The Eisenstein quotient of the Jacobian variety of a Drin’feld modular curve.

*(English)*Zbl 1045.11510Let \(J=J_0(N)\) denote the Jacobian of the modular curve \(X_0(N)\) of Hecke type, where \(N\) is a rational prime. B. Mazur in groundbreaking work [Publ. Math., Inst. Hautes Étud. Sci. 47, 33–186 (1977; Zbl 0394.14008)] figured out a quotient \(\widetilde{J}\) of \(J\) (the Eisenstein quotient) with remarkable properties. Investigation of \(\widetilde{J}\) finally led him to his famous theorem on the uniform boundedness of \(E(\mathbb Q)_{\text{tor}}\) for elliptic curves \(E\) over the rationals.

Already in the above paper, Mazur proposed to consider the analogous quotients in the context of Drinfel’d’s theory, where \(K = \mathbb Q\) has to be replaced by a global function field \(K\) (e.g., \(K=\mathbb{F}_q(T)\)), \(X_0(N)\) by some Drinfel’d modular curve \(X_0({\mathfrak n})\) with a divisor \({\mathfrak n}\) of \(K\), etc.

An important step in this program is made in the present article, where, among others things, the analogue of \(\widetilde{J}\) is defined and properties similar to those of \(\widetilde{J}\) are proved (although the results are weaker than Mazur’s, due to problems specific to positive characteristic).

In detail: Let \(K=\mathbb F_q(T)\) with its ring \(A=\mathbb F_q[ T ]\) of integers, and let \({\mathfrak n}\) be a nonzero ideal in \(A\). Then the Drinfel’d modular curve of Hecke type \(X_0({\mathfrak n})\) and its Jacobian \(J_0({\mathfrak n})\) are defined and have properties similar to their classical counterparts [E.-U. Gekeler, Compos. Math. 57, 219–236 (1986; Zbl 0599.14032)]. If \({\mathfrak n}\) is prime (which we suppose from now on), \(J=J_0({\mathfrak n})\) agrees with its new part. Define \(T\) to be the \(\mathbb Z\)-algebra generated by the Hecke operators \(T_{{\mathfrak p}}\) (\({\mathfrak n} \not= {\mathfrak p}\) prime) and the canonical involution \(w_{{\mathfrak n}}\). The splitting of the semisimple commutative ring \(T \otimes \mathbb Q\) into fields corresponds to the splitting (up to isogeny) of \(J\) into \(K\)-irreducible abelian varieties. (In fact, in the appendix it is shown, using ideas of Ribet, that \(T\otimes \mathbb Q = \text{End}_{\overline{K}}(J) \otimes \mathbb Q\).) The cuspidal divisor class group \(C\) of \(J\) is a finite cyclic group of order prime to \(q\), and the Eisenstein ideal \(I\) is defined as its annihilator in \(T\).

The Eisenstein quotient is then \(\widetilde{J} = J/{\mathfrak b}_I J\), where \({\mathfrak b}_I = \bigcap_{r \in \mathbb{N}} I^r\). Similarly, Eisenstein prime numbers \(l\) in \(\mathbb{N}\) (the divisors of \(\#C\)), Eisenstein prime ideals in \(T\) and the corresponding quotients \(\widetilde{J}^{(l)}\) of \(\widetilde{J}\) are defined. It is easy to see that \(l=2\) is an Eisenstein prime if and only if \(q\) is odd and \(d := \deg {\mathfrak n} \equiv 0\bmod 4\). If this is not the case then (Proposition 4.12) \(\widetilde{J}= \widetilde{J}^-\) (i.e., \(w_{{\mathfrak n}}\) acts as \(-1\) on \(\widetilde{J}\) as in the classical situation). Most likely this assertion remains true in the general case, but it cannot be proved with the present methods, due to the lack (?) of a reasonable theory of automorphic forms mod \(p\) (\(p = \text{char}\,K\)). We should point out that it is much more difficult here to work out significant numerical examples than in the classical case.

The central result of the paper is Theorem 5.7, which states that (i) \(\widetilde{J}^-(K)\) and (ii) Ш\((\widetilde{J}^-/K) \otimes \mathbb{Z}[q^{-1}]\) are finite (Ш = Tate-Shafarevich group). By Proposition 4.14, \(\widetilde{J}^- \not= 0\) whenever \(J \not= 0\), so we can hope that \(\widetilde{J}^-\) will play a similar role in the arithmetic of \(X_0({\mathfrak n})\) as does \(\widetilde{J}\) in the number field situation.

The proof of Theorem 5.7 relies on the study of \(H^1_{\text{ét}}(S,{\mathcal B}[l^n])\), where \(S=\mathbb P^1/\mathbb F_q\), \({\mathcal B}\) is the global Néron model of \(B=J/{\mathfrak p}J\) (\({\mathfrak p} =\) minimal prime ideal of \(T\) contained in some \(l\)-Eisenstein prime ideal), \({\mathcal B}[l^n]= l^n\)-division points, and on work of P. Schneider [Math. Ann. 260, 495–510 (1982; Zbl 0509.14022)]. The necessary prerequisites are given in the first three sections, where the author collects material which is more or less known but difficult to find elsewhere.

