What is the geometric meaning of $\omega=dx/(2y+a_1x+a_3)$ for an elliptic curve?
This question is an adjunct to MO Q1 on formal laws and L-series. Silverman (Q1) and Darmon (pg. 6) state:
The invariant holomorphic differential form (Neron differential) attached to an elliptic curve is
$\omega=dx/(2y+a_1x+a_3)$.
(Ancilliary question: Relation to Weierstrass's elliptic functions?)
I'd like to broaden the question as a community wiki to ask, "What are some interesting manifestations of this one-form in various families of elliptic curves?"
E.g., J. Hoffman in Topics in Elliptic Curves and Modular Forms (revised 2013) gives for the Jacobi quartic family of elliptic curves
$\omega=dx/(1+2\kappa x^{2}+x^{4})^{1/2}=\sum_{n=0}^{\infty}L_{n}(\kappa)x^{2n}dx$
with $L_{n}(\kappa)$ the Legendre polynomials.
Best Answer
A paper by John Tate (pg. 1 and 2) gives a clear derivation of the diff. form:
Reparametrize the elliptic curve
$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4 x+a_6$
with $p(z)=x+(a_1^2+4a_2)/12$ and $p^{'}(z)=2y+a_1x+a_3$ to obtain
$(p^{'})^2=4p^3-g_2p-g_3$, defining the Weierstrass elliptic fct., and
$\omega=dp(z)/p^{'}(z)=dz=dx/(2y+a_1x+a_3)$.
Per Dan's comment, a coordinate transformation of $x=u^2x^{'}+r$ and $y=u^3y^{'}+su^2x^{'}+t$
leaves $\omega^{'}=u\,\omega$.
Given $\sigma=p(z)$ and the inverse $z=p^{-1}(\sigma)$,
$dz=(p^{-1}(\sigma))^{'}d\sigma=(p^{-1}(\sigma))^{'}p^{'}(z)dz$, so
$(p^{-1}(\sigma))^{'}=1/p^{'}(z)$ and $dz=d\sigma/p^{'}(z)=\omega$.
The amplitwist interpretation of differentiation and inversion presented by Tristan Needham in his book Visual Complex Analysis provides a geometric interpretation of these differential relations.
Consider as an analogy $P(\theta)=sin(\theta), P^{'}(\theta)=cos(\theta),\,and\, P^2+(P^{'})^2=1$.
Update 3/13/21:
Delineating the relationships between a compositionally inverse pair of complex functions/formal power series $f(z)$ and $f^{-1}(w)$ with $f(0)=0$ collates the analogous relationships among the invariant differential above and the associated formal group law (FGL) and Dirichlet/L series as noted in Silverman's post and basic geometric constructs (the original intent of my question). I'll reference H below for "Three lectures on formal groups" by Hazewinkel.
For the inverse pair of $w=f(z)$ and $z=f^{-1}(w)$,
$$dz=(f^{-1}(w))' \; dw=(f^{-1})'(w) \cdot f'(z) \; dz,$$
so
$$(f^{-1})'(w)=\frac{1}{f'(z)}$$
for $(z,w) = (f^{-1}(w),f(z))$.
This analytic relation has the equivalent geometric interpretation in the case of real variables and curves that a tangent of the curve $y=f(x)$ at the point $(x,y)=(z,f(z))=(f^{-1}(w),w)$ is the reflection through the line $y=x$ of the tangent to the curve $y=f^{-1}(x)$ at the point $(x,y)=(f(z),z)= (w,f^{-1}(w))$; that is, more simply, the two curves are reflections of each other through the quadrant bisector $y=x$.
Let
$$ g(z) = \frac{1}{f'(z)} = (f^{-1})'(f(z)),$$
a morphed flow/evolution equation since
$$(f^{-1})'(z) = g(f^{-1}(z))$$
(see OEIS A145271 for flow equations and, in particular, a Riccati equation for $g(x)$ a quadratic in A008292).
Then
$$dw = df(z) = f'(z) \; dz = \frac{df^{-1}(w)}{(f^{-1})'(w)} = \frac{dz}{(f^{-1})'(f(z))} = \frac{dz}{g(z)} = \omega,$$
and
$$f(z) = \int_0^{z} \frac{1}{g(u)} du = \int_0^{z} \frac{1}{(f^{-1})'(f(u))} du = \int \omega .$$
Check with $ w = f(z) = \ln(1+z)$ and $z = f^{-1}(w) = e^w-1$ and, therefore, $g(z) = \frac{1}{f'(z)} = (f^{-1})'(f(u)) = 1+z$.
(For the invariant differential for elliptic curves above and in Silverman's post, see p. 61 of H.)
Now define the associated infinitesimal generator/vector, as suggested by the flow equation above,
$$ D_{w} = \frac{d}{dw} = \frac{d}{df(z)} = \frac{1}{f'(z)}\; \frac{d}{dz} = (f^{-1})'(f(z)) \; \frac{d}{dz} = g(z) \frac{d}{dz} $$
(all derivatives are partial derivatives w.r.t. the designated variable).
Then
$$e^{(t \; g(z) \; D_z )} \; z = e^{(t \; D_{w}) } \; f^{-1}(w) = f^{-1}(w + t) = f^{-1}(f(z) + t),$$
and
$$e^{(t \; g(z) \; D_z )} \; z \; |_{z=0}= f^{-1}(t).$$
Substitutions give the FGL
$$F(x,y) = f^{-1}(f(x) + f(y)),$$
so
$$ \frac{dF(x,y)}{dy} \; |_{y=0} = (f^{-1})'(f(x)) \cdot f'(0) = g(x)$$
if $f'(0) = 1$.
For $g(z)$ a quadratic, the elliptic FGL of Examples 3 and 63 of Buchstaber and Bunkova's "Elliptic formal group laws, integral Hirzebruch genera, and Krichever genera" apply (see also the A008292 formulas dated Sep 18 2014). The associated Riccati equation is related to soliton solutions of the Kdv equation presented in my response to the MO-Q "Is there an underlying explanation for the magical powers of the Schwarzian derivative?" The quadratic $g(z)$ is related to the ratio of the Schwarzian derivatives of an inverse pair of functions, one of which is the integral of the soliton solution, and to the velocity of the soliton.
Now characterize $f(z)$ as a logarithmic series (with $a_n =1$ for a canonical FGL),
$$f(z) = -\ln(1-a.z) = \sum_{n \ge 1} \frac{(a.z)^n}{n} = \sum_{n \ge 1} a_n \; \frac{z^n}{n}$$
(see p. 55 of H on the use of the term logarithm in the FGL formalism).
Then
$$qf'(q) = \frac{q}{g(q)} = \frac{a.q}{1-a.q} = \sum_{n \ge 1} a_n q^n \; ,$$
and, with $q = e^{-t}$, the normalized Mellin transform
$$\int_{0}^{\infty} \frac{e^{-t}}{g(e^{-t})} \; \frac{t^{s-1}}{(s-1)!} \; dt = \sum_{n \ge 1} \; a_n \; \int_{0}^{\infty} \;e^{-nt} \; \frac{t^{s-1}}{(s-1)!} \; dt = \sum_{n \ge 1} \; \frac{a_n}{n^s} $$
generates an associated Dirichlet series (see p. 53 of H). For the Riemann zeta function, $g(z) = 1-z$.
The tangent plays multiple roles in the multiplicative and compositional inversions above, so one should expect to encounter free probability, symmetric function theory, Hopf algebras, and the associated geometry and combinatorics as well. For more on this, see H, the recently revised "Formal Groups, Witt vectors and Free Probability" by Friedrich and McKay, and refs/links in the OEIS.