##
**The Lifted Root Number Conjecture and Iwasawa theory.**
*(English)*
Zbl 1002.11082

Mem. Am. Math. Soc. 748, 90 p. (2002).

The motivating idea behind this paper is to find an approach to the Lifted Root Number Conjecture, at least for abelian \(G\)-Galois extensions \(K/k\) of absolutely abelian fields, by an appropriate use of Iwasawa theory. The Lifted Root Number Conjecture (LRNC) for these extensions predicts that a certain element \(\Omega_\varphi\) in the relative class group K\(_0({\mathbb Z}[G],{\mathbb Q}[G])\) essentially comes from \(L\)-values. The element \(\Omega_\varphi\) is constructed from a Tate sequence \(E \to A \to B \to \Delta S\) and an isomorphism \(\varphi: {\mathbb Q}\otimes\Delta S \to {\mathbb Q}\otimes E\). Here \(S\) is a finite set of finite primes of \(K\) satisfying certain “largeness” assumptions, \(E\) is short for \(E_S\), the group of \(S\)-units of \(K\), and \(\Delta S\) is the kernel of augmentation on \({\mathbb Z} S\). (Side comment: The conjecture LRNC is equivalent to the vanishing of the invariant \(T\Omega(K/k)\) introduced by Burns and Flach and studied in particular by Burns and collaborators. \(T\Omega(K/k)\) is defined as a refined Euler characteristic [see D. Burns, Compos. Math. 129, 203-237 (2001; Zbl 1014.11070)]; even if this is more technical, it avoids the choice of \(\varphi\) above.)

In comparison to the non-lifted conjecture of which it is a sharpening, LRNC has the great advantage of being equivalent to the collection of “localized” conjectures at all primes \(l\). So without further notice we will assume we are already in a local situation at \(l\), and even that \(l\) is odd. A naive idea would now be to construct an element \(\Omega_\varphi^\infty\) at infinite level, that is for the \(l\)-cyclotomic tower \(K_\infty\) instead of \(K\), prove something about it, and then try to redescend to \(K\). This is not what the authors do. They construct a link from \(\Omega_\varphi\) (more precisely, \(\Omega_{\varphi_l}\)) to another invariant \(\Omega_\Phi\), which relates, very roughly, to \(\Omega_\varphi\) the same way as Chinburg’s third invariant relates to the first. The invariant \(\Omega_\Phi\) again comes about by a Tate sequence, now with end terms \(G(M/K)\) and \({\mathbb Z}_l\), and a certain isomorphism \(\Phi: {\mathbb Q}\otimes {\mathbb Z}_l\to {\mathbb Q}\otimes G(M/K)\). Here \(M\) is the maximal abelian \(S\)-ramified \(l\)-extension of \(K\). Now \(\Omega_\Phi\) turns out to be accessible by Iwasawa theory: the authors construct a companion element \(\mho\) at infinite level, more precisely in a relative \(K_0\) group over \({\mathbb Z}[\text{Gal}(K_\infty/k)]\), such that its image down in the \(K_0\) group over \({\mathbb Z}[G]\) is “almost” \(\Omega_\varphi\) (the correction terms are under control, Theorem B).

Thus there remain two tasks: (1) “Calculate” \(\mho\), more precisely: show that it comes from \(L\)-values. (2) Get a grip on the difference between \(\Omega_\Phi\) (which is known provided task (1) has been achieved) and the object of primary interest, \(\Omega_{\varphi_l}\). Both these tasks are very nontrivial indeed. Actually at least (1) has already developed into a topic in its own right, under the name of “Equivariant Iwasawa Theory”. We will discuss it below. As to (2), Theorem A states that the sum \(\Omega_{\varphi_L}= \Omega_{\varphi_l}+\Omega_\Phi\) is again obtained by an “Omega construction”, that is, by a Tate sequence, this time the one leading to Chinburg’s second invariant, plus a certain isomorphism on the rational level. (This result is a natural one, and should also be easily provable from the point of view of refined Euler characteristics.)

