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A pair of elliptic curves $E_1, E_2$ over a field $k$ are $n$-isogenous if there exists a non-constant rational map $\phi: E_1 \to E_2$ defined over the algebraic closure $\overline{k}$ of degree $n$ that preserves the group structure. *Usually* there is zero or one $n$-isogenies between two elliptic curves, but this is not always the case. The modular polynomial $\Phi_{n}(X,Y) \in \mathbb{Z}[X,Y]$ of level $n$ is a bivariate symmetric polynomial $\Phi_n(X, Y) \in \mathbb{Z}[X, Y]$ whose $k$-roots $(j_1, j_2)$ correspond to pairs of $n$-isogenous elliptic curves over $\overline{k}$ with $j$-invariants $j_1, j_2$. The modular curve $X_0(n)$ is a smooth projective curve which is a coarse moduli space for pairs of $n$-isogenous elliptic curves and is birationally equivalent to the affine curve defined by $\Phi_n(X, Y) = 0$. The singular points of $\Phi_n(X,Y)=0$ correspond to elliptic curves with more than one $n$-isogeny between them, and by resolving the singularity we can recover the full set of $n$-isogenies. In this talk we give background on $\Phi_n$ and $ X_0(n)$ and demonstrate how to recover isogenies from singular points of $\Phi_n(X,Y)$.
Fano mirror symmetry is a duality between Fano manifolds and Landau-Ginzburg models. Predictions include: Certain D-modules associated with given mirror pairs are isomorphic, and certain integral structures on their spaces of flat sections correspond to each other. I will discuss this picture using the projective line. Then I describe the current progress for a larger class of Fano manifolds called flag varieties.