Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields. (English) Zbl 0546.14027

Cremona, J. E.

Compos. Math. 51, 275-324 (1984).

Let \(f\) be a newform of weight two and trivial character for \(\Gamma_ 0(N)\), with rational coefficients, then there is an elliptic curve over \(\mathbb Q\), quotient of the Jacobian variety of \(X_ 0(N)\), whose \(L\)-function is that of \(f\). Conversely, Weil has conjectured that every elliptic curve over \({\mathbb Q}\) is isogenous to such a curve. In the present paper it is investigated, mostly numerically the case of a complex quadratic field \(K\) as a base field instead of \(\mathbb Q\). If \(\mathfrak a\) is an ideal of \(K\) and \(\Gamma_ 0(\mathfrak a)\) the associated congruence subgroup of \(\mathrm{SL}(2,\mathfrak G_ K)\), then cusp forms on \(\Gamma_ 0({\mathfrak a})\) can be seen as dual to the space \(V(\mathfrak a)=H_ 1(\Gamma_ 0({\mathfrak a})\backslash X,\mathbb C)\) where \(X\) is the completion of the hyperbolic 3-space \(H_ 3\) by cusps; then \(V(\mathfrak a)\) can be computed via modular symbols, using a tessellation of \(H_ 3\) relative to \(\Gamma_ 0(\mathfrak a)\). In case \(K\) is euclidean, these tessellations are studied in §2 and a generator and relation description is given; the action of Hecke operator is also expressed (§3). This results in extensive tables (see also work of Grunewald, Elstrodt, Mennicke when \(\mathfrak a=\mathfrak p\) is a prime) giving much evidence for the existence of a correspondence between rational newforms in \(V({\mathfrak a})^+\) (where \(\pm\) means \(\pm 1\) eigenspace for \(\left( \begin{matrix} \varepsilon\\ 0\end{matrix} \begin{matrix} 0\\ 1\end{matrix} \right)\) where \(\varepsilon\) is a generator for roots of unity in \(K\)), and isogeny classes of elliptic curves defined over \(K\), but with no complex multiplication by \(K\). However, as remarked in a forthcoming paper of the author [“Abelian varieties with extra twists and cusp forms for imaginary quadratic fields”] rational newforms in \(V(\mathfrak a)^+\) not corresponding to elliptic curves can be obtained from two-dimensional abelian varieties over \(\mathbb Q\), not splitting over \(K\), and with extra twist by the quadratic character defining \(K\).
The final §4 examines twisting by quadratic characters and its action on the decompositions \(V(\mathfrak a)=V(\mathfrak a)^+\otimes V(\mathfrak a)^-\).

Reviewer: Guy Henniart (Orsay)

MSC:

14K15	Arithmetic ground fields for abelian varieties
11R11	Quadratic extensions
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14G25	Global ground fields in algebraic geometry
11F11	Holomorphic modular forms of integral weight

Keywords:

modular symbols; hyperbolic tessellations; elliptic curve; complex quadratic field; twisting by quadratic characters

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