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Nonvanishing of the lattice discriminant

Statement

Let $Λ = Z ω_{1} + Z ω_{2}$ be a full complex lattice with oriented basis, with Weierstrass invariants $g_{2} = 60 G_{4}$ , $g_{3} = 140 G_{6}$ and discriminant $Δ (Λ) : = g_{2}^{3} - 27 g_{3}^{2}$ (Weierstrass p function, Weierstrass cubic differential equation). Then:

$Δ (Λ) \neq 0$ ;
the polynomial $4 x^{3} - g_{2} x - g_{3}$ has three distinct roots, namely the values $e_{j} = ℘_{Λ} (h_{j})$ at the three nonzero half-periods $h_{1} = ω_{1} / 2$ , $h_{2} = ω_{2} / 2$ , $h_{3} = (ω_{1} + ω_{2}) / 2$ (Degree two of ℘ and its four branch points);
consequently the projective cubic $C_{Λ} = {[X : Y : Z] \in {C P}^{2} : Y^{2} Z = 4 X^{3} - g_{2} X Z^{2} - g_{3} Z^{3}}$ is nonsingular in the Jacobian-rank sense at every point, including its unique point at infinity $O = [0 : 1 : 0]$ (Local holomorphic charts on nonsingular complex algebraic curves).

Facts & Assumptions

Given: A full complex lattice $Λ = Z ω_{1} + Z ω_{2}$ with oriented basis; the Weierstrass function $℘ = ℘_{Λ}$ with invariants $g_{2} = 60 G_{4}$ , $g_{3} = 140 G_{6}$ and $Δ = g_{2}^{3} - 27 g_{3}^{2}$ ; the half-periods $h_{1} = ω_{1} / 2$ , $h_{2} = ω_{2} / 2$ , $h_{3} = (ω_{1} + ω_{2}) / 2$ ; and the values $e_{j} = ℘ (h_{j})$ .

[F1]

$Λ \subseteq C$ is a discrete full lattice and $C / Λ = T_{Λ}$ is its torus (Complex lattice and quotient torus); $℘$ is the Weierstrass function of $Λ$ and $G_{4} = \sum_{ω \neq 0} ω^{- 4}$ , $G_{6} = \sum_{ω \neq 0} ω^{- 6}$ are absolutely convergent, with $g_{2} = 60 G_{4}$ , $g_{3} = 140 G_{6}$ (Weierstrass p function, Weierstrass cubic differential equation). The standard affine charts of ${C P}^{2}$ are the sets where one homogeneous coordinate is nonzero, with $[X : Y : Z] \leftrightarrow (X / Z, Y / Z)$ on ${Z \neq 0}$ and $[X : Y : Z] \leftrightarrow (X / Y, Z / Y)$ on ${Y \neq 0}$ (projective space points).

[F2]

Every nonzero half-period $h$ of $Λ$ , that is, $h \notin Λ$ with $2 h \in Λ$ , satisfies $℘^{'} (h) = 0$ ; the zeros of $℘^{'}$ are exactly the $Λ$ -translates of $h_{1}, h_{2}, h_{3}$ , each of order one; the classes $[h_{1}], [h_{2}], [h_{3}]$ and the values $e_{1}, e_{2}, e_{3} \in C$ are three distinct values each (Degree two of ℘ and its four branch points).

[F3]

$(℘^{'})^{2} = 4 ℘^{3} - g_{2} ℘ - g_{3}$ on $C ∖ Λ$ (Weierstrass cubic differential equation).

[F4]

(a) If $f$ is a monic polynomial of degree $n \geq 1$ over a field and $f (t) = \prod_{i = 1}^{n} (t - α_{i})$ in a splitting field, then $Disc (f) = \prod_{i < j} (α_{i} - α_{j})^{2}$ , and $Disc (f) = 0$ if and only if $f$ has a repeated root (The discriminant of a monic polynomial as the coefficient expression of $Δ_{n}^{2}$ , The discriminant is $\prod_{i < j} (α_{i} - α_{j})^{2}$ and vanishes exactly when a monic polynomial has a repeated root). (b) Every polynomial $f \in C [x]$ of degree $n \geq 1$ factors as $f = c \prod_{j = 1}^{r} (x - α_{j})^{m_{j}}$ with $c \in C^{\times}$ and $m_{1} + \dots + m_{r} = n$ , and these roots and multiplicities are uniquely determined (A complex polynomial of degree $n$ has exactly $n$ roots counted with multiplicity).

