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The zeros of E4 and E6 at the elliptic points

Statement

$E_{4}$ has a simple zero at the class of $ω = e^{2 π i / 3}$ and no other zeros; $E_{6}$ has a simple zero at the class of $i$ and no other zeros. In particular $E_{4}$ and $E_{6}$ are nonzero modular forms with $E_{4} (\infty) = E_{6} (\infty) = 1$ , and $E_{4}$ does not vanish at $i$ while $E_{6}$ does not vanish at $ω$ .

Facts & Assumptions

Given: The Eisenstein series $E_{4} \in M_{4}$ , $E_{6} \in M_{6}$ with $E_{4} (\infty) = E_{6} (\infty) = 1$ (Eisenstein series are modular forms; their Fourier coefficients, The level-one Eisenstein series E_k and the weight-two series E_2), and the valence formula (The level-one valence formula).

[F1]

In the valence formula $\sum_{P} {ord}_{P} (f) / ν_{P} + {ord}_{\infty} (f) = k / 12$ , each summand ${ord}_{P} (f) / ν_{P}$ is either $0$ or at least $1 / ν_{P}$ ; the elliptic classes have $ν_{i} = 2$ and $ν_{ω} = ν_{ω + 1} = 3$ , all other classes have $ν_{P} = 1$ (The level-one valence formula, The standard fundamental domain, boundary identifications and elliptic stabilisers).

Proof

1.1F1givenalgebra

For $E_{4}$ , $k / 12 = 1 / 3$ and ${ord}_{\infty} (E_{4}) = 0$ because $E_{4} (\infty) = 1 \neq 0$ , so $\sum_{P} {ord}_{P} (E_{4}) / ν_{P} = 1 / 3$ . Every nonzero summand is at least $1 / 3$ by [F1], with equality exactly for a simple zero at a class with $ν_{P} = 3$ , and the doubles $1 / 2, 2 / 3, \dots$ and the integers $1, 2, \dots$ are all strictly larger than $1 / 3$ . Hence exactly one summand is nonzero, namely a simple zero at a class with $ν_{P} = 3$ , and the only such class is that of $ω$ . So $E_{4}$ vanishes simply at the class of $ω$ and nowhere else; in particular it does not vanish at $i$ .

2.1F1step 1.1givenalgebra∎

For $E_{6}$ , $k / 12 = 1 / 2$ and ${ord}_{\infty} (E_{6}) = 0$ , so $\sum_{P} {ord}_{P} (E_{6}) / ν_{P} = 1 / 2$ . A nonzero summand at the class of $ω$ would be at least $1 / 3$ , leaving a total of at most $1 / 6$ to be supplied by the remaining summands, of which every nonzero one is at least $1 / 2$ (at $i$ ) or $1$ (non-elliptic); this is impossible, so the class of $ω$ is not a zero. The total $1 / 2$ must then be a single summand at $i$ (any non-elliptic zero contributes at least $1$ ), and it equals $1 / 2$ exactly for a simple zero at $i$ . Hence $E_{6}$ vanishes simply at the class of $i$ and nowhere else, in particular not at $ω$ .

Depends on

Used by

The dimension of the space of level-one modular forms Corollary
The modular discriminant and the j-invariant Definition
The elliptic points of the modular group and their images under j Example
The square and hexagonal tori have j-invariants 1728 and 0 Example
The graded ring of level-one modular forms Theorem
The j-invariant uniformizes X(1) Theorem

Dependency tree · two levels

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Sources

J. S. Milne, Modular Functions and Modular Forms (v1.31, 2017) (standard reference, not scraped)
D. Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms (Universitext, Springer, 2008) (standard reference, not scraped)