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If we consider the modular curve $X = X_0(N)$ as a curve over $\mathbb C$ then one can describe the jacobian $J(X)$ as $H^0(X,\Omega^1_X)^\vee/H_1(X,\mathbb Z)$ as one can do for any curve $X$. However when $X=X_0(N)$ there is more structure and we can describe $H^0(X,\Omega^1_X) \cong S_2(\Gamma_0(N),\mathbb C)$ in terms of modular forms and $H_1(X,\mathbb Z) \cong {\mathbb S}_2(\Gamma_0(N),\mathbb Z)$ in terms of integer-valued modular symbols.

Now since $J_0(N)$ is a jacobian, it has a principal polarization $\phi :J_0(N) \to J_0(N)^\vee$. This polarization hence also induces a linear map on modular symbols $\phi_*:{\mathbb S}_2(\Gamma_0(N),\mathbb Z) \to Hom({\mathbb S}_2(\Gamma_0(N),\mathbb Z), \mathbb Z)$. My question is:

Is there a nice (combinatorial) description of the principal polarization on $J_0(N)$ in terms of modular symbols? I.e. can one explicitly compute the map $\phi_*$ described above?

I'm also interested to know if some computer software can already compute $\phi_*$ for values of $N$ around 200 say.

Addendum/Edit:

I am aware of the paper Intersection sur des courbes modulaires by Loic Merel. That does compute intersections between certain homology spaces. However those are relative homology spaces that sometimes have presentations that are not compatible with the standard way of representing modular symbols and it seems to stop short of computing it on the spaces I am interested in.

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    $\begingroup$ There is a natural map $H_1(Y_0(N),\mathbb{Z}) \to H_1(X_0(N),\mathbb{Z})$ which is computed explicitly in Borisov-Gunnells, Toric modular forms and nonvanishing of L-functions, Section 2. Given two modular symbols $\gamma_1,\gamma_2$ in $H_1(X_0(N),\mathbb{Z})$, you can use linear algebra to compute $\widetilde{\gamma}_2$ in $H_1(Y_0(N),\mathbb{Z})$ mapping to $\gamma_2$, and then use Merel's formula to compute $\gamma_1 \bullet \widetilde{\gamma}_2$. $\endgroup$
    – François Brunault
    Commented Dec 4, 2022 at 16:57
  • $\begingroup$ Another approach (which I haven't looked at in detail) would be to use the fact that the Petersson scalar product on $S_2(\Gamma_0(N))$ is dual to the intersection pairing. Merel has proved an explicit formula for $\langle f_1, f_2 \rangle$ in terms of the modular symbols attached to $f_1$ and $f_2$, see Symboles de Manin et valeurs de fonctions L, Section 4. $\endgroup$
    – François Brunault
    Commented Dec 4, 2022 at 16:57
  • $\begingroup$ I was pointed also by Timo Keller via e-mail that William Stein has implemented the intersection pairing in magma a long time ago magma.maths.usyd.edu.au/magma/handbook/text/1675#18964 using the ideas from Loic Merel with a little bit of extra work. So this does answer my question for the purposes I was needing it. $\endgroup$
    – Maarten Derickx
    Commented Dec 7, 2022 at 16:38

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