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On Hecke’s decomposition of the regular differentials on the modular curve of prime level
 Benedict H. Gross^{1}Email author
https://doi.org/10.1007/s4068701801219
© SpringerNature 2018
Received: 19 April 2017
Accepted: 14 November 2017
Published: 22 January 2018
Abstract
In this paper, we review Hecke’s decomposition of the regular differentials on the modular curve of prime level p under the action of the group \({{\mathrm{SL}}}_2(p)/\langle {\pm 1}\rangle \). We show that his distinguished subspace corresponds to a factor of the Jacobian which decomposes as a product of conjugate, isogenous elliptic curves with complex multiplication.
1 Introduction
In a series of papers [15–18], Erich Hecke obtained the decomposition of the vector space of regular differentials (or in the language of the day, “Integrale 1. Gattung”) on the modular curve \(\Gamma (p)\backslash {\mathfrak {H}}^*\) of prime level p, under the action of the finite group \(G = \Gamma (1)/\Gamma (p) = {{\mathrm{SL}}}_2(p)/\langle {\pm 1}\rangle \). Hecke’s methods combined the geometric study of the curve with some explicit information contained in the character table of G. This table had been computed by Frobenius [8] in the first paper ever written on group characters, and Hecke’s decomposition of the space of regular differentials is one of the first applications of character theory (outside of the study of finite groups). It is also one of the first constructions of linear representations using cohomology. His ideas were immediately extended by Chevalley and Weil to finite group actions on the regular differentials of complex curves [4, Aus einem Briefe an E. Hecke], [33].
When \(p \equiv 3~~ (\mathrm{mod}~ 4)\) and \(p > 3\) Hecke described an invariant subspace of differentials of dimension \(h\cdot (p1)/2\) , where \(h = h(p)\) is the class number of the imaginary quadratic field \(K = {{\mathbb {Q}}}(\sqrt{p})\). The group G acts on this subspace by h copies of W, one of the two irreducible representations of G of dimension \((p1)/2\). This subspace is spanned by weighted binary theta series, and Hecke identified certain periods of these differentials as the periods of elliptic curves with complex multiplication by K. These elliptic curves appear as factors of the Jacobian of the modular curve of level p over \({{\mathbb {C}}}\).
In my PhD thesis [11], I described an elliptic curve A(p) with complex multiplication by the ring of integers of K, which is defined over its field of moduli \(F = {{\mathbb {Q}}}(j)\) and acquires complex multiplication over the Hilbert class field \(H = F\cdot K = K(j)\) of K. The curve A(p) is isogenous to all of its Galois conjugates over H. It appears as a factor of the Jacobian of the modular curve \(X_0(p^2)\), which is itself a quotient of the modular curve of level p. In that sense the curve A(p) had already made an appearance (some 50 years earlier) in Hecke’s work! When I mentioned this to my thesis advisor, he said that I shouldn’t worry—Hecke had anticipated some of the material in his thesis too.
A number of mathematicians have reconsidered and expanded on Hecke’s work. Shimura [29, 30] gave a different argument for the appearance of the class number \(h = h(p)\) and defined a simple factor B(p) of the Jacobian of \(X_0(p^2)\) over \({{\mathbb {Q}}}\), which decomposes as the product of h elliptic curves with complex multiplication by K over \({\overline{{{\mathbb {Q}}}}}\). This factor gives a refinement of Hecke’s invariant subspace on the cotangent space. In fact, Shimura’s abelian variety B(p) is isomorphic to the restriction of scalars of the elliptic curve A(p) from F to \({{\mathbb {Q}}}\), and decomposes as the product of A(p) and its conjugates over H [11, §15]. Hida [19] studied the abelian varieties with complex multiplication which appear as factors of the Jacobians of Shimura curves and Takei [31] generalized Hecke’s multiplicity formulas to the level p coverings of triangle groups with a cusp. Finally, Casselman [3] presented Hecke’s argument as a precursor of the work of Langlands and Labesse on endoscopy for \({{\mathrm{SL}}}_2 \).
In this paper, I will begin by reviewing Hecke’s main results. I will prove them using the character theory of G, the Lefshetz fixed point formula [21] (which dates from approximately the same period as Hecke’s work), and a holomorphic refinement of this fixed point formula. The general line argument is similar to that in Hecke’s original papers. I will then turn to the decomposition of the Jacobian, with particular attention to the factor of dimension \(h\cdot (p1)/2 \) corresponding to Hecke’s invariant subspace. I will give an explanation for the multiplicity h of the irreducible Gmodule W which occurs in Hecke’s invariant subspace, using the algebra generated by the Hecke operators at the primes \(\ell \) away from p, and will reinterpret that result using the language of automorphic representations. I will end with a summary of what is known about the arithmetic of the abelian varieties A(p) and B(p).
