# Noriko Yui, Modularity of certain Calabi-Yau threefolds over $\QQ$ (Wednesday, July 20)

:::{.remark}
Definition: smooth projective variety $X$ over $\CC$, $\dim_\CC X = 3$, and


- $H^i(X; \OO_X) = 0$ for $i=1,2$, and 
- $K_X \cong \OO_X$.

By conjugation, $h^{i, j} = h^{j, i}$, and by Serre duality $h^{i, j} = h^{3-i, 3-j}$.
Note $h^{1,1}, h^{2,2} > 0$ since Calabi-Yau threefolds are Kähler.
The outer edge of the Hodge diamond contains only zeros and ones and has 7 rows, up to 4 columns.
A typical example: quintic threefolds $\sum x_i^5 = 0$, or elliptic curves are dimension 1 CYs and K3s are dimension 2 CYs.
More generally, CYs can be constructed as intersection threefolds in weighted projective spaces.
Other examples:

- Double octics $w^2 = f_8(x,y,z,w)$
- $E\times S/\iota$ where $E$ is an elliptic curve, $S$ is a K3, $\iota$ is a non-symplectic involution
- Toric CYs

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:::{.remark}
Let $X\slice \QQ$ be Calabi-Yau iff $X\tensor_\QQ \CC$ is Calabi-Yau in the usual sense.
Suppose it has defining equations in $\ZZ\invert{m}$ for some $m$, pick a *good prime* $p$ coprime to $m$ so that $X$ has good reduction: $X_p$ is smooth over $\bar{\FF_p}$.
Define the zeta function $Z_p(X, T) \in \QQ\fps{T}$ as usual, then the Weil conjectures hold for $X$.
Since $P_1 = P_5 = 1$, 
\[
Z_p(X, T) = {P^3(T) \over (1-T) P^2(T) P^2(pT) (1-p^3 T)}
.\]
There is a compatible system of $\ell\dash$adic Galois representations on étale cohomology.
Define an $L\dash$function $L_i(X, s) = \prod_{p\neq \ell \text{ good}} P^i(p^{-s})\inv \times (\cdots)$, where the last factor accounts for when $\ell=p$.
Can compute $\size X(\FF_p)$ using the Lefschetz trace formula for $\Frob_p\actson H_\et(\bar X; \QQladic)$.
Here $\size X(\FF_p) = 1 + p^3 + (1+p ) t_2(p) - t_3(p)$ where $t_2, t_3$ are bounded above by $h^{1, 1}, \beta_?$ by RH.
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The modularity question: are there global functions or automorphic forms that determine $L(X, s)$?
This would be the Langlands philosophy.

Say $X$ is rigid if $h^{2, 1} = 0$ so $\beta_3 = 2$ and the Hodge diamond only has two unknowns, $h^{1, 1}, h^{2,2}$.
These resemble elliptic curves in terms of the characteristic polynomial.
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:::{.theorem title="?"}
Every rigid Calabi-Yau threefold is modular: there is some modular form $f$ of weight $4$ on some $\Gamma_0(N)$ such that $L(f, s) = L(X, s)$.
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:::{.remark}
There are 14 one-parameter families of CYs defined over $\QQ$ of hypergeometric type, and at conifold points each family as a rigid motive and corresponds to a special cusp form of weight 4 on some $\Gamma_0(N)$ where only $2,3,5\divides N$.
The next simplest case is when $h^{2, 1} = h^{1, 2} = 1$ and not zero, so $\beta_3 = 4$.
Constructing such CYs: Nygaard and van Geemen take a certain complete intersection whose singular locus has 16 ordinary double points and 4 plane conics intersecting in a square; take its resolution.
Van Geemen and Werner/Consani and Scholten construct another using the affine equation of a skew pentagon, yielding a quintic Calabi-Yau threefold defined over $\QQ$ with bad primes $2,3,5$.
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:::{.theorem title="?"}
For $X$ the Calabi-Yau coming from a complete intersection as above,
\[
L(X, s) = L(\chi^3, s) \cdot L(\chi_{(-1)}, s)
,\]
where $\chi$ is a Hecke character for $\QQ(i)$?
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:::{.remark}
Note that something similar happens for elliptic curves with CM.
Idea of proof: split $H^3$ into two rank 2 motives to decompose the L function.

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Since $X$ is covered by some $E^3$ where $E: y^2 = 1+x^4$ which has CM by $\QQ(i)$, so $L(X, s) \divides L(E^3, s)$ and we can use this to define when a Calabi-Yau has CM in terms of the cover.
General expectation: if $X$ has CM for an imaginary quadratic number field $K$, then $L$ splits into factors involving Hecke characters of $K$.
To avoid Hecke characters, define *real* multiplication instead: if $X$ is Calabi-Yau with $h^{p, q} = 1$ for $p+q=3$ and $\beta_3 = 4$, set $K = \QQ(\sqrt d)$ and say $X$ has RM with $K$ if

- There is an algebraic correspondence $\Psi:X\to X$ defined over $K$.
- There is an automorphism of $H^3$ preserving the Hodge structure,
- There is an automorphism of $H^3$ preserving Galois representations.

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:::{.remark}
See Hilbert modular forms, Adrianov $L\dash$functions.
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