[F5]

(Implicit function theorem.) If $f$ is holomorphic near $(u_{0}, v_{0}) \in C^{2}$ , $f (u_{0}, v_{0}) = 0$ and $\partial f / \partial v (u_{0}, v_{0}) \neq 0$ , then near $u_{0}$ there is a unique holomorphic $φ$ with $φ (u_{0}) = v_{0}$ and $f (u, φ (u)) = 0$ , and locally $f (u, v) = 0$ if and only if $v = φ (u)$ (The holomorphic implicit function theorem).

[F6]

A curve in a standard affine chart of ${C P}^{2}$ that near a point is the common zero set of one holomorphic function of two variables, with nonzero complex gradient at that point, is nonsingular there in the Jacobian-rank sense of the chart lemma, whose Jacobian has rank $N - 1 = 1$ ; the lemma then supplies a local parameter for the curve (Local holomorphic charts on nonsingular complex algebraic curves).

Proof

technique · direct

1.1F2F3givenalgebra

(The half-period values are roots of the cubic.) For each $j$ the point $h_{j}$ satisfies $h_{j} \notin Λ$ and $2 h_{j} \in Λ$ , so [F2] gives $℘^{'} (h_{j}) = 0$ . Since $h_{j} \notin Λ$ , the differential equation [F3] may be evaluated at $h_{j}$ : $0 = ℘^{'} (h_{j})^{2} = 4 ℘ (h_{j})^{3} - g_{2} ℘ (h_{j}) - g_{3} = 4 e_{j}^{3} - g_{2} e_{j} - g_{3}$ . Hence each $e_{j}$ is a root of $p (x) : = 4 x^{3} - g_{2} x - g_{3}$ .

1.2F2given

(The three values are distinct.) By [F2] the three nonzero half-period classes $[h_{1}], [h_{2}], [h_{3}]$ and the three values $e_{1}, e_{2}, e_{3}$ are distinct.

2.1F4givenstep 1.1step 1.2algebra

(The discriminant does not vanish.) Put $P : = - \frac{g_{2}}{4}$ and $Q : = - \frac{g_{3}}{4}$ , so that $q (x) : = x^{3} + P x + Q$ is the monic cubic with $p = 4 q$ . By step 1.1 each $e_{j}$ is a root of $q$ , and by step 1.2 the three roots $e_{1}, e_{2}, e_{3}$ are distinct; since $\deg q = 3$ , F4 gives $q (x) = (x - e_{1}) (x - e_{2}) (x - e_{3})$ — the leading coefficient is $1$ and the three roots exhaust the multiplicities — and comparing coefficients with $x^{3} + P x + Q$ gives $e_{1} + e_{2} + e_{3} = 0, e_{1} e_{2} + e_{1} e_{3} + e_{2} e_{3} = P, e_{1} e_{2} e_{3} = - Q .$ By F4 applied to this split form, $Disc (q) = \prod_{i < j} (e_{i} - e_{j})^{2}$ , which is nonzero because no factor $e_{i} - e_{j}$ with $i < j$ vanishes. I claim $Disc (q) = - 4 P^{3} - 27 Q^{2} .$ Indeed put $u : = e_{1} + e_{2}$ and $v : = e_{1} e_{2}$ ; the coefficient relations give $e_{3} = - u$ , hence $P = e_{1} e_{2} + e_{1} e_{3} + e_{2} e_{3} = v - u^{2}$ and $Q = - e_{1} e_{2} e_{3} = u v$ , that is $u^{2} = v - P$ and $Q^{2} = u^{2} v^{2} = (v - P) v^{2}$ . Moreover $(e_{1} - e_{2})^{2} = u^{2} - 4 v = (v - P) - 4 v = - 3 v - P,$ and $e_{1} - e_{3} = 2 e_{1} + e_{2}$ , $e_{2} - e_{3} = e_{1} + 2 e_{2}$ , so $(e_{1} - e_{3}) (e_{2} - e_{3}) = 2 e_{1}^{2} + 5 e_{1} e_{2} + 2 e_{2}^{2} = 2 (u^{2} - 2 v) + 5 v = 2 (v - P) + v = 3 v - 2 P .$ Therefore $Disc (q) = (e_{1} - e_{2})^{2} (e_{1} - e_{3})^{2} (e_{2} - e_{3})^{2} = (- 3 v - P) (3 v - 2 P)^{2} = - (3 v + P) (9 v^{2} - 12 P v + 4 P^{2}) = - (27 v^{3} - 27 P v^{2} + 4 P^{3}) = - 4 P^{3} - 27 (v^{3} - P v^{2}) = - 4 P^{3} - 27 Q^{2} .$ Substituting $P = - g_{2} / 4$ and $Q = - g_{3} / 4$ gives $Disc (q) = - 4 (- \frac{g_{2}}{4})^{3} - 27 (- \frac{g_{3}}{4})^{2} = \frac{g_{2}^{3}}{16} - \frac{27 g_{3}^{2}}{16} = \frac{Δ}{16},$ so $Δ = 16 Disc (q) \neq 0$ .