2 Representations of \(G = {{\mathrm{SL}}}_2(p)/\langle {\pm 1}\rangle \)
Let p be a prime with \(p > 3\), and let G be the finite group \({{\mathrm{SL}}}_2(p)/\langle {\pm 1}\rangle \). The group G is simple of order \(p(p^21)/2\). Every nontrivial element in G is either contained in a split torus \(S = \langle s^a\rangle \), which is cyclic of order \((p1)/2\), a nonsplit torus \(T = \langle t^b \rangle \), which is cyclic of order \((p+1)/2\), or a unipotent subgroup \(U = \langle u^c\rangle \), which is cyclic of order p. Since these orders are relatively prime, the type of subgroup containing a nontrivial element is unique. The normalizers of these subgroups are semidirect products: \(N(S) = S.2,N(T) = T.2,N(U) = U.S\) and are maximal subgroups of G.
Every semisimple element is conjugate to its inverse (so gives a real conjugacy class, in the sense of [26, Ch 13]), but is not conjugate to any other element in the cyclic group it generates. Hence, for every divisor \(d > 2\) of \((p1)/2\) or \((p+1)/2\), there are \(\phi (d)/2\) conjugacy classes of order d in G. When \(d = 2\) or \(d =3\), there is a unique conjugacy class of order d.
When \(p \equiv 3 ~(\mathrm{mod}~4)\), we may take \(e = 1\). Hence \(v = u^{1}\) is not conjugate to u in this case, and two conjugacy classes defined by u and v are not real, in the sense of [26, Ch 13]. All of the semisimple conjugacy classes are real, and when \(p \equiv 1~(\mathrm{mod}~4)\), all of the conjugacy classes are real.
When \(p \equiv 1 ~(\mathrm{mod}~4)\) and \(\alpha = \alpha ^{1}\) is quadratic, the corresponding principal series representation is not irreducible, but decomposes into two irreducible pieces W and \(W^{\prime }\) of dimensions \((p+1)/2\). The characters of W and \(W^{\prime }\) take values in the real quadratic field \({{\mathbb {Q}}}(\sqrt{p})\) and are conjugate.
When \(p \equiv 1 ~(\mathrm{mod}~4)\), all of the conjugacy classes in G are real and all of the irreducible characters take real values. In general, whenever an irreducible character of G takes values in \({{\mathbb {R}}}\), Hecke proved that the corresponding representation of G can be realized over \({{\mathbb {R}}}\) [18], so the duality is symmetric.
3 Regular differentials on the Riemann surface \(PX(p)^+\)
For example, in the simplest case when \(p = 7\), the curve \(PX(7)^+\) has genus 3 and is isomorphic to the Klein quartic with equation \(xy^3 + yz^3 + zx^3 = 0\) in \({\mathbb {P}}^2\). In this case, Hecke shows that \(m(W) = 1\), for W the distinguished representation of dimension \(3 = (7  1)/2\) of G described above, and \(m(U) = 0\) for all other irreducible representations. When \(p = 11\) the curve \(PX(11)^+\) has genus 26. In this case, Hecke shows that the multiplicity m(U) is 1 for \(U = W\) of dimension 5, the Steinberg representation U of dimension 11, and the discrete series representation U of dimension 10 corresponding to a cubic character \(\beta \) of T. We have \(m(U) = 0\) for all other irreducible representations U of G.
4 Using the Lefschetz fixed point formula
Proposition 3
Let g be an element of order 2, 3, or p in G, and let C be the cyclic subgroup generated by g. Then the number of fixed points \(\#Fix(g)\) of g acting on the curve X is equal to the trace of g on the induced representation \({{\mathrm{Ind}}}_C^G({{\mathbb {C}}})\).
Although we won’t need the number of fixed points explicitly, we can calculate this number by determining the order of the normalizer \(N_G(C)\) of each cyclic subgroup C in G. For the class of order 2, the normalizer of C is the normalizer of a nonsplit torus, and there are \(\#N_G(C)/C = (p+1)/2\) fixed points. For the class of order 3, the normalizer of C is either the normalizer of a split torus or the normalizer of a nonsplit torus, depending on whether \(p\equiv \pm 1~(\mathrm{mod}~3)\), and there are \(\#N(C)/C = (p1)/3\) or \(\#N(C)/C = (p+1)/3\) fixed points. For the class of order p the normalizer of C is the Borel subgroup and there are \(\#N(C)/C = (p1)/2\) fixed points.
This geometric data, together with the genus of X, determines the trace of every element in G on \(H^1(X)\). Using this, and the fact that the isomorphism class of a virtual complex representation of G is determined by its character, we can determine the isomorphism class of \(H^1(X)\).
