# Adriana Salerno and Ursula Whitcher, Diagonal pencils and Hasse-Witt invariants (Wednesday, July 20)

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Calabi-Yau threefolds generalize elliptic curves, and K3s are somewhere in-between. 
Idea: use mirror symmetry to prove arithmetic properties of higher dimensional varieties, generalizing things we know about elliptic curves.
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Consider the Legendre family $X_\psi: y^2 = x(x-1)(x-\psi)$ satisfies Picard-Fuchs, and over $\FF_p$ the trace of Frobenius is $a_p = 1 + p - \size X_\psi(\FF_p)$.
Igusa noticed that the Picard-Fuchs equations for the holomorphic form is hypergeometric, and $a_p$ satisfies a truncated hypergeometric formula.
Some progress on arithmetic properties of the Fermat quintic.
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What is the mirror of smooth quintics in $\PP^4$?
Start with the Fermat quintic pencil $\sum x_i^5 - \psi \prod x_i = 0$, which admits a $C_5^3$ action.
Taking the quotient and resolving singularities yields the mirror $Y$.
For $\psi \in \ZZ$, the point counts agree: $\size X(\FF_q) \cong \size Y(\FF_q)\mod q$.
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Generalized hypergeometric functions: 
\[
{}_A F_V(\alpha, \beta | z) = \sum_k {\alpha_{1, k} \cdots \alpha_{A, k} \over \beta_{1, k} \cdots \beta_{B, k}} z^k
.\]

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Daqing-Wang: for the Fermat pencil $X_\psi$ and its Greene-Plesser mirror $Y_\psi$, the unit roots of their zeta functions coincide, yielding the congruence of point counts.
Kadir claims something similar works for generalize Fermat pencils of CYs in a Gorenstein Fano weighted projective space and its Greene-Plesser mirror.
Recent work by (lots of people) uses BHK mirror symmetry to generalize to invertible pencils in $\PP^n$.
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Batyrev's construction: mirror families of CYs using reflexive polytopes.
Look at polar polytopes, lattice polytopes $\Delta$ which contains $0$ in its interior.
The facet equation can be normalized to $\sum a_i x_i = -1$, and there is an associated **polar polytope** $\Delta^\circ$ associated to the coefficients $(a_i)$.
The polytope $\Delta$ is **reflexive** if $\Delta^\circ$ is again a lattice polytope.
Note that there is no classification of reflexive polytopes in dimensions $d\geq 5$.
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Building hypersurfaces: for each lattice point choose a parameter $\alpha$ and define $p_\alpha = \sum_{M \intersect } \alpha_m \prod x^{\cdots}$.
Can use this to somehow cook up mirror pairs of CYs?
Using that CYs can be realized as hypersurfaces in toric varieties.
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Kernel pairs: for $\Delta, \Gamma$ combinatorially equivalent polytopes, put vertices as columns of matrix and see if they have equal kernels.
These are resolutions of quotients of a common toric variety, and the corresponding vertex pencils satisfy the same Picard-Fuchs equation.
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An alternative to counting points for small primes: for $X$ a smooth variety, Frobenius induces a $p\dash$linear operator $H^n(X; \OO_X)\selfmap$, and the matrix for this map in some basis is the **Hasse-Witt matrix** $HW_p$.
Katz: for CYs, this determines the point counts mod $p$?
A theorem of Huang-Lian-Yau-Yu shows that $HW_p$ for $X$ a Calabi-Yau satisfies a truncated relation and shows that the period integral has a certain power series expansion.
For smooth CYs, we have a good idea of what $HW_p$ is.
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Idea: for mirror kernel pairs, can sometimes replace a very complicated equation with a much simpler equation for its mirror, e.g. if one wants to do point counts or study the Picard-Fuchs equation.
Can classify K3 surfaces by their $HW_p$.
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Picard-Fuchs for rank 19 lattice-polarized K3s can be written as the symmetric square of a 2nd order linear Fuchsian differential equation.
For hypergeometric functions, can use Clausen's formula.
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Let $\Gamma \subseteq \PSL_2(\RR)$, and define a modular curve as $\bar{\dcosetl{\Gamma}{\HH}}$.
See Atkin-Lehner map.
Group I in the classification is related to Monstrous Moonshine, and the classification shows it's associated to $\Gamma_0(3)^+$ where plus means retaining all Atkin-Lehner involutions.
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