3.1F1F5F6step 2.1step 1.2algebra

(Smoothness of the projective cubic.) The curve $C_{Λ}$ meets the chart ${Z = 0}$ only in points with $0 = Y^{2} \cdot 0 = 4 X^{3}$ , that is $X = 0$ , so $O = [0 : 1 : 0]$ is its unique point at infinity; in the chart ${Y \neq 0}$ with coordinates $u = X / Y$ , $v = Z / Y$ , the curve is the zero set of $G (u, v) = v - 4 u^{3} + g_{2} u v^{2} + g_{3} v^{3}$ , and $\partial G / \partial v (0, 0) = 1 + 2 g_{2} u v + 3 g_{3} v^{2} ∣_{(0, 0)} = 1 \neq 0$ , so by [F5] there is a holomorphic $φ$ with $v = φ (u)$ on $G = 0$ near $O$ : the curve is nonsingular at $O$ with local parameter $u$ , by [F6]. Every other point of $C_{Λ}$ lies in the chart ${Z \neq 0}$ , where the curve is the zero set of the polynomial $f (x, y) = y^{2} - 4 x^{3} + g_{2} x + g_{3}$ in the affine coordinates $x = X / Z$ , $y = Y / Z$ ; its gradient $\nabla f = (- 12 x^{2} + g_{2}, 2 y)$ is nonzero at every point of $f = 0$ : if $y = 0$ and $f (x, 0) = 0$ , then $0 = 4 x^{3} - g_{2} x - g_{3} = p (x) = 4 (x - e_{1}) (x - e_{2}) (x - e_{3})$ by step 2.1, so $x = e_{j}$ for some $j$ and $- 12 x^{2} + g_{2} = - p^{'} (e_{j}) = - 4 \prod_{k \neq j} (e_{j} - e_{k}) \neq 0$ by the distinctness in step 1.2 — a contradiction; hence $\nabla f \neq 0$ on the affine curve, which by [F6] is nonsingular in the Jacobian-rank sense at each of its points. Thus every point of $C_{Λ}$ , including the unique point at infinity $O = [0 : 1 : 0]$ , is nonsingular in the Jacobian-rank sense.

4.1

(Conclusion.) Step 1.1 exhibits $e_{1}, e_{2}, e_{3}$ as roots of $4 x^{3} - g_{2} x - g_{3}$ with distinct classes of half-periods, step 1.2 makes them distinct values, step 2.1 proves $Δ = 16 Disc (q) \neq 0$ , and step 3.1 proves that the projective cubic $Y^{2} Z = 4 X^{3} - g_{2} X Z^{2} - g_{3} Z^{3}$ is nonsingular in the Jacobian-rank sense at all its points, including its unique point at infinity $O = [0 : 1 : 0]$ . ∎

Remarks

The three distinct roots are the branch values of the degree-two map $℘ : T_{Λ} \to \hat{C}$ , and the nonvanishing of $Δ$ is the statement that this cubic is a smooth elliptic curve rather than a nodal or cuspidal degeneration; the smoothness at infinity is checked in the chart where $Z / Y$ is the dependent variable, since $O$ is never in the chart $Z \neq 0$ . The proof uses no elliptic integral and no Riemann-Roch theorem: the distinctness of the roots comes from the fibre description of $℘$ , and the algebraic discriminant is computed directly from the three roots by comparing coefficients, without naming the roots. This is the theorem that rules out the degenerate cubic of A singular cubic outside the lattice family ↗ for coefficients coming from a lattice.

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Sources

C. T. McMullen, Advanced Complex Analysis, Math 213a course notes, Ch. 5 §5.1, pp. 79-90 (standard reference, not scraped)
J. S. Milne, Modular Functions and Modular Forms, Ch. 3, pp. 41-47 (standard reference, not scraped)
NIST Digital Library of Mathematical Functions, §23.2 (standard reference, not scraped)