Theorem 4.1
Corollary 4
Indeed, for any irreducible representation U, the multiplicity of U in the permutation representation \({{\mathrm{Ind}}}_C^G({{\mathbb {C}}})\) is equal to the dimension \(U^C\) of the C invariant subspace, by Frobenius reciprocity [26, Ch 7].
5 The real case
6 The complex case
Hecke completed the determination of m(W) by finding an amazing formula for the difference of multiplicities (which must also be odd).
Theorem 6.1
The difference \(m(W)  m(W^{\vee }) = h(p)\), where \(h(p)\) is the class number of the imaginary quadratic field \(K = {{\mathbb {Q}}}(\sqrt{p}).\)
There are a few cases where one finds that \(m(W) + m(W^{\prime }) = m(W)  m(W^{\prime })\), so \(m(W^{\prime }) = 0\) and \(m(W) = h(p)\). These are \(p = 7,11,19\) where \(h(p) = 1,p = 23,31\) where \(h(p) = 3,p = 47\) where \(h(p) = 5\), and \(p = 71\) where \(h(p) = 7\) [16]. In all other cases, both W and \(W^{\prime }\) appear with positive multiplicity in V.
Proposition 5
Let p be a prime with \(p > 3\) and \(p \equiv 3~(\mathrm{mod}~4)\) and let \(\zeta = e^{2\pi i/p}\) in \({{\mathbb {C}}}\). Then the trace of the element \(1/(1\zeta )\) from the cyclotomic field \({{\mathbb {Q}}}(\zeta )\) in \({{\mathbb {C}}}\) to its quadratic subfield \(K = {{\mathbb {Q}}}(\sqrt{p})\) is equal to the algebraic integer \(\frac{1}{2} ((p1)/2 + h\sqrt{p})\), where \( h = h(p)\) is the class number of K and \(\sqrt{p}\) has positive imaginary part in \({{\mathbb {C}}}\).
7 The Shimura variety PX(p)
Henceforth in this paper, we let G be the reductive group scheme \({{\mathrm{PGL}}}_2\) over \({{\mathbb {Z}}}\) and let p be an odd prime. The group \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \) that we have previously called by this letter is a subgroup of index two in the points \(G(p) = {{\mathrm{PGL}}}_2(p)\) of G over the field of p elements.
When \(p \equiv 3~ (\mathrm{mod} ~4)\), the homomorphism \({{\mathrm{PGL}}}_2({{\mathbb {Z}}}) \rightarrow {{\mathrm{PGL}}}_2(p)\) is surjective, and we still have only one double coset: \(G({{\mathbb {A}}}_f) = G({{\mathbb {Q}}})\cdot \prod G(Z_{\ell }) \times K_1\). The intersection is now the subgroup \(\Gamma (p)\) of \({{\mathrm{SL}}}_2({{\mathbb {Z}}})/\langle \pm 1 \rangle \), and \(S({{\mathbb {C}}})\) is isomorphic to the orbit space \(\Gamma (p) \backslash {\mathfrak {H}}^{\pm }\) (which can be compactified by the addition of a finite number of cusps). In this case, the two components are antiisomorphic over \({{\mathbb {C}}}\).
When \(p \equiv 1~ (\mathrm{mod} ~4)\), the homomorphism \({{\mathrm{PGL}}}_2({{\mathbb {Z}}}) \rightarrow {{\mathrm{PGL}}}_2(p)\) is not surjective, but has image the subgroup \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \) where the determinant is a square modulo p. In this case we have two double cosets of \(G({{\mathbb {Q}}})\) and \(\prod G({{\mathbb {Z}}}_{\ell }) \times K_1\) in \(G({{\mathbb {A}}}_f)\), with each discrete subgroup \(\Gamma \) in the intersection containing elements of determinant \(\pm 1\). Here each component \(\Gamma \backslash {\mathfrak {H}}^{\pm }\) of \(S({{\mathbb {C}}})\) has an antiholomorphic involution and descends to \({{\mathbb {R}}}\).