Chapters 6 and 7 give calculations of this so-called semilocal class \(\Omega_{\varphi_L}\). We will not give any details of these and only say that these calculations suffice to prove Theorem F in Section 8 which says: If Task 1 is successfully completed (see below), then LRNC is true at \(l\) for all real cyclotomic fields \(K\) in which the prime \(l\) is at most tamely ramified.

Comments: Theorem F is a formula giving an expression for the quantity \(\omega^{(l)}\) which is by definition the “default of validity” of the \(l\)-part of LRNC, thus one has to show that (in the terminology of the paper) it is represented by a unit of \({\mathbb Z}_l[G]\). The numerator on the right hand side in Theorem F, \(\chi(\Upsilon)\), will be discussed below because it comes from Task (1). The denominator is actually represented by a unit of \({\mathbb Z}_l[G]\) as pointed out by the authors in their sequel paper [Towards equivariant Iwasawa theory, Manuscr. Math., in press], so one may simply omit it. The element \(\mho\) (more precisely \(\mho_S\); for reasons of space and simplicity we just have to neglect issues like dependency on \(S\) in this review) arises as follows: let \(G_\infty = \text{Gal}(K_\infty/k)\) and \(X_\infty=\text{Gal}(M_\infty/K_\infty)\) (this is a “classical” Iwasawa module). The central ingredient in the construction is again a 4-term sequence \[ 0 \to X_\infty \to Y_\infty \to {\mathbb Z}_l[G_\infty] \to {\mathbb Z}_l \to 0 \] which has a cohomologically trivial \(Y_\infty\)-term and which represents the canonical class at infinite level. One method to get this sequence is to use the theorem of Shafarevich-Weil and the so-called translation functor which transforms any group extension \(X \to \widetilde Y \to G\) into an extension of modules \(X \to Y \to \Delta G\). This four-term sequence has to be modified slightly in such a way that \(Y_\infty\) is replaced by a torsion module \(Y_\infty'\) over \({\mathbb Z}_l[G_\infty]\), and the third module \({\mathbb Z}_l[G_\infty]\) is replaced by a torsion quotient. This modification entails an auxiliary element \(c_\infty \in {\mathbb Z}_l[G_\infty]\), and \(\mho\) is finally defined as the class of \(Y_\infty'\) minus a correction term \(\delta(c_\infty)\) in the relative group K\(_0({\mathbb Z}_l[G_\infty],Q)\) where \(Q\) is the full quotient ring of \({\mathbb Z}_l[G_\infty]\). This is then independent of the choice of \(c_\infty\). The equivariant Main Conjecture, in the sense of this paper, then says that \(\mho=\mho_S\) comes from a Wiles power series \(\Theta_S\), more formally: \(\mho_S=\delta(\Theta_S)\).

In Section 5 it is shown that for \(K/k\) totally real (and abelian as always) the image of the equation \(\mho_S=\delta(\Theta_S)\) in K\(_0({\mathcal M},Q)\) follows from the “usual” main conjecture, that is, from the work of Wiles. Here \(\mathcal M\) is the integral closure of \({\mathbb Z}_l[G_\infty]\) in \(Q\). This can be restated as follows: \(\mho_S=\delta(\Upsilon) \cdot \delta(\Theta_S)\), where \(\Upsilon\) is a unit in \(\mathcal M\). The Equivariant Main Conjecture (EMC) is then equivalent to \(\Upsilon\in{\mathbb Z}_l[G_\infty]\). (Small comment: if \(\Upsilon\in {\mathbb Z}_l[G_\infty]\cap {\mathcal M}^\times\), then it already is a unit in \({\mathbb Z}_l[G_\infty]\).) In the sequel mentioned above, the authors prove EMC under the assumption \(\mu=0\).

Thus, putting everything together, we now have the validity of LRNC for odd \(l\) and absolutely abelian \(K\) in which \(l\) is at most tamely ramified (this latter condition was needed in the treatment of the semilocal invariant \(\Omega_{\varphi_L}\)). Since the authors also announce a sequel paper on \(\Omega_{\varphi_L}\), in which the tameness condition on \(l\) is eliminated, the authors’ results imply the validity of LRNC without the tameness condition. In a preprint (10/2000, submitted for publication), D. Burns and the reviewer proved the Equivariant Tamagawa Number Conjecture for Tate motives, odd \(l\) and absolutely abelian \(K\) by different methods, and this implies LRNC.