Besides its important result, the article is highly laudable for its lucidity of exposition and intelligent choice of notation.

Already in the above paper, Mazur proposed to consider the analogous quotients in the context of Drinfel’d’s theory, where \(K = \mathbb Q\) has to be replaced by a global function field \(K\) (e.g., \(K=\mathbb{F}_q(T)\)), \(X_0(N)\) by some Drinfel’d modular curve \(X_0({\mathfrak n})\) with a divisor \({\mathfrak n}\) of \(K\), etc.

An important step in this program is made in the present article, where, among others things, the analogue of \(\widetilde{J}\) is defined and properties similar to those of \(\widetilde{J}\) are proved (although the results are weaker than Mazur’s, due to problems specific to positive characteristic).

In detail: Let \(K=\mathbb F_q(T)\) with its ring \(A=\mathbb F_q[ T ]\) of integers, and let \({\mathfrak n}\) be a nonzero ideal in \(A\). Then the Drinfel’d modular curve of Hecke type \(X_0({\mathfrak n})\) and its Jacobian \(J_0({\mathfrak n})\) are defined and have properties similar to their classical counterparts [E.-U. Gekeler, Compos. Math. 57, 219–236 (1986; Zbl 0599.14032)]. If \({\mathfrak n}\) is prime (which we suppose from now on), \(J=J_0({\mathfrak n})\) agrees with its new part. Define \(T\) to be the \(\mathbb Z\)-algebra generated by the Hecke operators \(T_{{\mathfrak p}}\) (\({\mathfrak n} \not= {\mathfrak p}\) prime) and the canonical involution \(w_{{\mathfrak n}}\). The splitting of the semisimple commutative ring \(T \otimes \mathbb Q\) into fields corresponds to the splitting (up to isogeny) of \(J\) into \(K\)-irreducible abelian varieties. (In fact, in the appendix it is shown, using ideas of Ribet, that \(T\otimes \mathbb Q = \text{End}_{\overline{K}}(J) \otimes \mathbb Q\).) The cuspidal divisor class group \(C\) of \(J\) is a finite cyclic group of order prime to \(q\), and the Eisenstein ideal \(I\) is defined as its annihilator in \(T\).

The Eisenstein quotient is then \(\widetilde{J} = J/{\mathfrak b}_I J\), where \({\mathfrak b}_I = \bigcap_{r \in \mathbb{N}} I^r\). Similarly, Eisenstein prime numbers \(l\) in \(\mathbb{N}\) (the divisors of \(\#C\)), Eisenstein prime ideals in \(T\) and the corresponding quotients \(\widetilde{J}^{(l)}\) of \(\widetilde{J}\) are defined. It is easy to see that \(l=2\) is an Eisenstein prime if and only if \(q\) is odd and \(d := \deg {\mathfrak n} \equiv 0\bmod 4\). If this is not the case then (Proposition 4.12) \(\widetilde{J}= \widetilde{J}^-\) (i.e., \(w_{{\mathfrak n}}\) acts as \(-1\) on \(\widetilde{J}\) as in the classical situation). Most likely this assertion remains true in the general case, but it cannot be proved with the present methods, due to the lack (?) of a reasonable theory of automorphic forms mod \(p\) (\(p = \text{char}\,K\)). We should point out that it is much more difficult here to work out significant numerical examples than in the classical case.

The central result of the paper is Theorem 5.7, which states that (i) \(\widetilde{J}^-(K)\) and (ii) Ш\((\widetilde{J}^-/K) \otimes \mathbb{Z}[q^{-1}]\) are finite (Ш = Tate-Shafarevich group). By Proposition 4.14, \(\widetilde{J}^- \not= 0\) whenever \(J \not= 0\), so we can hope that \(\widetilde{J}^-\) will play a similar role in the arithmetic of \(X_0({\mathfrak n})\) as does \(\widetilde{J}\) in the number field situation.

The proof of Theorem 5.7 relies on the study of \(H^1_{\text{ét}}(S,{\mathcal B}[l^n])\), where \(S=\mathbb P^1/\mathbb F_q\), \({\mathcal B}\) is the global Néron model of \(B=J/{\mathfrak p}J\) (\({\mathfrak p} =\) minimal prime ideal of \(T\) contained in some \(l\)-Eisenstein prime ideal), \({\mathcal B}[l^n]= l^n\)-division points, and on work of P. Schneider [Math. Ann. 260, 495–510 (1982; Zbl 0509.14022)]. The necessary prerequisites are given in the first three sections, where the author collects material which is more or less known but difficult to find elsewhere.

Besides its important result, the article is highly laudable for its lucidity of exposition and intelligent choice of notation.

##### MSC:

11G09 | Drinfel’d modules; higher-dimensional motives, etc. |

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

14K15 | Arithmetic ground fields for abelian varieties |

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\textit{A. Tamagawa}, Publ. Res. Inst. Math. Sci. 31, No. 2, 203--246 (1995; Zbl 1045.11510)

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