The theory of canonical models of Shimura varieties shows that the algebraic curve S and its compactification descend to \({{\mathbb {Q}}}\) as the coarse moduli space PX(p) of (generalized) elliptic curves with a full level p structure, up to scaling [5, 6]. This means that a basis \(\{e_1,e_2\}\) of the ptorsion is fixed, up to the scaling of \(e_1\) and \(e_2\) by the same element of \(({{\mathbb {Z}}}/p{{\mathbb {Z}}})^*\). We use the notation PX(p) as X(p) is the moduli space of (generalized) elliptic curves with a full level p structure. The curve X(p) is defined over \({{\mathbb {Q}}}\) and has an action of the group \({{\mathrm{GL}}}_2(p)/\langle \pm 1 \rangle \) [12]. It is not geometrically connected: the \((p1)\) geometric components of X(p) are defined over the \(p^{th}\) cyclotomic field \({{\mathbb {Q}}}(\zeta )\) The curve PX(p) is the quotient of X(p) by the central subgroup \(({{\mathbb {Z}}}/p{{\mathbb {Z}}})^*/\langle \pm 1 \rangle \) of \({{\mathrm{GL}}}_2(p)/\langle \pm 1 \rangle \), and the quotient group \({{\mathrm{PGL}}}_2(p)\) acts on PX(p) over \({{\mathbb {Q}}}\). The two geometric components of X(p) are defined over the real quadratic field \({{\mathbb {Q}}}(\sqrt{p})\) when \(p \equiv 1~(\mathrm ~mod ~4)\) and over the imaginary quadratic field \(K = {{\mathbb {Q}}}(\sqrt{p})\) when \(p \equiv 3~(\mathrm{mod}~ 4)\). In the latter case, which is of primary interest to us, \(PX(p)({{\mathbb {C}}})\) is the compactification of the quotient \(\Gamma (p) \backslash {\mathfrak {H}}^{\pm }\), and complex conjugation gives an antiisomorphism from the component \(X = \Gamma (p) \backslash {\mathfrak {H}}^*\) to the component Y uniformized by the lower half plane. Hence the action of \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \) on the regular differentials of Y gives the dual of the module studied by Hecke on the regular differentials of X.
Since the group \({{\mathrm{PGL}}}_2(p)\) acts on the curve PX(p) over \({{\mathbb {Q}}}\), one can ask for the structure of the representation on the \({{\mathbb {Q}}}\)vector space \(H^0(PX(p),\Omega ^1)\). To determine the multiplicities of irreducible representations, it suffices to do this over \({{\mathbb {R}}}\), where all of the irreducible complex representations V of \({{\mathrm{PGL}}}_2(p)\) can be defined.
Every irreducible representation U of the subgroup \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \) which is defined over the real numbers extends to \({{\mathrm{PGL}}}_2(p)\) in two ways: R and \(R \otimes \chi \), where \(\chi \) is the nontrivial quadratic character of \({{\mathrm{PGL}}}_2(p)\) with kernel \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \). In this case, the induced representation of U decomposes as a direct sum \(R \oplus (R \otimes \chi )\). The two irreducible representations W and \(W^{\prime }\) of dimensions \((p1)/2\) of \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \) which cannot be defined over the real numbers combine to give a single irreducible representation R of dimension \((p1)\) of \({{\mathrm{PGL}}}_2(p)\). In this case, the induced representation of W (or \(W^{\prime }\)) is irreducible, and isomorphic to R. This representation R is the discrete series corresponding to a character \(\psi \) of order 4 of a nonsplit torus of \({{\mathrm{PGL}}}_2(p)\) and can be defined over \({{\mathbb {Q}}}\). The restriction of R to the subgroup \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \) decomposes as the direct sum of the conjugate irreducible representations \(W + W^{\prime }\) over K.
For any irreducible complex representation U of \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \), let m(U) denote the multiplicity of U in Hecke’s module \(H^0(X, \Omega ^1)\). Since the representation of \({{\mathrm{PGL}}}_2(p)\) on the differentials of PX(p) is induced from the representation of the subgroup \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \) of index 2 on the differentials of the component X, we find the following.
Proposition 6
The multiplicity of an irreducible representation R of \({{\mathrm{PGL}}}_2(p)\) over \({{\mathbb {R}}}\) in the module \(H^0(PX(p),\Omega ^1) \otimes {{\mathbb {R}}}\) is equal to m(U) if the restriction of R to \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \) is isomorphic to the irreducible representation U, and is equal to \(m(W) + m(W^{\prime })\) if the restriction of R to \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \) is isomorphic to the direct sum \(W + W^{\prime }\) over \({{\mathbb {C}}}\).
In particular, Hecke’s distinguished subspace of \(H^0(X,\Omega ^1)\) of dimension \(h\,{\cdot }\, (p1)/2\) which is spanned by weighted binary theta series gives a distinguished subspace V of dimension \(h\,{\cdot }\, (p1)\) in \(H^0(PX(p),\Omega ^1)\) over \({{\mathbb {Q}}}\). The group \({{\mathrm{PGL}}}_2(p)\) acts on the distinguished subspace by \(h = h(p)\) copies of the unique irreducible representation R of dimension \((p1)\) over \({{\mathbb {Q}}}\) which decomposes over K. We will give some explanation for this multiplicity in the next section.
