Title:  What is Gauss-Manin Connection in Disguise?
Author: Hossein Movasati
Let
$$
P(x):= 4(x-t_1)^3+t_2(x-t_1)+t_3.
$$
 We have
$$
\begin{pmatrix}
d\left(\int\frac{dx}{\sqrt{P(x)}}\right)\\
d\left(\int\frac{xdx}{\sqrt{P(x)}}\right)\
\end{pmatrix}
=
\frac{1}{\Delta}\begin{pmatrix}
{-\frac{3}{2}t_1\alpha-\frac{1}{12}d\Delta,} &
{\frac{3}{2}\alpha} \\
{\Delta dt_1-\frac{1}{6}t_1d\Delta-(\frac{3}{2}t_1^2+\frac{1}{8}t_2)\alpha}, &
{\frac{3}{2}t_1\alpha+\frac{1}{12}d\Delta}
\end{pmatrix}
\begin{pmatrix}
\int\frac{dx}{\sqrt{P(x)}}\\
\int\frac{xdx}{\sqrt{P(x)}}\
\end{pmatrix}
$$
where
$$
\Delta:=27t_3^2-t_2^3,\ \alpha:=3t_3dt_2-2t_2dt_3.
$$
The above data is the Gauss-Manin connection of the family of elliptic curves \(y^2=P(x)\) before the invention of cohomology theories (before 1900)!
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Let \(A\)  be the 2 times  2 matrix above. There are unique vector fields \(e,f,h\) in the parameter space  \(t=(t_1,t_2,t_3)\) such that
$$
A(h)= \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}, \quad A(e)=\begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix},
\quad A(f)=\begin{bmatrix} 0 & 0\\ 1 & 0 \end{bmatrix}
$$
In fact
\begin{equation}
\label{ramanve}
e=-(t_1^2-\frac{1}{12}t_2)\frac{\partial}{\partial t_1}-(4t_1t_2-6t_3)\frac{\partial}{\partial t_2}-
(6t_1t_3-\frac{1}{3}t_2^2)\frac{\partial}{\partial t_3}.
\end{equation}
$$
h=-6t_3\frac{\partial }{\partial t_3}-4t_2\frac{\partial }{\partial t_2}-2t_1\frac{\partial}{\partial t_1},\ f=\frac{\partial}{\partial t_1}
$$
and we get the  Lie Algebra \(\mathfrak{sl}_2\):
$$
[h,e] = 2e, \quad [h,f] = -2f, \quad [e,f] = h.
$$
After a letter of P. Deligne for my lecture notes, I call these vector fields the Gauss-Manin connection in disguise.
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S. Ramanujan in 1916 proved that the Eisenstein series
$$
g_{k}(\tau)=a_k\left (1+(-1)^k\frac{4k}{B_k}\sum_{n\geq
1}\sigma_{2k-1}(n)e^{2\pi i \tau n}\right ),\ \  k=1,2,3, \ \tau\in{\mathbb H},
$$
$$
B_1=\frac{1}{6},\
B_2=\frac{1}{30},\ B_3=\frac{1}{42},\ \ldots,\
\sigma_i(n):=
\sum_{d\mid n}d^i, \ \
(a_1,a_2,a_3)=(\frac{2\pi i}{12},12(\frac{2\pi i }{12})^2 ,
8(\frac{2\pi i}{12})^3),
$$
satisfy the ODE's
\begin{equation}
\label{raman}
 \left \{ \begin{array}{l}
\dot t_1=t_1^2-\frac{1}{12}t_2 \\
\dot t_2=4t_1t_2-6t_3 \\
\dot t_3=6t_1t_3-\frac{1}{3}t_2^2
\end{array} \right.,\ \ \ \ \ \ \ \ \ \ \ \ \dot t:= \frac{\partial}{\partial \tau}
\end{equation}
This is the vector field \(-e\) written in the ODE form.
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Let
$$
P(x):= 4(x-t_1)(x-t_2)(x-t_3)
$$ 
We have
$$
\begin{pmatrix}
d(\int\frac{dx}{\sqrt{P(x)}})\\
d(\int\frac{xdx}{\sqrt{P(x)}})\
\end{pmatrix}
=
\left( \frac{dt_1}{2(t_1-t_2)(t_1-t_3)}
\begin{pmatrix}
{-t_1}&{1}\\
{t_2t_3-t_1(t_2+t_3)}&{t_1}
\end{pmatrix}+\cdots
\right )
\begin{pmatrix}
\int\frac{dx}{\sqrt{P(x)}}\\
\int\frac{xdx}{\sqrt{P(x)}}\
\end{pmatrix}
$$
Let \(A\)  be the 2 times  2 matrix above. There is a unique vector field \(e\) in the parameter space such that
$$
A(e)=\begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix}
$$
We have
\begin{equation}
e:= (t_1(t_2+t_3)-t_2t_3)\frac{\partial }{\partial t_1}+
(t_2(t_1+t_3)-t_1t_3)\frac{\partial }{\partial t_2}+
 ( t_3(t_1+t_2)-t_1t_2)\frac{\partial }{\partial t_3}
\end{equation}
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In 1881, G. Halphen considered the non-linear differential system
\begin{equation}
{\rm H}:\label{halphen}
\left \{ \begin{array}{l}
\dot t_1=t_1(t_2+t_3)-t_2t_3\\
\dot  t_2= t_2(t_1+t_3)-t_1t_3 \\
 \dot  t_3= t_3(t_1+t_2)-t_1t_2
\end{array} \right. .
\end{equation}
which originally appeared in G. Darboux's work in 1878 on triply orthogonal  surfaces in \(\mathbb R^3\).  Halphen expressed a solution of the system (\ref{halphen}) in terms of the logarithmic derivatives  of the null theta functions,
\begin{eqnarray*}
u_1&=& 2(\ln \theta_4(0|\tau))',\\
u_2&=&2(\ln \theta_2(0|\tau))', \ \ \ \ \  '=\frac{\partial}{\partial \tau}\\
u_3&=&2(\ln \theta_3(0|\tau))'.
\end{eqnarray*}
where
$$
\left \{ \begin{array}{l}
\theta_2(0|\tau):=\sum_{n=-\infty}^\infty q^{\frac{1}{2}(n+\frac{1}{2})^2}
\\
\theta_3(0|\tau):=\sum_{n=-\infty}^\infty q^{\frac{1}{2}n^2}
\\
\theta_4(0|\tau):=\sum_{n=-\infty}^\infty (-1)^nq^{\frac{1}{2}n^2}
\end{array} \right.,
\ q=e^{2\pi i \tau},\  \tau\in \mathbb H.
$$
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I want to generalize the above picture beyond elliptic curves and modular forms:
  1. \(t_1,t_2,t_3\) are coordinates on a moduli space.
  2. Hodge theory back to its origin (study of multiple integrals)
  3. Differential modular/automorphic forms.
  4. Beyond classical automorphic forms (generating function of Gromov-witten invariants)
  5. Arithmetic and dynamics of Darboux-Halphen-Ramanujan type differential equations
  6. Moduli spaces inside moduli spaces and holomorphic foliations.
  7. The study of the Noether-Lefschetz and Hodge loci, Deligne's absolute Hodge cycles etc.
  8. ...........

