##
**Analogies between function fields and number fields.**
*(English)*
Zbl 0531.12015

Whereas Iwasawa’s theory of p-cyclotomic extensions was inspired by Weil’s theory of the characteristic polynomial of the Frobenius endomorphism of a function field over a finite field of constants, the authors of the present paper in turn take Iwasawa’s theory as a sample for an analogous theory in the setting of function fields resp. curves over finite fields. The paper arose as a byproduct of the famous Mazur- Wiles proof of a conjecture (”Hauptvermutung”) of Iwasawa [the authors, Class fields of Abelian extensions of \({\mathbb{Q}}\), Invent. Math., 179-330 (1984)]. This conjecture states that, for a 1-dimensional, p-adic valued odd character \(\chi\) of the absolute Galois group \(G({\bar {\mathbb{Q}}}/{\mathbb{Q}})\) of the field \({\mathbb{Q}}\) of rational numbers having a conductor not divisible by \(p^ 2\), the zeroes of the Iwasawa characteristic polynomial are given, after a change of variables, by the zeroes of the Kubota-Leopoldt p-adic L-function \(L_ p(\omega \chi^{- 1},s)\) in the extended s-disc \(| s|_ p<p^{(p-2)/(p-1)}\), where \(\omega\) denotes the Teichmüller character and \(||_ p\) the normalized p-adic valuation of the p-adic completion \({\mathbb{Q}}_ p\) of \({\mathbb{Q}}.\)

The authors, terming their paper a ”preliminary investigation”, consider a so-called tower of Igusa curves \(X_ n\) of level \(p^ n\) for \(n\in {\mathbb{N}}_ 0\) over the prime field \(k={\mathbb{F}}_ p\) of characteristic p and, based on these \(X_ n\), introduce, for every non-trivial character \(\chi\) : \({\mathbb{F}}^{\times}_ p\to {\mathbb{Z}}^*_ p\) with values in the p-adic units \({\mathbb{Z}}^*_ p\) an Iwasawa-function \(D_ p(\chi,s)\), i.e. a certain p-adic analytic function convergent in the extended s-disc and coming from a characteristic element D(k,\(\chi)\) of the Iwasawa ring \(\Lambda\). Their main result established by virtue of methods from the above-mentioned Mazur-Wiles proof enunciates that for \(\chi \neq \omega^{-2}\) the zeroes of \(D_ p(\chi,s)\) comprise those of the Kubota-Leopoldt function \(L_ p(\chi \omega^ 2,-1-s)\) and moreover that \(L_ p(\chi \omega^ 2,-1-s)\) divides \(D_ p(\chi,s)\) when both functions are identified with their corresponding elements of the Iwasawa ring \(\Lambda\). This implies in particular that the quotient of the two functions is again an Iwasawa function (Proposition 9 of Section 3).

Specializing to the simplest case in which the character \(\chi\) is an even power \(\chi =\omega^ i\) for \(i\not\equiv 0 \bmod p-1\) of the Teichmüller character \(\omega\), the authors also connect their function \(D_ p(\chi,s)\) to the Kubota-Leopoldt function \(L_ p(\omega^ j,s)\) for an even integer in the range \(4\leq j\leq p-1\) such that \(i\not\equiv j-2 \bmod p-1\) via the following observation. Call a pair \((p,\chi)\) Igusa-regular if D(k,\(\chi)\in \Lambda\) is a unit and classically regular if \(L_ p(\omega^ j,s)\) has no zeroes in the extended s-disc. Then classically irregular implies Igusa-irregular (Proposition 8 of Section 3). Igusa-regularity can also be described in terms of the Hecke operator \(T_ p\) (Proposition 6 of Section 3). The element D(k,\(\chi)\in \Lambda\) enjoys the important property of being proalgebraic, i.e. being mapped onto an algebraic integer in the algebraic closure \({\bar {\mathbb{Q}}}_ p\) of \({\mathbb{Q}}_ p\) under certain homomorphisms h: \(\Lambda\to {\bar {\mathbb{Q}}}_ p\), and furthermore, under a special assumption, \(D(k,\chi)+1\) is proalgebraic of Weil type (cf. Section 1), i.e. is proalgebraic and such that \(h(D(k,\chi)+1)\) assumes under all imbeddings into the complex numbers \({\mathbb{C}}\) the absolute value \(p^ r\) for half- integral \(r\in frac{1}{2}{\mathbb{Z}}\). Finally D(k,\(\chi)\) can be expressed, up to a unit in \(\Lambda\), as the norm to \(\Lambda\) of a certain element in a certain Hecke algebra, the latter arising as a faithful ring of endomorphisms generated by the Hecke operators \(T_{\ell}\) for \(\ell \neq p\), the Frobenius endomorphism \(\Phi\) and the so-called ”diamond” operators (Corollary 2 of Section 4). Many interesting open questions are posed such as the one about the zeroes of the quotient \(D_ p(\chi,s)/L_ p(\chi \omega^ 2,-1-s)\), the vanishing of the \(\mu\)-invariant of \(D_ p(\chi,s)\) (cf. Sections 1 and 3) and the number of occurrences of Igusa-irregular pairs \((p,\chi)\) that are classically regular.

The paper is written in a lucid and condensed style with somewhat sketchy proofs. (The explanation gives on p. 516 for the subgroup of the Jacobian \(J_ 1({\mathbb{F}}_ p)[p]\) would actually be needed already for \(J_ n(k)[p^ m]\) on p. 512.) Unfortunately, essential parts of the reference section consist of preprints not easily accessible to the interested reader who does not belong to an apparently restricted circle of insiders. This seems to be a wide-spread problem of mathematical communication in these present times.