8 An irreducible module for the Hecke algebra
Let \(\ell \) be a prime with \(\ell \ne p\). The Hecke operator \(T_{\ell }\) is defined by a correspondence on the modular curve PX(p) over \({{\mathbb {Q}}}\), which maps an elliptic curve E with a level p structure up to scaling to the \((\ell +1)\) elliptic curves \(E^{\prime }\) which are isogenous to E by an isogeny of degree \(\ell \). Since \(\ell \) is prime to p, and \(\ell \) isogeny maps a level p structure up to scaling on E to a level p structure up to scaling on \(E^{\prime }\). This correspondence induces an endomorphism of the Jacobian of PX(p), as well as a linear endomorphism of the rational vector space \(H^0(PX(p), \Omega ^1)\) of regular differentials on PX(p). We denote the linear endomorphism of regular differentials by \(T(\ell )\).
The operators \(T_{\ell }\) for different primes commute with each other, and we let \({\mathbb {T}}\) be the commutative \({{\mathbb {Q}}}\)algebra that they freely generate. The algebra \({\mathbb {T}}\) acts on the regular differentials through the quotient algebra generated by the \(T(\ell )\) and commutes with the action of the group \({{\mathrm{PGL}}}_2(p)\). It therefore acts on each isotypic component. We shall see that it also preserves Hecke’s distinguished subspace, which lies in the isotypic component where \({{\mathrm{PGL}}}_2(p)\) acts through the representation R.
We now define a CM field E, which is an extension of \(K = {{\mathbb {Q}}}(\sqrt{p})\) of degree \(h(p)\). Write the class group of K as a direct sum of cyclic groups, and for each summand, let \({\mathfrak {a}}\) be an ideal prime to p whose class gives the generator of order m. Then \(({\mathfrak {a}})^m = (\alpha )\) is a principal ideal, with a unique generator \(\alpha \) which is a square modulo \((\sqrt{p})\). Indeed, the two generators of this ideal are \(\alpha \) and \(\alpha \), and \(1\) is not a square modulo p. We obtain the field E from K by adjoining an \(m^{th}\) root \(\beta \) of \(\alpha \), for each cyclic summand which is isomorphic to \({{\mathbb {Z}}}/m{{\mathbb {Z}}}\) of the class group. For example, if the class group is cyclic and \({\mathfrak {a}}\) is a generator which is prime to p, then \(E = K(\beta )\), where \(\beta ^h\) is an element of \(K^*\) which generates the ideal \({\mathfrak {a}}^h\).
Theorem 8.1
The map taking \(T_{\ell }\) to the elements \(\phi (T_{\ell }) = t_{\ell }\) in the totally real field \(E^+\) gives a surjective homomorphism of \({{\mathbb {Q}}}\)algebras \(\phi :{\mathbb {T}} \rightarrow E^+\), which factors through the quotient algebra generated by the linear endomorphisms \(T(\ell )\) on the regular differentials on PX(p).
Let \(M(E^+)\) be a vector space over \(E^+\) of dimension 1 (which is a simple \({\mathbb {T}}\)  module via the homomorphism \(\phi \)). Then Hecke’s invariant subspace in \(H^0(PX(p),\Omega ^1)\) is isomorphic to the simple module \(M(E^+) \otimes R\) of dimension \(h(p)\cdot (p1)\) over \({{\mathbb {Q}}}\), under the action of \({\mathbb {T}} \times {{\mathbb {Q}}}[{{\mathrm{PGL}}}_2(p)]\).
We will prove this result in the two sections, after reviewing the relationship between the regular differentials on the curves PX(p) and \(X_0(p^2)\) over \({{\mathbb {Q}}}\). Here we note that just as we have an action of the subgroup \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \) of \({{\mathrm{PGL}}}_2(p)\) on each component \(X^{\pm }\) of PX(p) over \(K = {{\mathbb {Q}}}(\sqrt{p})\), we note that the Hecke operator \(T(\ell )\) preserves the regular differentials on each component whenever \(\ell \) is a square modulo p. Since the Hecke operators \(T_{\ell }\) at all primes \(\ell \) which are not squares modulo p act as zero on the module \(M(E^+)\) defined above, we find that \(M(E^+)\) is also a simple module for the subalgebra of the Hecke algebra that preserves each component of the curve, and Hecke’s invariant subspace in the regular differentials on the complex curve \(X(p)^+ = \Gamma (p) \backslash {\mathfrak {H}}^*\) is isomorphic to the simple module \(M(E^+) \otimes W\) of dimension \(h\,{\cdot }\, (p1)/2\) over \({{\mathbb {C}}}\).
