Abstract
We prove that p-determinants of a certain class of differential operators can be lifted to power series over \(\mathbb {Q}\). We compute these power series in terms of monodromy of the corresponding differential operators.
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Notes
In principle, we could have a function \(\lambda (t)\) depending analytically on \(\sqrt{t}\). However, one can show by direct calculations that \(\lambda (t)\) is analytic in t, see a formula (1.4) below and Example 1 in the last Section.
See [1] for general definition and discussion of p-determinants.
This means that \(c_0,c_1,\dots ,c_{p-1}\) are rational numbers without p in denominator.
Our previous concrete operator D can be reduced to this canonical form after a multiplication by a function of t. See Example 1 in the last Section for details.
For example \(R=\mathbb {Z}[t]\) and \(I=(t)\). In this case \(\hat{R}_I=\mathbb {Z}[[t]]\).
Here \(R[\varepsilon ]\) is a ring of polynomials in an independent variable \(\varepsilon \) with coefficients in R. Later variable \(\varepsilon \) will appear as a parameter for the twist (conjugation) by \(x^\varepsilon \) of a differential operator.
This statement might be well known to the experts but we were unable to find it in the literature. A very similar but not suitable for us statement can be found in Bourbaki, Commutative Algebra, Chapter 7, Section 3.8. The Bourbaki’s proof is similar to our Proof 1.
In the sequel we will refer to the polynomial \(\varepsilon ^n-w_1\varepsilon ^{n-1}+\cdots +(-1)^nw_n\) as to the Weierstrass polynomial of \(q(\varepsilon )\).
More precisely, \(\overline{R}=\hat{R}_I\hat{\otimes }_{\mathbb {Z}}\mathbb {Q}[\varepsilon _1,\ldots ,\varepsilon _n]/\{\text {coefficients of}~\varepsilon ^j,~j=0,\ldots ,n-1~~\text {in}~~ w(\varepsilon )-\prod _{j=1}^n(\varepsilon -\varepsilon _j)\}\).
One can show that in this case g(x) is proportional to the solution \(g_j(x)\) but we do not need this for our purposes.
One can prove the existence of this limit in the same way as in the proof of Lemma 2.
This can also be proved by constructing an inverse matrix \(B(\delta )^{-1}\) in the case \(w(\delta )\ne 0\), see Remark 1 in the end of Sect. 3.
More formally, we have \(D_z=\mu (D)\).
One can show that solutions of the equation \(D_z g(x)=0\) are equal to \(g_j^{\text {an}}=\overline{\mu }(g_j)=x^{\overline{\mu }(\varepsilon _j)}\sum _{l\in \mathbb {Z}}\overline{\mu }(r_{j,l})x^l,\; j=1,\ldots ,n\), up to multiplication by a constant depending on z and j, where \(g_1,\ldots ,g_n\) are defined in Lemma 6.
Our methods give the proof of this identity modulo \(p,~t^{\lceil \frac{p}{2}\rceil }\).
This is a well known fact in the theory of Heun functions. See for example https://dlmf.nist.gov/31 and references therein.
The following formulas has exceptions in the case \(\alpha _3(n)\), \(n=6,7,8\).
References
Kontsevich, Maxim: Holonomic D-modules and positive characteristic. Jpn. J. Math. 4(1), 1–25 (2009)
Manin, Yu.I.: The Hasse-Witt matrix of an algebraic curve. AMS Trans. Ser. 2 45, 245–264 (1965)
Acknowledgements
We are grateful to Don Zagier for useful discussions in the early stages of this project. A.O. is grateful to IHES for invitations and excellent working atmosphere.
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Kontsevich, M., Odesskii, A. p-Determinants and monodromy of differential operators. Sel. Math. New Ser. 28, 52 (2022). https://doi.org/10.1007/s00029-022-00770-6
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DOI: https://doi.org/10.1007/s00029-022-00770-6