Solutions of Calabi-Yau Differential Operators as Truncated p-adic Series and Efficient Computation of Zeta Functions

Abstract

Recently, a version of the deformation method developed in arXiv:2104.07816 has been used to great effect to compute the local zeta functions of Calabi-Yau threefolds by computing their periods as series with rational coefficients and using this to find a matrix representing the Frobenius action on a $p$-adic cohomology. However, this method rapidly becomes inefficient as the prime $p$ grows, due to the rational period coefficients growing quickly. In this paper, we point out that this problem can be circumvented by a simple process that we call $p$-adically truncated recurrence. This is a recurrence relation whose solutions are $p$-adic numbers modulo $p^A$ for a given $A \in \mathbb{N}$ and thus grow only slowly as $p$ grows. We show that the $p$-adic accuracy $A$ can be chosen such that all $p$-adic digits which contribute to the final result are kept, and therefore we are able to obtain the correct result by using these solutions. The improvements to speed and memory usage allow for computing the local zeta functions for tens of thousands of primes on a desktop computer, and make computing local zeta functions possible even for primes of size $10^6$ to $10^7$. Previously such computations were practically possible for around 1000 first primes. We have implemented this method in a Sage-compatible Python package PFLFunction.

Figures (4)

• Figure 1: Comparing Euler factor peak memory usage for the mirror quintic using rational periods, $p$-adically truncated rational periods, and $p$-adically truncated recurrence.
• Figure 2: The normalized histogram displaying the distribution of the normalized Frobenius traces $a_p^{(1)}/p$ for the order 3 degree 2 Calabi--Yau operator \ref{['eq:HV_PF_operator']} at $\varphi = 1$ (upper figure) and $\varphi = 1/8$ (lower figure). We include the traces corresponding to the first 10000 primes in 70 bins. We see that the figures follow approximately the "flying Batman" \ref{['eq:flying_Batman_distribution']} and shifted semicircle \ref{['eq:semicircle_measure']} distributions, respectively, which we have overlaid on the histogram. These distributions are expected when the corresponding K3 surface has complex multiplication. Note that the figures do not display the bars fully at $\pm 1$, which together account for approximately half of the traces of the respective figures.
• Figure 3: The normalized histogram displaying the distribution of the normalized Frobenius traces $a_p^{(1)}/p$ for the order 3 degree 2 Calabi--Yau operator \ref{['eq:HV_PF_operator']} at $\varphi = 3$ (upper figure) and $\varphi = 2/7$ (lower figure). We include the traces corresponding to the first 10000 primes in 70 bins. We see that the figures follow approximately the "Batman" and "wing" distributions of eq. \ref{['eq:non-CM_distributions']}, respectively, which we have overlaid on the histogram. These distributions are the pushforwards of $\text{O}(3)$ and $\text{SO}(3)$ Haar measures, and correspond to cases where the K3 surface does not have complex multiplication.
• Figure 4: The binary accuracy $\eta_0$ to which the approximate $L$-function \ref{['eq:approximate_L-function']} satisfies the functional equation \ref{['eq:L_functional_equation']}. The horizontal axis displays the maximal prime $p_{\max}$ included in the approximate $L$-function, and the vertical axis the corresponding accuracy. We display the data for every 20th prime. Note that the accuracy increases as the number of Euler factors is increased. This strongly indicates that the Euler factors computed using our method are correct, as the accuracy would be expected to plateau after an incorrect Euler factor.

Theorems & Definitions (1)

• Conjecture 1