9 The modular curve \(X_0(p^2)\) as a quotient of PX(p)
In this section, we show how the Hecke module \(M(E^+)\) appears (with multiplicity one) in the regular differentials on the modular curve \(X_0(p^2)\) over \({{\mathbb {Q}}}\). Recall that the curve \(X_0(N)\) classifies (generalized) elliptic curves together with a cyclic Nisogeny [6]. The regular differentials on \(X_0(N\)) correspond to the cusp forms of weight 2 for the group \(\Gamma _0(N)\) with rational Fourier coefficients.
By a theorem of Weil (cf. [27]) the Dirichlet series \(\sum a_n n^{s}\) determined by the Hecke character \(\chi \) is the Lfunction of a cusp form \(\sum a_n q^n\) of weight 2 for the group \(\Gamma _0(p^2)\) with coefficients in the totally real field \(E^+\). Taking the distinct embeddings of \(E^+\) into \({{\mathbb {R}}}\), we obtain an \(h(p)\) dimensional subspace of \(H^0(X_0(p^2), \Omega ^1) \otimes {{\mathbb {R}}}\). This subspace descends to \({{\mathbb {Q}}}\) and consists of the forms of weight 2 and level \(p^2\) with complex multiplication by \({{\mathbb {Q}}}(\sqrt{p})\). Indeed, by a theorem of Serre [27] this Hecke module is determined by the condition that \(T_{\ell } = 0\) for all primes \(\ell \) which are inert in \({{\mathbb {Q}}}(\sqrt{p})\). This realizes the simple Hecke module \(M(E^+)\) in the space of regular differentials on \(X_0(p^2)\) over \({{\mathbb {Q}}}\). To realize \(M(E^+)\) in the regular differentials on PX(p), we need to recognize \(X_0(p^2)\) as a quotient of PX(p) over \({{\mathbb {Q}}}\), by the action of a split torus in the group \({{\mathrm{PGL}}}_2(p)\).
Let T be a split torus in the group \({{\mathrm{GL}}}_2(p)\) and let \(B = TU\) be a Borel subgroup containing T. These groups act on the curve X(p) over \({{\mathbb {Q}}}\), and we let X(p) / B and X(p) / T be the quotient curves. Since both subgroups contain the center Z of \({{\mathrm{GL}}}_2(p)\), both curves are also quotients of the curve X(p), by the subgroups (B / Z) and (T / Z) in \({{\mathrm{PGL}}}_2(p)\), respectively, and X(p) / T is a cyclic cover of X(p) / B of degree p. Since both subgroups map surjectively to \(({{\mathbb {Z}}}/p{{\mathbb {Z}}})^*\) under the determinant homomorphism, both of the quotient curves are geometrically connected over \({{\mathbb {Q}}}\). We will identify them with the modular curves \(X_0(p)\) and \(X_0(p^2)\), which classify pairs \(E \rightarrow E^{\prime }\) of elliptic curves with a cyclic isogeny of degree p and \(p^2\), respectively [6].
Since X(p) is the moduli of generalized elliptic curves with a full level structure at p, the quotient curve X(p) / B is the moduli of generalized elliptic curves E with a fixed line C in the ptorsion E[p]. Then C is the kernel of a cyclic pisogeny \(\phi : E \rightarrow E^{\prime } = E/C\), which identifies X(p) / B with \(X_0(p)\).
With these identifications, we obtain the following.
Proposition 7
The space regular differentials \(H^0(X_0(p),\Omega ^1)\) on the curve \(X_0(p)\) is isomorphic to the subspace of \(H^0(PX(p), \Omega ^1)\) which is fixed by (B / Z).
The space of regular differentials \(H^0(X_0(p^2),\Omega ^1)\) on the curve \(X_0(p^2)\) is isomorphic with the subspace of \(H^0(PX(p), \Omega ^1)\) which is fixed by (T / Z).
The case of \(X_0(p^2)\) is more interesting, as every irreducible representation U of \({{\mathrm{PGL}}}_2(p)\), except for the nontrivial determinant representation \(\chi \) of dimension 1, has a nonzero fixed vector under the split torus T. In fact, it follows easily from an examination of the character table of \({{\mathrm{PGL}}}_2(p)\) that every irreducible representation with a fixed vector has \(\dim U^T = 1\), except for the Steinberg representation S which has \(\dim S^T = 2\). Since every irreducible representation U of \({{\mathrm{PGL}}}_2(p)\) which occurs in the regular differentials on PX(p) has at least one nonzero fixed vector under the split torus, the Hecke modules M(U) which occur in the decomposition \(\oplus _{U} U \otimes M(U)\) of regular differentials on PX(p) must all occur in the decomposition of regular differentials of \(X_0(p^2)\): \(\oplus _U \dim U^T \otimes M(U)\).
