into stable terms since all terms in a L-packet have the same Plancherel measure [13]. We write π ϕ for the sum of the representations in an L-packet ϕ and dπ ϕ = dπ for the corresponding measure. We then obtain Now we use transfer again. Indeed, Clozel [10] shows that if ϕ is a tempered L-packet and π ϕ the sum of the twisted representations of G associated to ϕ by base-change, we have: We conclude: We want to apply this to the function φ = k ρ,σ t . Since it is not compactly supported we have to explain why (7.12.2) still holds for functions φ in the Harish-Chandra Schwartz space. We first note we have already checked (in §4.8) that the distribution extends continuously to the Harish-Chandra Schwartz space, i.e. it defines a tempered distribution. Now for φ compactly supported Bouaziz [8,Théorème 4.3] proves that we have (recall that we suppose that H 1 (σ, G) = {1}): We refer to [8] for all undefined notations and simply note that • the Π f,τ are tempered (twisted) representation, and • if φ belongs to Harish-Chandra Schwartz space, the (twisted) characters Θ f,τ (φ) = trace Π f,τ (φ) define rapidly decreasing functions of f . Bouaziz does not explicitly compute the function Q σ (f, τ ) but proves however that it grows at most polynomially in f . It therefore follows that the distribution defined by the right hand side of (7.12.3) also extends continuously to the Harish-Chandra Schwartz space. 5 We conclude that (7.12.3) still holds when φ belongs to Harish-Chandra Schwartz space.
We may now group the characters Θ f,τ into finite packets to get (all) stable tempered characters, as in [8, §7.3] and denoted Θ λ there. Then the right hand side of (7.12.3) becomes . 5 In fact, we need that (twisted) tempered characters are rapidly decreasing in the parameters "Schwartz-uniformly" in φ. But this holds because of known properties of discrete series characters combined with the fact that the constant term operator φ → φ (P ) are all Schwartz-continuous.
8.3. Now let E be anétale R-algebra; concretely, E = R r × C s . Fix σ ∈ Aut(E/R). The automorphism σ permutes the factors of E and so induces a decomposition of the factors of E into its set of orbits O : E = o∈O E o . Each orbit is either (a) a product of real places acted on by cyclic permutation, (b) a product of complex places acted on by cyclic permutation, or (c) a single complex place acted on by complex conjugation. Let G be a semisimple group over R. Let ρ be a representation of G/R and ρ o the corresponding representation of Res Eo/R . In particular, ρ = ρ ⊗ρ is the corresponding representation of Res C/R G. The automorphism σ induces a corresponding automorphism of the group Res E/R G. There is a decomposition Res Eo/R G with respect to which σ acts as a product automorphism.
• Theorem 6.3 shows that t Lef X G(R) (ρ) for the orbits of type (a).
• Theorem 6.3 shows that t X G(C) ( ρ) for the orbits of type (b). • Theorem 7.13 proves that t Lef X G(R) (ρ) for the orbits of type (c). The aggregate of these three examples, together with Theorem 8.2, allows us to compute the twisted L 2 -torsion for arbitrary base change. Proof. Suppose that there are n orbits. Using Lemma 8.2, we expand τ (2)σ X G(E) ( ρ E ) as a sum of n terms. Each summand is a product of Lef  We let s A • denote the preimage of 1 under the above isomorphism 9.3. Definition (Reidemeister torsion of a complex with volume forms). The Reidemeister torsion It is readily checked that if µ ′ i = c i µ i for some non-zero constants c i , then (9.3.1) RT (A • , ω, µ ′ ) = c 0 c 2 · · · c 1 c 3 · · · · RT (A • , ω, µ).

Example.
Suppose N is any finite free abelian group. We can define a volume ω form on A C by the formula ω(e 1 ∧ ... ∧ e n ) = 1 for any basis e 1 , ..., e n . This is well-defined up to sign. Endow each A • C and H i (A • C ) = H i (A • ) C with the above volume forms, ω Z and µ Z . Then which clearly does not depend on the triangulation K. By (9.3.1), we see that Remark. Illman [15] proves that any two smooth, Γ-equivariant triangulations of a smooth manifold M acted on by a finite group Γ admit a common equivariant refinement. Therefore, Definition 9.8 is independent of of the choice of (transversal) gradient vector field. 9.10. The equivariant Cheeger-Müller theorem on locally symmetric spaces. Let G/Q be a semisimple group and σ be an automorphism of G of prime degree p. Let G = G(R) and X G = G/K for K a σ-stable maximal compact subgroup of G. Let ρ : G → GL(V ) be a homomorphism of algebraic groups over Q. Let Γ ⊂ G(Q) be a σ-stable cocompact lattice and let O ⊂ V be Γ-stable Z-lattice. Let M = Γ\G/K and L ρ → M the σ-equivariant local system associated to ρ.
9.11. Theorem. Let L → M be an equivariant, metrized local system of free abelian groups over a locally symmetric space M acted on equivariantly and isometrically by σ ∼ = Z/pZ. Suppose further that the restriction to the fixed point set L| M σ = L ⊗p for L → M σ self-dual and that M σ is odd dimensional. It follows that log τ σ (M, L) = log RT σ (M, L).
Proof. The essense of this theorem is the Bismut-Zhang formula [5,Theorem 0.2]. In [20,Corollary 5.5], the aforementioned formula is massaged to prove that the error term is zero under the hypotheses of the theorem. Proof. This follows immediately by combining Theorem 9.9 with Theorem 9.11. The hypotheses imply that the p-power torsion error can be ignored.
9.14. Corollary. Enforce the same notation and hypotheses as in Corollary 9.13, with no a priori cohomology growth assumptions. Furthermore, assume that t Since (M n , L) has no rational cohomology by assumption, the desired conclusion follows by the universal coefficient theorem. Otherwise, both a priori cohomology growth hypothesis from Corollary 9.13 are satisfied. We thus apply Corollary 9.13, whose conclusion is more refined.
Remark. Corollaries 9.13 and 9.14 were one major source of inspiration for this paper. We sought to understand when t (2) XG ( ρ) = 0 in order to detect torsion cohomology growth. Note that it follows from Theorems 6.3 and 7.13 (and the computations in the non-twisted case done in [4]) that t (2)σ XG ( ρ) = 0 whenever δ(G σ ) = 1. Theorem 1.4 of the Introduction therefore follows from Corollary 9.14. 6 Recall the Smith sequence [9, § III.3.1], from which (9.14.1) follows: where K is a sufficiently fine σ-stable triangulation of M. The first map is the sum of the obvious inclusions. Bredon's argument proving exactness of the above sequence when L is the trivial local system carries over to any situation where C•(M σ , L) is a direct sum of copies of trivial Fp[σ] modules and a free Fp[σ]-module of finite rank. This holds in our context because L| M σ is isomorphic to L ⊗p . The "standard basis of a tensor product" (relative to a fixed basis of the original vector space) realizes the required direct sum decomposition.