\(\{\) Reviewer’s remark: It might be worth mentioning that recently K. Lamprecht in her Diploma Thesis written at the University of Saarbrücken has computed zeroes of Kubota-Leopoldt p-adic L-functions thus extending earlier calculations by S. S. Wagstaff jun. [Number theory related to Fermat’s last theorem, Proc. Conf., Prog. Math. 26, 297-308 (1982; Zbl 0498.12015)].}

The authors, terming their paper a ”preliminary investigation”, consider a so-called tower of Igusa curves \(X_ n\) of level \(p^ n\) for \(n\in {\mathbb{N}}_ 0\) over the prime field \(k={\mathbb{F}}_ p\) of characteristic p and, based on these \(X_ n\), introduce, for every non-trivial character \(\chi\) : \({\mathbb{F}}^{\times}_ p\to {\mathbb{Z}}^*_ p\) with values in the p-adic units \({\mathbb{Z}}^*_ p\) an Iwasawa-function \(D_ p(\chi,s)\), i.e. a certain p-adic analytic function convergent in the extended s-disc and coming from a characteristic element D(k,\(\chi)\) of the Iwasawa ring \(\Lambda\). Their main result established by virtue of methods from the above-mentioned Mazur-Wiles proof enunciates that for \(\chi \neq \omega^{-2}\) the zeroes of \(D_ p(\chi,s)\) comprise those of the Kubota-Leopoldt function \(L_ p(\chi \omega^ 2,-1-s)\) and moreover that \(L_ p(\chi \omega^ 2,-1-s)\) divides \(D_ p(\chi,s)\) when both functions are identified with their corresponding elements of the Iwasawa ring \(\Lambda\). This implies in particular that the quotient of the two functions is again an Iwasawa function (Proposition 9 of Section 3).

Specializing to the simplest case in which the character \(\chi\) is an even power \(\chi =\omega^ i\) for \(i\not\equiv 0 \bmod p-1\) of the Teichmüller character \(\omega\), the authors also connect their function \(D_ p(\chi,s)\) to the Kubota-Leopoldt function \(L_ p(\omega^ j,s)\) for an even integer in the range \(4\leq j\leq p-1\) such that \(i\not\equiv j-2 \bmod p-1\) via the following observation. Call a pair \((p,\chi)\) Igusa-regular if D(k,\(\chi)\in \Lambda\) is a unit and classically regular if \(L_ p(\omega^ j,s)\) has no zeroes in the extended s-disc. Then classically irregular implies Igusa-irregular (Proposition 8 of Section 3). Igusa-regularity can also be described in terms of the Hecke operator \(T_ p\) (Proposition 6 of Section 3). The element D(k,\(\chi)\in \Lambda\) enjoys the important property of being proalgebraic, i.e. being mapped onto an algebraic integer in the algebraic closure \({\bar {\mathbb{Q}}}_ p\) of \({\mathbb{Q}}_ p\) under certain homomorphisms h: \(\Lambda\to {\bar {\mathbb{Q}}}_ p\), and furthermore, under a special assumption, \(D(k,\chi)+1\) is proalgebraic of Weil type (cf. Section 1), i.e. is proalgebraic and such that \(h(D(k,\chi)+1)\) assumes under all imbeddings into the complex numbers \({\mathbb{C}}\) the absolute value \(p^ r\) for half- integral \(r\in frac{1}{2}{\mathbb{Z}}\). Finally D(k,\(\chi)\) can be expressed, up to a unit in \(\Lambda\), as the norm to \(\Lambda\) of a certain element in a certain Hecke algebra, the latter arising as a faithful ring of endomorphisms generated by the Hecke operators \(T_{\ell}\) for \(\ell \neq p\), the Frobenius endomorphism \(\Phi\) and the so-called ”diamond” operators (Corollary 2 of Section 4). Many interesting open questions are posed such as the one about the zeroes of the quotient \(D_ p(\chi,s)/L_ p(\chi \omega^ 2,-1-s)\), the vanishing of the \(\mu\)-invariant of \(D_ p(\chi,s)\) (cf. Sections 1 and 3) and the number of occurrences of Igusa-irregular pairs \((p,\chi)\) that are classically regular.

The paper is written in a lucid and condensed style with somewhat sketchy proofs. (The explanation gives on p. 516 for the subgroup of the Jacobian \(J_ 1({\mathbb{F}}_ p)[p]\) would actually be needed already for \(J_ n(k)[p^ m]\) on p. 512.) Unfortunately, essential parts of the reference section consist of preprints not easily accessible to the interested reader who does not belong to an apparently restricted circle of insiders. This seems to be a wide-spread problem of mathematical communication in these present times.

\(\{\) Reviewer’s remark: It might be worth mentioning that recently K. Lamprecht in her Diploma Thesis written at the University of Saarbrücken has computed zeroes of Kubota-Leopoldt p-adic L-functions thus extending earlier calculations by S. S. Wagstaff jun. [Number theory related to Fermat’s last theorem, Proc. Conf., Prog. Math. 26, 297-308 (1982; Zbl 0498.12015)].}

Reviewer: H.G.Zimmer

### MSC:

11S40 | Zeta functions and \(L\)-functions |

11R58 | Arithmetic theory of algebraic function fields |

14G15 | Finite ground fields in algebraic geometry |

11R42 | Zeta functions and \(L\)-functions of number fields |

14H05 | Algebraic functions and function fields in algebraic geometry |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14H40 | Jacobians, Prym varieties |