Thus, the memoir under review is an important piece of work; it develops still more strength if two sequel publications are also taken into account. Also, it builds on earlier publications of the same authors in an essential way, and this does not mitigate its difficulty. On the other hand, the writing is very clear and well done, and the notation, even if it is not everybody’s notation, is efficient and very consistent. Moreover, the introduction gives an excellent overview. To conclude, the reviewer wants to point out that many of the more technical parts of this long paper are not mentioned in this review, and he apologizes for possible oversights due to over-simplification.

In comparison to the non-lifted conjecture of which it is a sharpening, LRNC has the great advantage of being equivalent to the collection of “localized” conjectures at all primes \(l\). So without further notice we will assume we are already in a local situation at \(l\), and even that \(l\) is odd. A naive idea would now be to construct an element \(\Omega_\varphi^\infty\) at infinite level, that is for the \(l\)-cyclotomic tower \(K_\infty\) instead of \(K\), prove something about it, and then try to redescend to \(K\). This is not what the authors do. They construct a link from \(\Omega_\varphi\) (more precisely, \(\Omega_{\varphi_l}\)) to another invariant \(\Omega_\Phi\), which relates, very roughly, to \(\Omega_\varphi\) the same way as Chinburg’s third invariant relates to the first. The invariant \(\Omega_\Phi\) again comes about by a Tate sequence, now with end terms \(G(M/K)\) and \({\mathbb Z}_l\), and a certain isomorphism \(\Phi: {\mathbb Q}\otimes {\mathbb Z}_l\to {\mathbb Q}\otimes G(M/K)\). Here \(M\) is the maximal abelian \(S\)-ramified \(l\)-extension of \(K\). Now \(\Omega_\Phi\) turns out to be accessible by Iwasawa theory: the authors construct a companion element \(\mho\) at infinite level, more precisely in a relative \(K_0\) group over \({\mathbb Z}[\text{Gal}(K_\infty/k)]\), such that its image down in the \(K_0\) group over \({\mathbb Z}[G]\) is “almost” \(\Omega_\varphi\) (the correction terms are under control, Theorem B).

Thus there remain two tasks: (1) “Calculate” \(\mho\), more precisely: show that it comes from \(L\)-values. (2) Get a grip on the difference between \(\Omega_\Phi\) (which is known provided task (1) has been achieved) and the object of primary interest, \(\Omega_{\varphi_l}\). Both these tasks are very nontrivial indeed. Actually at least (1) has already developed into a topic in its own right, under the name of “Equivariant Iwasawa Theory”. We will discuss it below. As to (2), Theorem A states that the sum \(\Omega_{\varphi_L}= \Omega_{\varphi_l}+\Omega_\Phi\) is again obtained by an “Omega construction”, that is, by a Tate sequence, this time the one leading to Chinburg’s second invariant, plus a certain isomorphism on the rational level. (This result is a natural one, and should also be easily provable from the point of view of refined Euler characteristics.)

Chapters 6 and 7 give calculations of this so-called semilocal class \(\Omega_{\varphi_L}\). We will not give any details of these and only say that these calculations suffice to prove Theorem F in Section 8 which says: If Task 1 is successfully completed (see below), then LRNC is true at \(l\) for all real cyclotomic fields \(K\) in which the prime \(l\) is at most tamely ramified.