In particular, the Hecke module \(M(E^+)\), which we saw occurred on the regular differentials of \(X_0(p^2)\), occurs in the action of the Hecke algebra \({\mathbb {T}}\) on the regular differentials of PX(p). Moreover, since the simple Hecke module \(M(E^+)\) occurs with multiplicity one on the differentials of the curve \(X_0(p^2)\) [1], there is a unique irreducible representation U of \({{\mathrm{PGL}}}_2(p)\) (which is not the trivial or the Steinberg representation) such that \(U \otimes M(E^+)\) occurs as a simple submodule of the differentials on the curve PX(p). To finish the proof of Theorem 9.1, we need to show that U is isomorphic to R, the unique discrete series associated to a quartic character of the nonsplit torus, which decomposes as \(W \oplus W^{\prime }\) when restricted to the subgroup \({{\mathrm{SL}}}_2(p)/\langle \pm 1 \rangle \). This follows from the fact that Hecke’s distinguished subspace of dimension \(h\cdot (p1)\) in \(R \otimes M(R)\) contains the forms with complex multiplication by \(K = {{\mathbb {Q}}}(\sqrt{p})\). Hence it is isomorphic to \(R \otimes M(E^+)\) as claimed.
10 Local components of some automorphic representations of \({{\mathrm{PGL}}}_2({{\mathbb {A}}})\)
The identification of the simple module \(M(E^+) \otimes R\) in the regular differentials on PX(p) over \({{\mathbb {Q}}}\) has a nice interpretation in the language of automorphic representations. We present this here.
Since F(q) has weight 2, the local component \(\pi _{\infty }\) is the discrete series for \({{\mathrm{PGL}}}_2({{\mathbb {R}}})\) of weight 2 and infinitesimal character \(\rho \). For all \(\ell \ne p\) the local component \(\pi _{\ell }\) has a vector fixed by the maximal compact subgroup \({{\mathrm{PGL}}}_2({{\mathbb {Z}}}_{\ell })\). Since it is an unramified representation of \({{\mathrm{PGL}}}_2({{\mathbb {Q}}}_{\ell })\), its isomorphism class is determined by its FrobeniusHecke parameter under the Satake isomorphism [13]. If we use the “Hecke normalization” of this isomorphism, the parameter of \(\pi _{\ell }\) is the semisimple conjugacy class in \({{\mathrm{GL}}}_2({{\mathbb {C}}})\) with characteristic polynomial \(x^2  \iota (a_{\ell })x + \ell .\) The one local representation that is difficult to read off of the classical data is the representation \(\pi _p\) of \({{\mathrm{PGL}}}_2({{\mathbb {Q}}}_p)\). This has conductor \(p^2\), so has a vector fixed by the congruence subgroup \(\Gamma _0(p^2)\) of \({{\mathrm{PGL}}}_2({{\mathbb {Z}}}_p)\).
Since the Langlands parameter of the automorphic representation \(\pi \) is induced from the Hecke character \(\chi \) of the imaginary quadratic field \(K ={{\mathbb {Q}}}(\sqrt{p})\) (choosing an embedding \(E \rightarrow {{\mathbb {C}}}\) extending the real embedding \(\iota : E^+ \rightarrow {{\mathbb {R}}}\)), the local Langlands parameter of \(\pi _p\) is obtained by inducing the local character \(\chi _p\) of \(K_p^*\). That character is tamely ramified, and its restriction to the units has order 2. Since \(\,1\) is not a square modulo p, we must have \(\chi _p(1) = 1\). Since the character \(\chi _p\) is conjugate symplectic, \(\chi _p(p) = 1\), as p is a norm from the quadratic extension \({{\mathbb {Q}}}_p(\sqrt{p})\). Hence \(\chi (p) = 1\) and \(\chi (\sqrt{p})\) is a fourth root of unity. This determines \(\chi _p\) completely, as it is trivial on the squares of units. The corresponding cyclic extension of degree 4 over \(K_p\) is Galois over \({{\mathbb {Q}}}_p\) with Galois group the quaternion group of order 8. The Langlands parameter of \(\pi _p\) is the twodimensional symplectic representation of this quaternion group.
From the local Langlands correspondence, which is known explicitly for \({{\mathrm{PGL}}}_2\), we obtain the following (which explains the appearance of the representations R and W in Hecke’s invariant subspace of forms with complex multiplication). Note that the local representations \(\pi _{\infty }\) and \(\pi _p\) are independent of the choice of embedding \(\iota : E^+ \rightarrow {{\mathbb {R}}}\).
Theorem 10.1
The local component \(\pi _p\) of \(\pi \) is the depth zero supercuspidal representation compactly induced from the irreducible representation R of \({{\mathrm{PGL}}}_2({{\mathbb {Z}}}_p)\), where R is the discrete series of dimension \((p1)\) of \({{\mathrm{PGL}}}_2(p)\) associated to a quartic character of the nonsplit torus.
