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FALSE: the weight-two Eisenstein series E_2 is a modular form

Statement

The function $E_{2} (τ) = 1 - 24 \sum_{n \geq 1} σ_{1} (n) q^{n}$ satisfies the weight-two transformation law $f (τ + 1) = f (τ)$ , and its completion $E_{2}^{*} (τ) = E_{2} (τ) - \frac{3}{π ℑ τ}$ is real-analytic and transforms like a weight-two form, so one might expect $E_{2}$ itself to be a modular form of weight $2$ . This is false.

Facts & Assumptions

Given: $E_{2} (τ) = 1 - 24 \sum_{n \geq 1} σ_{1} (n) q^{n}$ on $H$ (The level-one Eisenstein series E_k and the weight-two series E_2), and the weight-two transformation law of Level-one modular forms and cusp forms.

[F1]

$E_{2} (- 1 / τ) = τ^{2} E_{2} (τ) - \frac{6 i}{π} τ$ for every $τ \in H$ (The transformation law of the weight-two Eisenstein series E_2, The modular group and its action on the upper half-plane).

[F2]

A modular form $f$ of weight $2$ satisfies $f (- 1 / τ) = τ^{2} f (τ)$ (Level-one modular forms and cusp forms).

Refutation

1.1F1givenalgebra

Evaluating [F1] at $τ = i$ , where $- 1 / i = i$ and $i^{2} = - 1$ , gives $E_{2} (i) = i^{2} E_{2} (i) - \frac{6 i}{π} i = - E_{2} (i) + \frac{6}{π}$ , because $1 / i = - i$ and $i^{2} = - 1$ ; hence $2 E_{2} (i) = \frac{6}{π}$ and therefore $E_{2} (i) = \frac{3}{π} \neq 0$ .

2.1F1F2step 1.1givenalgebra∎

A weight-two modular form would satisfy, by [F2] at $τ = i$ , the equation $f (i) = i^{2} f (i) = - f (i)$ , hence $f (i) = 0$ ; but $E_{2} (i) = 3 / π \neq 0$ by 1.1. Moreover [F1] shows directly that $E_{2} (- 1 / i) = E_{2} (i) = 3 / π$ while $i^{2} E_{2} (i) = - 3 / π$ , so $E_{2} (- 1 / τ) \neq τ^{2} E_{2} (τ)$ at $τ = i$ . Hence $E_{2}$ is not a modular form of weight $2$ ; the correctly transforming object is the non-holomorphic completion $E_{2}^{*}$ .

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Sources

D. Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms (Universitext, Springer, 2008) (standard reference, not scraped)
C. T. McMullen, Advanced Complex Analysis, Math 213a course notes (Harvard, 2010) (standard reference, not scraped)