During the last years I have generalized the above picture in many different ways. For articles before 2014 see my publications and after 2014 see the webpage of the project Gauss-Manin connection in disguise.
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My favorite generalization
\begin{equation}
\label{lovely}
\left \{ \begin{array}{l}
\dot t_0=\frac{1}{t_5}
(6\cdot 5^4t_0^5+t_0t_3-5^4t_4)
\\
\dot t_1=\frac{1}{t_5}
(-5^8t_0^6+5^5t_0^4t_1+5^8t_0t_4+t_1t_3)
 
\\
\dot t_2=\frac{1}{t_5}
(-3\cdot 5^9t_0^7-5^4t_0^5t_1+2\cdot 5^5t_0^4t_2+3\cdot 5^9 t_0^2t_4+5^4t_1t_4+2t_2t_3)
\\
\dot t_3=\frac{1}{t_5}
(-5^{10}t_0^8-5^4t_0^5t_2+3\cdot 5^5t_0^4t_3+5^{10}t_0^3t_4+5^4t_2t_4+3t_3^2)

\\
\dot t_4=\frac{1}{t_5}
(5^6t_0^4t_4+5t_3t_4)
\\
\dot t_5=\frac{1}{t_5}
(-5^4t_0^5t_6+3\cdot 5^5t_0^4t_5+2t_3t_5+5^4t_4t_6)
\\
\dot t_6=\frac{1}{t_5}
(3\cdot 5^5t_0^4t_6-5^5t_0^3t_5-2t_2t_5+3t_3t_6)
 
\end{array} \right.
\end{equation}
The computation of this ODE is done in the article Modular-type functions attached to mirror quintic Calabi-Yau varieties.
I turned this article into a 200 pages monograph, see here.





q expansion for mirror quintic






I would love to find some enumerative meaning or some arithmetic  for these numbers!!!!!











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The Yukawa coupling
\begin{eqnarray*}
Y &=& \frac{5^8(t_4-t_0^5)^2}{t_5^3}= (5+2875 \frac{q}{1-q}+ 609250\cdot 2^3\frac{q^2}{1-q^2}+ 317206375\cdot 3^2 \frac{q^3}{1-q^3} +
\cdots+ n_d d^3\frac{q^d}{1-q^d}+\cdots
\end{eqnarray*}
Genus one topological partition \(F_1:=\ln(t_4^{\frac{25}{12}}(t_4-t_0^5)^{ \frac{-5}{12}}t_5^{\frac{1}{2}})\):
$$
q\frac{\partial}{\partial q} F_1=
\frac{25}{12}-\sum_{n=1}(\sum_{r|n}d_r)\frac{ nq^ {n}}{1-q^ {n}}-\frac{1}{12}\sum_{s=1}^\infty\frac{sn_s q^ s}{(1-q^ s)}
$$
Here  \(n_d\) is the virtual number of rational curves in a generic quintic threefold
$$
(n_d)_{d\in \mathbb N}=(2875,609250,317206375,242467530000,229305888887625,\cdots)
$$
and \(d_r\)  is the virtual number of elliptic curves of degree $r$ in a generic quintic threefold
$$
(d_r)_{r\in \mathbb N}=(0,0,609250,3721431625,12129909700200,31147299733286500,\cdots).
$$
We have also genus \(g\) topological string partition functions, see here.