$X$ to 0, is explained by the fact the classical Frobenius map behaves “multiplicatively” (it belongs naturally to some Galois group) while the $p$-curvature behaves “additively” (it belongs naturally to some Lie algebra).

In the classical setting, Kronecker's theorem is obtained as a corollary of Chebotarev's density theorem, which is itself proved by means of Artin's $L$-functions. Unfortunately, similar tools do not seem to be available so far in the differential context; developing them might then sound as an exciting project.

As mentioned above, Honda proved that the Grothendieck conjecture for order-1 differential equations is equivalent to Kronecker's theorem. In [30, §4], the Chudnovsky brothers gave an elementary (although “extravagant”) proof of these equivalent statements; their approach is based on Hermite's explicit Hermite-Padé approximants to binomial functions. More precisely, they proved that if $y'(x) = \frac{x}{\alpha}y(x)$ has zero $p$-curvature for almost all primes $p$, then for all primes ideals $\mathfrak{p}$ of $\mathbb{Q}(\alpha)$ all the binomial coefficients $\binom{\alpha}{n}$ are $p$-integral for all $n$. From there, it is shown that Hermite-Padé approximants to $1, x^{\alpha}, \dots, x^{(m-1)\alpha}$ at $x = 1$ with weights $(N, \dots, N)$ are trivial for large $m$ and $N$. This in turn implies that $1, x^{\alpha}, \dots, x^{(m-1)\alpha}$ are linearly dependent over $\mathbb{Q}(x)$, that is $x^{\alpha}$ is an algebraic function, which is equivalent to $\alpha \in \mathbb{Q}$.

## 3.3 Computation of the $p$-curvature

After what we have done previously, it is clear that the $p$-curvature is an invariant of primary importance of linear differential equations in characteristic $p$. In this subsection, we outline some algorithms for computing it (or other quantities associated to it) efficiently.

### 3.3.1 Operators of order 1

In the case of differential equations of order 1 of the form (18):

$$y' + b(x)y = 0$$

we have seen in Theorem 3.12 that the $p$-curvature is explicitly given by the formula

$$b_p(x) = b^{(p-1)}(x) + b(x)^p.$$

Furthermore, computing explicitly the latter is a quite easy task which directly reduces to writing the partial fraction decomposition of $b(x)$. Indeed, we have seen that if $b(x)$ decomposes as

$$b(x) = P(x) + \sum_{i=1}^{m} \sum_{j=1}^{r_i} \frac{\alpha_{i,j}}{(x - a_i)^j}$$

then

$$b_p(x) = P^{(p-1)}(x) + P(x)^p - \sum_{i=1}^{m} \sum_{\substack{1 \le j \le r_i \\ j \equiv 1 \pmod p}} \frac{\alpha_{i,j}}{(x - a_i)^{j+p-1}} + \sum_{i=1}^{m} \sum_{j=1}^{r_i} \frac{\alpha_{i,j}^p}{(x - a_i)^{pj}}.$$

Importantly for algorithmic purposes, we observe that the size of $b_p(x)$ is roughly the same as the same of the input $b(x)$, although the degree of (the numerator and the denominator of) the former is $p$ times larger the degree of the latter. This apparent contradiction is explained by the fact that $b_p(x)$ is actually a function of $x^p$; it is then a sparse rational function, in the sense that a large proportion of its coefficients vanishes.
It follows from this observation that, writing $E_i = (0, \dots, 0, 1, 0, \dots, 0)$ with the coordinate 1 in $i$-th position, the coordinates of $\Delta^p(E_i)$ are exactly the coefficients of the remainder in the division of $\partial_x^{p+i}$ by $\mathcal{L}$. Hence $\Delta^p(E_i)$ vanishes if and only if $\mathcal{L}$ divides $\partial_x^{p+i}$. The equivalence (2) $\iff$ (3) follows immediately. $\square$

**Remark 3.20.** The three assertions of Theorem 3.19 are also equivalent to

(4) $\mathcal{L}$ admits $n$ power series solutions in $\mathbb{F}_p[[x]]$, linearly independent over $\mathbb{F}_p((x^p))$;

(5) $\mathcal{L}$ admits $n$ polynomial solutions in $\mathbb{F}_p[x]$, linearly independent over $\mathbb{F}_p((x^p))$.

The implication (4) $\implies$ (5) is proved in [53, Lemma 1], while (5) $\implies$ (1) and (1) $\implies$ (4) are trivial. Moreover, under the equivalent assertions (1)–(5), Proposition 1 in [23] shows that there exists a full basis of polynomial solutions in $\mathbb{F}_p[x]$, each of them having degree less than $pd$, where $d$ is the maximal degree of the numerators/denominators of the coefficients $b_i(x)$ of $\mathcal{L}$ in (26).