Comments: Theorem F is a formula giving an expression for the quantity \(\omega^{(l)}\) which is by definition the “default of validity” of the \(l\)-part of LRNC, thus one has to show that (in the terminology of the paper) it is represented by a unit of \({\mathbb Z}_l[G]\). The numerator on the right hand side in Theorem F, \(\chi(\Upsilon)\), will be discussed below because it comes from Task (1). The denominator is actually represented by a unit of \({\mathbb Z}_l[G]\) as pointed out by the authors in their sequel paper [Towards equivariant Iwasawa theory, Manuscr. Math., in press], so one may simply omit it. The element \(\mho\) (more precisely \(\mho_S\); for reasons of space and simplicity we just have to neglect issues like dependency on \(S\) in this review) arises as follows: let \(G_\infty = \text{Gal}(K_\infty/k)\) and \(X_\infty=\text{Gal}(M_\infty/K_\infty)\) (this is a “classical” Iwasawa module). The central ingredient in the construction is again a 4-term sequence \[ 0 \to X_\infty \to Y_\infty \to {\mathbb Z}_l[G_\infty] \to {\mathbb Z}_l \to 0 \] which has a cohomologically trivial \(Y_\infty\)-term and which represents the canonical class at infinite level. One method to get this sequence is to use the theorem of Shafarevich-Weil and the so-called translation functor which transforms any group extension \(X \to \widetilde Y \to G\) into an extension of modules \(X \to Y \to \Delta G\). This four-term sequence has to be modified slightly in such a way that \(Y_\infty\) is replaced by a torsion module \(Y_\infty'\) over \({\mathbb Z}_l[G_\infty]\), and the third module \({\mathbb Z}_l[G_\infty]\) is replaced by a torsion quotient. This modification entails an auxiliary element \(c_\infty \in {\mathbb Z}_l[G_\infty]\), and \(\mho\) is finally defined as the class of \(Y_\infty'\) minus a correction term \(\delta(c_\infty)\) in the relative group K\(_0({\mathbb Z}_l[G_\infty],Q)\) where \(Q\) is the full quotient ring of \({\mathbb Z}_l[G_\infty]\). This is then independent of the choice of \(c_\infty\). The equivariant Main Conjecture, in the sense of this paper, then says that \(\mho=\mho_S\) comes from a Wiles power series \(\Theta_S\), more formally: \(\mho_S=\delta(\Theta_S)\).

In Section 5 it is shown that for \(K/k\) totally real (and abelian as always) the image of the equation \(\mho_S=\delta(\Theta_S)\) in K\(_0({\mathcal M},Q)\) follows from the “usual” main conjecture, that is, from the work of Wiles. Here \(\mathcal M\) is the integral closure of \({\mathbb Z}_l[G_\infty]\) in \(Q\). This can be restated as follows: \(\mho_S=\delta(\Upsilon) \cdot \delta(\Theta_S)\), where \(\Upsilon\) is a unit in \(\mathcal M\). The Equivariant Main Conjecture (EMC) is then equivalent to \(\Upsilon\in{\mathbb Z}_l[G_\infty]\). (Small comment: if \(\Upsilon\in {\mathbb Z}_l[G_\infty]\cap {\mathcal M}^\times\), then it already is a unit in \({\mathbb Z}_l[G_\infty]\).) In the sequel mentioned above, the authors prove EMC under the assumption \(\mu=0\).

Thus, putting everything together, we now have the validity of LRNC for odd \(l\) and absolutely abelian \(K\) in which \(l\) is at most tamely ramified (this latter condition was needed in the treatment of the semilocal invariant \(\Omega_{\varphi_L}\)). Since the authors also announce a sequel paper on \(\Omega_{\varphi_L}\), in which the tameness condition on \(l\) is eliminated, the authors’ results imply the validity of LRNC without the tameness condition. In a preprint (10/2000, submitted for publication), D. Burns and the reviewer proved the Equivariant Tamagawa Number Conjecture for Tate motives, odd \(l\) and absolutely abelian \(K\) by different methods, and this implies LRNC.

Thus, the memoir under review is an important piece of work; it develops still more strength if two sequel publications are also taken into account. Also, it builds on earlier publications of the same authors in an essential way, and this does not mitigate its difficulty. On the other hand, the writing is very clear and well done, and the notation, even if it is not everybody’s notation, is efficient and very consistent. Moreover, the introduction gives an excellent overview. To conclude, the reviewer wants to point out that many of the more technical parts of this long paper are not mentioned in this review, and he apologizes for possible oversights due to over-simplification.

Reviewer: Cornelius Greither (Neubiberg)

### MSC:

11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |

11R23 | Iwasawa theory |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11R29 | Class numbers, class groups, discriminants |

11R37 | Class field theory |