The finite dimensional space of \(K_1\)invariants in \(\pi _p\) (where \(K_1\) is the first congruence subgroup of \({{\mathrm{PGL}}}_2({{\mathbb {Z}}}_p)\)) is isomorphic to the representation R of \({{\mathrm{PGL}}}_2(p)\).
Since the representations \(\pi _{\infty }\) and \(\pi _p\) are both in the discrete series, the global Jacquet–Langlands correspondence [20] shows that (for each real embedding \(\iota \) of \(E^+\)) there is an automorphic representation \(\pi ^* = \prod \pi _v^*\) of the adèlic group \((D \otimes {{\mathbb {A}}})^*/{{\mathbb {A}}}^*\), with D the quaternion algebra over \({{\mathbb {Q}}}\) ramified at \(\infty \) and p. The local components \(\pi _{\ell }^*\) are all unramified, and isomorphic to the components \(\pi _{\ell }\). The local component \(\pi _{\infty }^*\) is the trivial representation of the compact group \(D_{\infty }^*/{{\mathbb {R}}}^*\). To describe the local component of \(\pi \) at the prime p, let \(R_p\) be the maximal order in the division algebra \(D_p = D \otimes {{\mathbb {Q}}}_p\) and let P be the maximal (twosided) ideal of R. Then the local component \(\pi _p^*\) is the twodimensional representation of the finite dihedral group \(D_p^*/{{\mathbb {Q}}}_p^*(1 + P)\) of order \(2(p+1)\), induced from a quartic character of the cyclic subgroup [23, Appendix].
Note that the dimension of the space \(V_0^+\) in [23, Prop 12] is equal to the integer \(m(W) + m(W^{\vee })\) in Sect. 6. This is the number of irreducible automorphic representations \(\pi \) of \({{\mathrm{PGL}}}_2({{\mathbb {A}}})\) with local components \(\pi _{\infty }\) isomorphic to the discrete series of weight \(2,\pi _p\) isomorphic to the depth zero supercuspidal representation which is compactly induced from the representation R of \({{\mathrm{PGL}}}_2({{\mathbb {Z}}}_p)\), and \(\pi _{\ell }\) unramified for all \(\ell \ne p\).
11 Elliptic curves with complex multiplication as quotients of the Jacobians of \(X_0(p^2)\) and \(PX(p)^+\)
As Shimura observed in [29], certain regular differentials in Hecke’s distinguished subspace give elliptic curve quotients of the Jacobian \(J_0(p^2)\) of \(X_0(p^2)\) over \({{\mathbb {C}}}\). Associated to the simple module \(M(E^+)\) for the Hecke algebra, he defined an abelian variety B(p) of dimension \(h = h(p)\) which was a quotient of \(J_0(p^2)\) over \({{\mathbb {Q}}}\). In my PhD thesis [11], I showed that, over the Hilbert class field H of K, Shimura’s abelian variety B(p) decomposes as a product of h conjugate elliptic curves \(A(p)^{\sigma }\) with complex multiplication by the ring of integers of K. I will review this theory here and show how one can define a quotient of the Jacobian of \(PX(p)^+\) related to Hecke’s distinguished subspace.
The curve \(B(7) = A(7) = X_0(49)\) was the first elliptic curve for which the full conjecture of Birch and SwinnertonDyer was proved. Since \(A(7)({{\mathbb {Q}}}) = {{\mathbb {Z}}}/2{{\mathbb {Z}}}\) has rank zero and \(L(A(7)/{{\mathbb {Q}}}, 1) = \Omega _{{{\mathbb {R}}}}/2\), where \(\Omega _{{{\mathbb {R}}}}\)is the fundamental real period, the conjecture of Birch and SwinnertonDyer becomes equivalent to the statement that the Tate–Shafarevich group of A(7) over \({{\mathbb {Q}}}\) is trivial. In 1978 I was able to prove that [11, §22], and in the fall of 1986, Rubin was able to prove that for all primes \(\ell \) which were not equal to 2 or 7 [25].

The order of vanishing of \(L(B(p)/{{\mathbb {Q}}},s)\) at the point \(s=1\) is equal to 0 if \(p \equiv 7~(\mathrm{mod}~8)\) and is equal to \(h = h(p)\) if \(p \equiv 3~(\mathrm{mod}~8)\).

The MordellWeil group \(B(p)({{\mathbb {Q}}})\) is isomorphic to \({{\mathbb {Z}}}/2{{\mathbb {Z}}}\) if \(p \equiv 7~(\mathrm{mod}~8)\) and is isomorphic to \({{\mathbb {Z}}}^h\) if \(p \equiv 3~(\mathrm{mod}~8)\).
Declarations
Authors’ Affiliations
References
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