**Remark 3.21.** Assume that $\mathcal{L}$ has $p$-curvature zero. An easy calculation shows that

$$U_0(x) = \sum_{k=0}^{p-1} (-1)^k \frac{x^k}{k!} B_k(x) \in M_n(\mathbb{F}_p(x))$$

is a solution of $Y' + B(x)Y = 0$. If, moreover, $B(x)$ has no pole at 0, then $U_0(x)$ has no pole at 0 as well and we have $U_0(0) = I_n$, so $U_0(x)$ is a fundamental matrix of rational solutions of $Y' + B(x)Y = 0$. If $B(x)$ has a pole at 0, then $U_0(x)$ is not necessarily invertible. Note that, more generally, if $a \in \mathbb{F}_p$ is not a pole of $B(x)$, then

$$U_a(x) = \sum_{k=0}^{p-1} (-1)^k \frac{(x-a)^k}{k!} B_k(x-a)$$

is a fundamental matrix of rational solutions of $Y' + B(x)Y = 0$.

Putting together all that precedes, we obtain a simple algorithm to determine whether (26) has a full basis of rational solutions: compute inductively $B_p(x)$ and, then, check whether $B_p(x)$ vanishes. Note however that no extension of the simple formula of Theorem 3.12 is known for higher order differential equations. Roughly speaking, this is due to the fact that, contrarily to $\mathbb{F}_p(x)$, the ring of $n \times n$ matrices over $\mathbb{F}_p(x)$ is noncommutative as soon as $n \ge 2$. Computing the $p$-curvature is then much more complicated in this case but rather efficient algorithms for this task are nevertheless available; we will discuss them in Section 4.

Using Theorem 3.19 (Cartier's lemma), we get the following reformulation of Grothendieck's conjecture.

**Conjecture 3.22** (Grothendieck's conjecture in terms of $p$-curvature). For a differential operator $\mathcal{L}$ as in equation (25), the following properties are equivalent:

(1) $\mathcal{L}$ has a full basis of algebraic solutions;

(2) for almost all primes $p$, the $p$-curvature of $\mathcal{L}_p$ vanishes;

(3) for almost all primes $p$, $\mathcal{L}_p$ divides $\partial_x^p$ in the ring of differential operators $\mathbb{F}_p(x)\langle\partial_x\rangle$.
**Remark 3.25.** Another option for computing explicitly the *p*-curvature of differential operators of order 1 is presented in [21, Thm. 2]; it avoids the computation of partial fraction decomposition and to factor the denominator of $b(x)$. Let us briefly sketch it with the differential operator

$$ \partial_x - \frac{1}{x^2 + 1} $$

of Example 3.4. We write $b(x) = -1/(x^2 + 1)$ and assume $p > 2$ for simplicity. In order to compute $b_p(x)$, we expand $b(x)$ in power series:

$$ b(x) = - \sum_{n=0}^{\infty} (-1)^n x^{2n}. $$

The $(p-1)$-st derivative of $x^{2n}$ is 0 when $2n \neq -1 \pmod p$, and it is $-x^{2n-p+2}$ otherwise thanks to Wilson's theorem. Writing $2n = p-1 + pk$ and noticing that $k$ has to be even, $k = 2\ell$, we end up with

$$ b^{(p-1)}(x) = - \sum_{\ell=0}^{\infty} (-1)^{\ell - \frac{p-1}{2}} x^{2\ell p} $$

On the other hand, it is clear that $b(x)^p = - \sum_{n=0}^{\infty} (-1)^n x^{2np}$. Adding both sums, we find

$$ b_p(x) = - \sum_{n=0}^{\infty} (-1)^n \cdot \left(1 - (-1)^{\frac{p-1}{2}}\right) \cdot x^{2np}. $$

When $p \equiv 1 \pmod 4$, the exponent $\frac{p-1}{2}$ is even and the term in the parenthesis vanishes. Therefore $b_p(x) = 0$ in this case. On the contrary, when $p \equiv 3 \pmod 4$, we have

$$ b_p(x) = -2 \cdot \sum_{n=0}^{\infty} (-1)^n \cdot x^{2np} = - \frac{2}{1 + x^{2p}} = - \frac{2}{(1 + x^2)^p}. $$

and we recover the result of Example 3.14.

