##
**Drinfeld-Moduln und modulare Formen über rationalen Funktionenkörpern.**
*(German)*
Zbl 0446.14018

Bonn. Math. Schr. 119, 142 S. (1980).

The purpose of this paper is to study spaces of cusp forms over an algebraic function field \(K\) and its application to the theory of classification of elliptic curves over \(K\) by reviewing the theory of Drinfeld module [V. G. Drinfel’d, Math. USSR, Sb. 23 (1974), 561–592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594–627 (1974; Zbl 0321.14014)].

The paper also contains several interesting examples. Let \(K\) be an algebraic function field of one variable with a finite field \(\mathbb F_q\), as the exact constant field of \(K\). Let \(A\) be the adele ring of \(K\) and let \(\mathfrak o\) be the ring of integral elements in \(A\). For an arbitrary ring \(R\), let us denote \(\mathrm{PGL}_2(R)\) by \(G_R\). Let \(\mathcal K\) be the maximal compact subgroup of \(G_0\) and let \(\mathcal H\) be an open subgroup of \(\mathcal K\). Furthermore let \(\mathcal K(n)\) \((\mathcal K_0(n))\) be the homogeneous (resp. inhomogeneous) congruence subgroup of level \(n\) (any positive divisor of \(K)\) in \(G_0\). Then let \(W = C_0(G_K\backslash G_A/\mathcal H)\) be the subspace of all complex-valued cusp forms in \(L^2(G_K\backslash G_A)\).

In §1, following the theory of G. Harder [Ann. Math. (2) 100, 249–306 (1974; Zbl 0309.14041)] and one of T. Schleich [Bonn. Math. Schr. 71 (1976; Zbl 0345.10013)], the dimension of \(W\) is calculated when the group \(\mathcal H\) is \(\mathcal K_0(n)\) with \(\deg \le 4\) and \(K\) is rational, and when \(K\) is elliptic over \(\mathbb F_q\) and \(n\) is unit or a prime divisor of degree \(1\), respectively. For some divisors of degree 4 in the rational \(K\) over \(\mathbb F_q\), it is shown that all eigenforms of \(W\) under the Hecke algebra come from the construction theory of A. Weil [“Dirichlet series and automorphic forms”, Lect. Notes Math. 189 (1971; Zbl 0218.10046)]. From now on let us assume \(K = \mathbb F_q(T)\) rational.

In §2, the theory of Drinfeld module is reviewed to treat cusp forms with their values in a field of characteristic 0 not necessarily equal to \(\mathbb C\). Let \(\overline{Q}_\ell\) be the algebraic closure of the \(\ell\)-adic field of \(\mathbb Q\) \((\ne\) the characteristic of \(K)\), \(K_\infty\) be the \(\infty\)-completion, and let \(\mathcal H^f\) be the subgroup of \(\mathcal H\)consisting of elements without the \(\infty\)-component (\(\infty\): a fixed place of \(K)\). Let us define a space

\[ W_0 = C_0 (\mathrm{GL(2,K)}\backslash \mathrm{GL}(2,A)/\mathcal H^f, \overline{Q}_\ell) \]

of \(\overline{Q}_\ell\)-valued functions analogously to \(W\). Combining “the special representation of \(\mathrm{GL}(2,K_\infty)\) in any field \(C''\) and \(W_0\), the author obtains a subspace \(W_{ap}\) of \(W\) by using a renewed theorem of Drinfeld module. Let \(R\) be the subring \(\mathbb F_q[T]\) and let \(C_\infty\) be the \(\infty\)-completion of the algebraic closure of \(K_\infty\). Let us consider the group \(G_{K_\infty}\) as the fractional linear transformation of \(\Omega = P_1(C_\infty)/P_1(K_\infty)\).

In §3, developing the theory of Drinfeld module of rank 2, the following discussion is done:

(1) Completion of \(\Delta\backslash \Omega\) by adjoining some cusps of \(\Delta\backslash \Omega\).

(2) Construction of modular forms on \(\Omega\).

(3) Calculation of the genera of the algebraic curves \(F(M)\) and \(F_0(M)\) of the groups \(\Gamma(M)\) and \(\Gamma_0(M)\) respectively by the formula of Hurwitz.

Here \(\Delta\) is a discrete subgroup \(\mathrm{GL}(2,R)\) of \(\mathrm{GL}(2,K_\infty)\) or a congruence subgroup of \(\mathrm{GL}(2,R)\), \(\Gamma(M)\) \((\Gamma_0(M)\) is the homogeneous (inhomogeneous) congruence subgroup of \(\mathrm{GL}(2,R)\) of level \(M\) (ideal of \(R)\). This study is analogous to the classical case, but the situation is very different, for example, there appear the nonzero characteristic of \(K\), wild ramification in the field extensions and so on.

§4 is devoted to investigate the Fourier coefficients of cusp forms of \(W_{ap}\) and the matrix representation of some Hecke operators with respect to a nicely chosen basis of \(W_{ap}\) for the group \(\Gamma_0(M)\).

In §5 some relation with the theory of elliptic curves is explained. The theory of Drinfeld is applied to the modular curve \(F_0(M)\) with the assumption that the space \(W_{ap}\) on \(G_K\backslash G_A/\Gamma_0(M\cdot \infty)\) has a basis of new forms. Then a bijection between the set of \(K\)-isogenous elliptic curves of the conductor \(M\cdot \infty\) with some condition at \(\infty\) and the set of normalized eigenforms with the rational eigenvalues under the Hecke algebra is given.

