Derived Hecke Algebra for Weight One Forms

ABSTRACT

We study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit. We verify the truth of the conjecture numerically, for the weight one forms of level 23 and 31, and many derived Hecke operators at primes less than 200. Our computation depends in an essential way on Merel’s evaluation of the pairing between the Shimura and cuspidal subgroups of J₀(q).

KEYWORDS:

• Modular forms
• number theory

Notes

1 We apologize for the perhaps pedantic distinction between Ad*ρ and Ad⁰ρ. Since we will shortly be localizing at a prime larger than 2, one could identify them by means of the pairing trace(AB). However, when working in a general setting, one really needs to use Ad*,  and following this convention makes it easier to compare with [Venkatesh nd].

2 It seems likely that the two sides are actually equal in ${(\mathbb{Z}/q)}_p^{*}$ but we do not prove this.

3 Recently, Lecouturier has proposed a very interesting generalization of the conjectural equality (5–10(5--12) $${\mathrm{U}}_g\otimes \mathbb{Z}\left[\frac{1}{6}\right]\simeq {\mathcal{O}}_K^{\left(1\right)}\otimes \mathbb{Z}\left[\frac{1}{6}\right],$$(5--12) ) to the case when ϖ_Merel is zero modulo p and has verified it numerically in some cases.

4 To be absolutely clear, we write out the meaning of this statement. We understand $$\begin{array}{c}\hfill L:=a_1\left(G_{\mathfrak{I}}^{\mathrm{proj}}\right)\otimes {\left({\varpi }_{\mathrm{Merel}}\right)}_p,R:={\theta }_q\left(u\right)\end{array}$$as elements of $\mathcal{O}\otimes {(\mathbb{Z}/q)}_p^{*}$; and the statement above means that if we reduce $\bar{L},\bar{R}$ to $\mathcal{O}/p\otimes {(\mathbb{Z}/q)}_p^{*}$, then $\bar{L}=\alpha \bar{R}$, in the sense of (3–4(3--4) $$x=\alpha y,$$(3--4) ).

5 The primes q for which ρ(Frob_q) is a 3-cycle also are Taylor–Wiles primes, but it is then easy to see that T_{q, z}g = 0 for such q. To verify this, one can use the fact—notation as in (5–3(5--5) $$⟨\mathfrak{S},G^{\mathrm{proj}}⟩\propto {\theta }_q\left(u\right),\text{equality}\text{in}{(\mathbb{Z}/q)}^{*}\otimes k,$$(5--5) —that the Atkin–Lehner involution at q for X₀₁(qN) acts by − 1 on ${\mathfrak{S}}_X$, but it acts by χ(q) on G, where χ is the quadratic Nebentypus character for g.

6 with an apology to 21st century readers, see below...

Additional information

Funding

M.H.’s research received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 290766 (AAMOT). M.H. was partially supported by NSF Grant DMS-1404769. A.V. research was partially supported by the NSF and by the Packard foundation.