On Hecke’s decomposition of the regular differentials on the modular curve of prime level. (English) Zbl 1455.11083

Gross, Benedict H.

Res. Math. Sci. 5, No. 1, Paper No. 1, 19 p. (2018).

Let \(p\) be a prime number and \(\mathbb F_p\) the prime field of \(p\) elements. The class number of the field \(K=\mathbb Q(\sqrt{-p})\) is denoted by \(h\). The group \(G=\mathrm{SL}_2(\mathbb F_p)/\{\pm 1\}\) acts on the modular curve \(X\) associated with the principal congruence subgroup \(\Gamma(p)\). Erich Hecke decomposed the space \(H^0(X,\Omega^1)\) of regular differentials of \(X\) in the irreducible subspaces under the action of \(G\). Henceforth, let \(p\equiv 3 \mod 4, p>3\). Hecke found the \(G\)-invariant subspace \(V_0\) of \(H^0(X,\Omega^1)\) of dimension \(h(p-1)/2\) isomorphic to the \(h\)- copies of \(W\), where \(W\) is one of two irreducible representations of \(G\) of dimension \((p-1)/2\). Further, he identified certain periods of these differentials as the periods of elliptic curves with complex multiplication by \(K\). On the other hand, the author has obtained an elliptic curve \(A(p)\) defined over the Hilbert class field \(H\) of \(K\) as a factor of the Jacobian of the modular curve \(X_0(p^2)\), and also G. Shimura [J. Math. Soc. Japan 25, 523–544 (1973; Zbl 0266.14017)] has obtained an abelian variety \(B(p)\) of dimension \(h\) as a simple factor of the same Jacobian.
In this article, the author proves the above Hecke’s results using the character theory of \(G\), the Lefshetz fixed point formula, and the holomorphic fixed point formula. He explains how \(A(p)\) and \(B(p)\) relate to Hecke’s distinguished subspace \(V_0\). Let \(\mathcal{U}\) be an irreducible representation of \(G\) appeared in \(H^0(X,\Omega^1)\) of multiplicity \(m(\mathcal{U})\) and \(\mathcal{U}^\vee\) the dual representation of \(\mathcal{U}\). Since the singular cohomology \(H^1(X)\) is \(G\)-isomorphic to deRham cohomology \(H^1(X,\mathbb C)\) which is the extension of \(H^0(C,\Omega^1)\) by \(H^1(X,\mathcal{O})=H^0(X,\Omega^1)^\vee\), one can compute the sum of the multiplicities \(m(\mathcal{U})+ m(\mathcal{U^\vee})\) by the Lefshetz fixed point formula. Further, the result that \(m(W)-m(W^\vee)=h\) is proved by the holomorphic trace formula. This implies that \(H^0(X,\Omega^1)\) contains \(h\) copies of \(W\). Let \(PX(p)\) be the Shimura variety which is the coarse moduli space of generalized elliptic curves with a full level \(p\) structure, up to scaling. Let \(Y\) be the compactification of \(\Gamma(p)\backslash \mathfrak{H}^-\),where \(\mathfrak{H}^-\) is the lower half plane of \(\mathbb C\). The group \(\mathrm{PGL}_2(\mathbb F_p)\) acts on \(PX(p)\) over \(\mathbb Q\). The curve \(PX(p)(\mathbb C)\) has two components \(X\) and \(Y\) and \(H^0(Y,\Omega^1)=H^0(X,\Omega^1)^\vee\). Further, the Hecke operator \(T_\ell (\ell:\text{prime}\ne p)\) acts on \(PX(p)\) and \(H^0(PX(p),\Omega^1)\). The action of \(T_\ell\) commutes with that of \(\mathrm{PGL}_2(\mathbb F_p)\). Let \(\mathbb T\) be the commutative algebra over \(\mathbb Q\) generated by \(T_\ell\). The author constructs a CM-field \(E\) over \(K\) of degree \(h\), a Hecke character \(\chi:{\mathbb A}_K ^\times\rightarrow E^*\) and a surjective homomorphism of \(\mathbb T\) to the maximal real subfield \(E^+\) of \(E\). Let \(W'\) be the conjugate representation of \(W\). Since \(W+W'\) is extended to the representation \(R\) of \(\mathrm{PGL}_2(\mathbb F_p)\), \(V_0\) gives a distinguished subspace \(V\) of dimension \(h(p-1)\) in \(H^0(PX(p),\Omega^1)\) over \(\mathbb Q\). Let \(f\) be the new form of weight \(2\) with coefficients in \(E^+\) with respect to \(\Gamma(p^2)\) determined by \(\chi\). Let \(M(E^+)\) be the subspace of dimension \(1\) over \(E^+\) of \(H^0(X_0(p^2),\Omega^1)\) spanned by all conjugates of \(f\) over \(\mathbb Q\). Since \(X_0(p^2)\) is the quotient of \(PX(p)\) by the action of a split torus in \(\mathrm{PGL}_2(\mathbb F_p)\), \(M(E^+)\) is realized as a subspace of \(H^0(PX(p),\Omega^1)\),which is denoted by the same notation. The author shows that \(V\) is isomorphic to the simple module \(M(E^+)\otimes R\) over \(\mathbb Q\) under the action of \(\mathbb T\times\mathbb Q[\mathrm{PGL}_2(\mathbb F_p)]\) (See Theorem 8). In particular \(V_0\) is isomorphic to \(M(E^+)\otimes W\). Let \(\rho_A\) be the Hecke character of \(H\) obtained as the composition of \(\chi\) and the norm map from \(H\) to \(K\). Then \(\rho_A\) determines an isogeny class of elliptic curves defined over \(H\) with complex multiplication by the integers of \(K\). The elliptic curve \(A(p)\) is defined as the elliptic curve of the minimal discriminant \(-p^3\) in this class and \(B(p)\) is the abelian variety associated with the module \(M(E^+)\) by Shimura theory, and \(B(p)\) is obtained from \(A(p)\) by restriction of scalars. In the last place, the author gives a summary of what is known about the arithmetic of \(A(p)\) and \(B(p)\).

Reviewer: Noburo Ishii (Kyoto)

MSC:

11G05	Elliptic curves over global fields
14G35	Modular and Shimura varieties

Keywords:

Shimura varieties; modular forms

Citations:

Zbl 0266.14017

References:

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