The paper also contains several interesting examples. Let \(K\) be an algebraic function field of one variable with a finite field \(\mathbb F_q\), as the exact constant field of \(K\). Let \(A\) be the adele ring of \(K\) and let \(\mathfrak o\) be the ring of integral elements in \(A\). For an arbitrary ring \(R\), let us denote \(\mathrm{PGL}_2(R)\) by \(G_R\). Let \(\mathcal K\) be the maximal compact subgroup of \(G_0\) and let \(\mathcal H\) be an open subgroup of \(\mathcal K\). Furthermore let \(\mathcal K(n)\) \((\mathcal K_0(n))\) be the homogeneous (resp. inhomogeneous) congruence subgroup of level \(n\) (any positive divisor of \(K)\) in \(G_0\). Then let \(W = C_0(G_K\backslash G_A/\mathcal H)\) be the subspace of all complex-valued cusp forms in \(L^2(G_K\backslash G_A)\).

In §1, following the theory of G. Harder [Ann. Math. (2) 100, 249–306 (1974; Zbl 0309.14041)] and one of T. Schleich [Bonn. Math. Schr. 71 (1976; Zbl 0345.10013)], the dimension of \(W\) is calculated when the group \(\mathcal H\) is \(\mathcal K_0(n)\) with \(\deg \le 4\) and \(K\) is rational, and when \(K\) is elliptic over \(\mathbb F_q\) and \(n\) is unit or a prime divisor of degree \(1\), respectively. For some divisors of degree 4 in the rational \(K\) over \(\mathbb F_q\), it is shown that all eigenforms of \(W\) under the Hecke algebra come from the construction theory of A. Weil [“Dirichlet series and automorphic forms”, Lect. Notes Math. 189 (1971; Zbl 0218.10046)]. From now on let us assume \(K = \mathbb F_q(T)\) rational.

In §2, the theory of Drinfeld module is reviewed to treat cusp forms with their values in a field of characteristic 0 not necessarily equal to \(\mathbb C\). Let \(\overline{Q}_\ell\) be the algebraic closure of the \(\ell\)-adic field of \(\mathbb Q\) \((\ne\) the characteristic of \(K)\), \(K_\infty\) be the \(\infty\)-completion, and let \(\mathcal H^f\) be the subgroup of \(\mathcal H\)consisting of elements without the \(\infty\)-component (\(\infty\): a fixed place of \(K)\). Let us define a space

\[ W_0 = C_0 (\mathrm{GL(2,K)}\backslash \mathrm{GL}(2,A)/\mathcal H^f, \overline{Q}_\ell) \]

of \(\overline{Q}_\ell\)-valued functions analogously to \(W\). Combining “the special representation of \(\mathrm{GL}(2,K_\infty)\) in any field \(C''\) and \(W_0\), the author obtains a subspace \(W_{ap}\) of \(W\) by using a renewed theorem of Drinfeld module. Let \(R\) be the subring \(\mathbb F_q[T]\) and let \(C_\infty\) be the \(\infty\)-completion of the algebraic closure of \(K_\infty\). Let us consider the group \(G_{K_\infty}\) as the fractional linear transformation of \(\Omega = P_1(C_\infty)/P_1(K_\infty)\).

In §3, developing the theory of Drinfeld module of rank 2, the following discussion is done:

(1) Completion of \(\Delta\backslash \Omega\) by adjoining some cusps of \(\Delta\backslash \Omega\).

(2) Construction of modular forms on \(\Omega\).

(3) Calculation of the genera of the algebraic curves \(F(M)\) and \(F_0(M)\) of the groups \(\Gamma(M)\) and \(\Gamma_0(M)\) respectively by the formula of Hurwitz.

Here \(\Delta\) is a discrete subgroup \(\mathrm{GL}(2,R)\) of \(\mathrm{GL}(2,K_\infty)\) or a congruence subgroup of \(\mathrm{GL}(2,R)\), \(\Gamma(M)\) \((\Gamma_0(M)\) is the homogeneous (inhomogeneous) congruence subgroup of \(\mathrm{GL}(2,R)\) of level \(M\) (ideal of \(R)\). This study is analogous to the classical case, but the situation is very different, for example, there appear the nonzero characteristic of \(K\), wild ramification in the field extensions and so on.

§4 is devoted to investigate the Fourier coefficients of cusp forms of \(W_{ap}\) and the matrix representation of some Hecke operators with respect to a nicely chosen basis of \(W_{ap}\) for the group \(\Gamma_0(M)\).

In §5 some relation with the theory of elliptic curves is explained. The theory of Drinfeld is applied to the modular curve \(F_0(M)\) with the assumption that the space \(W_{ap}\) on \(G_K\backslash G_A/\Gamma_0(M\cdot \infty)\) has a basis of new forms. Then a bijection between the set of \(K\)-isogenous elliptic curves of the conductor \(M\cdot \infty\) with some condition at \(\infty\) and the set of normalized eigenforms with the rational eigenvalues under the Hecke algebra is given.

Reviewer: Tetsuo Kodama

### MSC:

11F52 | Modular forms associated to Drinfel’d modules |

14H52 | Elliptic curves |

11R56 | Adèle rings and groups |

14H45 | Special algebraic curves and curves of low genus |

11F11 | Holomorphic modular forms of integral weight |

14G15 | Finite ground fields in algebraic geometry |

14G25 | Global ground fields in algebraic geometry |