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The Eisenstein quotient of the Jacobian variety of a Drin’feld modular curve. (English) Zbl 1045.11510
Let \(J=J_0(N)\) 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.

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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[1] Barsotti, I., Structure theorems for group-varieties, Ann. Mat. Pura Appl, 38 (1955), 77-119. · Zbl 0068.34301
[2] Bruhat, F., Lectures on some aspects ofp-adic analysis, Lectures on Mathematics and Physics, 27, Tata Institute, 1963.
[3] Casselman, W., On some results of Atkin and Lehner, Math. Ann., 201 (1973), 301-314. · Zbl 0239.10015
[4] Curtis, C. W., Reiner, L, Representation theory of finite groups and associative algebras, Interscience, 1962. · Zbl 0131.25601
[5] Deligne, P., Les constantes des equations fonctionelles des fonctions L, in Modular Functions of One Variable II, Lecture Notes in Math., 349, Springer- Verlag, 1973, 501-597. · Zbl 0271.14011
[6] - ^ La conjecture de Weil n, Publ. Math. IHES, 52 (1980), 137-252.
[7] Deligne, P., Husemoller, D., Survey of Drinfeld modules, Contemp. Math., 67 (1987). · Zbl 0627.14026
[8] Demazure, M., Gabriel, P., Groupes Algebriques, I, North-Holland, 1970.
[9] Drinfeld, V. G., EUiptic modules, Math. USSR Sbornik, 23 (1974), 4, 561-592.
[10] , Two dimensional /-adic representations of the Galois group of a global field of characteristic/? and automorphic forms on GL(2), J. Soviet Math., 36, No. 1 (1987), 93-105. · Zbl 0609.12010
[11] Gekeler, E.-U., Drinfeld-Moduln und modulare Formen iiber rationalen Funktionenkb’rpern, Banner Math. Schriften, 119 (1980). · Zbl 0446.14018
[12] , Automorphe Formen iiber F,(D mit kleinem Fiihrer, Abh. Math. Sem. Univ. Hamburg, 55 (1985), 111-146.
[13] Gekeler, E.-U., tiber Drinfeld’sche Modulkurven vom Hecke-Type, Compositio Math., 57 (1986), 219-236. · Zbl 0599.14032
[14] ^ Drinfeld modular curves, Lecture Notes in Math., 1231, Springer-Verlag, 1986.
[15] Gelbart, S. S., Automorphic forms on adele groups, Ann. of Math. Stud., 83, Princeton, 1975. · Zbl 0329.10018
[16] Goss, D., 7r-adic Eisenstein series for function fields, Compositio Math., 41 (1980), 3-38. · Zbl 0388.10020
[17] Jacquet, H., Langlands, R. P., Automorphic forms on (r£(2), Lecture Notes in Math., 114, Springer-Verlag, 1970. · Zbl 0236.12010
[18] Katz, N. M., Mazur, B., Arithmetic moduli of elliptic curves, Ann. of Math. Stud., 108, Princeton, 1985. · Zbl 0576.14026
[19] Mazur, B., Modular curves and the Eisenstein ideal, Publ Math. IHES, 47 (1978), 35-193.
[20] , Rational points on modular curves, in Modular Functions of One Variable V, Lecture Notes in Math., 601, Springer-Verlag, 1977, 107-148. · Zbl 0357.14005
[21] Milne, J. S., fitale cohomology, Princeton, 1980. · Zbl 0433.14012
[22] Mumford, ”D., Abelian varieties, Oxford, 1970. · Zbl 0223.14022
[23] Ribet, K. A., Galois action on division points of abelian varieties with real multiplications, Amer. J. Math., 98 (1976), 751-804. · Zbl 0348.14022
[24] , Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math., 101 (1975), 555-562. · Zbl 0305.14016
[25] Robert, A., Formes automorphes sur GL2(Travaux de H. Jacquet et R. P. Langlands), Sem. Bourbaki 1971/72, 415. · Zbl 0254.10022
[26] Rosenlicht, M., Some basic theorems on algebraic groups, Amer. J. Math., 78 (1956), 2, 401-443. · Zbl 0073.37601
[27] Schneider, P., Zur Vermutung von Birch und Swinnerton-Dyer iiber globalen Funk- tionenkorpern, Math. Ann., 260 (1982), 495-510. · Zbl 0509.14022
[28] Serre, J.-P., Tate, J., Good reduction of abelian varieties, Ann. of Math., 88 (1968), 3, 492-517. · Zbl 0172.46101
[29] Shimura, G., Algebraic number fields and symplectic discontinuous groups, Ann. of Math., 86 (1967), 503-592. · Zbl 0205.50601
[30] Shimura, G., Taniyama, Y., Complex multiplication of abelian varieties and its application to number theory, 2nd printing corrected, Math. Soc. of Japan, 1975.
[31] Teitelbaum, J. T., Modular symbols for \?q(T Duke Math. J., 68 (1992), 271-295. · Zbl 0777.11021
[32] Weil, A., Dirichlet series and automorphic forms, Lecture Notes in Math., 189, Springer-Verlag, 1971. · Zbl 0218.10046
[33] , Basic Number Theory, 3rd ed., Springer-Verlag, 1974.
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