The same idea applies actually to any differential operator $\mathcal{L} = \partial_x - b(x)$. Indeed, as already noticed its *p*-curvature is a rational function in $x^p$. Besides, it is of the form $f(x)/\mathrm{denom}(b(x))$, where the numerator $f(x)$ is a polynomial of degree at most $d = \deg(b(x))$. Hence, it is enough to determine the power series expansion of $(b(x)^{(p-1)})^{1/p}$ at precision $x^d$, starting from the power series expansion of $b(x)$ at the same precision $d$. If $b(x) = \sum_{n \ge 0} u_n x^n$, then by Wilson's theorem we have $(b(x)^{(p-1)})^{1/p} = - \sum_{n \ge 1} u_{np-1} x^{n-1}$, and hence it is enough to be able to compute the terms $u_{p-1}, \dots, u_{dp-1}$. Since $b(x)$ is a rational function, the sequence $(u_n)_{n \ge 0}$ satisfies a linear recursion of order at most $d$, with coefficients in $\mathbb{F}_p$ (given by the coefficients of $\mathrm{denom}(b)$). As the $N$-th term of such a linear recurrence with constant coefficients can be computed using $O(d \log(d) \log(N))$ operations in $\mathbb{F}_p$ using the technique of binary powering combined with fast polynomial multiplication in $\mathbb{F}_p[x]$, we conclude that the *p*-curvature $b_p(x)$ can be computed by an algorithm that uses $O(d^2 \log(d) \log(p))$ operations in $\mathbb{F}_p$.

Note that the reasoning above shows that the *p*-curvature $b_p(x)$ of $\mathcal{L} = \partial_x - b(x)$ is zero if and only if the following infinite systems of congruences holds

$$ u_n \equiv u_{(n+1)p-1} \pmod p \quad \text{for all } n \ge 0. $$
## 3.3 Progresses toward Grothendieck's conjecture

### 3.3.1 A known case: the generalized hypergeometric equations

It is in general very difficult to determine whether a given differential equation has a full basis of algebraic solutions. In their celebrated work [10], Beukers and Heckman managed to do this for an important class of differential equations, omnipresent in the mathematical and physical literature, namely the generalized hypergeometric equations. In fact, Beukers and Heckman extended the Landau-Errera criterion mentioned in §2.1.3. Let $\mathbf{a} = (a_1, \dots, a_{s+1})$ and $\mathbf{b} = (b_1, \dots, b_s, b_{s+1} = 1)$ be two $(s+1)$-tuples of rational parameters, assumed disjoint modulo $\mathbb{Z}$. This assumption is equivalent to the irreducibility in $\mathbb{Q}(x)\langle\partial\rangle$ of the "generalized hypergeometric operator" defined by

$$ \mathcal{H}(\mathbf{a}, \mathbf{b}) := (x\partial_x + b_1 - 1) \cdots (x\partial_x + b_s - 1) x\partial_x - x(x\partial_x + a_1) \cdots (x\partial_x + a_{s+1}). $$

It is easy to check that $\mathcal{H}(\mathbf{a}, \mathbf{b})$ admits in its solution space the generalized hypergeometric function $s+1 F_s([a_1, \dots, a_{s+1}], [b_1, \dots, b_s]; x)$ defined in (13). The Beukers-Heckman result then reads as follows.

**Theorem 3.23** ("interlacing criterion", Beukers-Heckman, [10]). Given two $(s+1)$-tuples of rational numbers $\mathbf{a} = (a_1, \dots, a_{s+1})$ and $\mathbf{b} = (b_1, \dots, b_s, b_{s+1} = 1)$, assumed to be disjoint modulo $\mathbb{Z}$, let $D$ be the common denominator of their elements. Then, the following assertions are equivalent:

1. the hypergeometric function $s+1 F_s([a_1, \dots, a_{s+1}], [b_1, \dots, b_s]; x)$ is algebraic;

2. the operator $\mathcal{H}(\mathbf{a}, \mathbf{b})$ admits a full basis of algebraic solutions;

3. for all $1 \le l < D$ with $\text{gcd}(l, D) = 1$ the $(s+1)$-tuples $(e^{2\pi i l a_j})_{1 \le j \le s+1}$ and $(e^{2\pi i l b_j})_{1 \le j \le s+1}$ interlace on the unit circle.

The interlacing condition mentioned in the latter result is defined as follows. We say that two $(s+1)$-tuples $(u_j)_{1 \le j \le s+1}$ and $(v_j)_{1 \le j \le s+1}$ of elements of the unit circle interlace if, up to renumbering the $u_j$ and the $v_j$, we have $u_1 = e^{2\pi i a_1}, \dots, u_{s+1} = e^{2\pi i a_{s+1}}$ and $v_1 = e^{2\pi i b_1}, \dots, v_{s+1} = e^{2\pi i b_{s+1}}$ with either

$$ 0 \le \alpha_1 < \beta_1 < \alpha_2 < \beta_2 \le \cdots < \alpha_{s+1} < \beta_{s+1} < 1 $$

or

$$ 0 \le \beta_1 < \alpha_1 < \beta_2 < \alpha_2 \le \cdots < \beta_{s+1} < \alpha_{s+1} < 1. $$

**Example 3.24.** The Beukers-Heckman criterion immediately implies that the operator $\mathcal{H}(\mathbf{a}, \mathbf{b})$ admits a full basis of algebraic solutions for the choice

$$ \mathbf{a} = \left\{ \frac{1}{30}, \frac{7}{30}, \frac{11}{30}, \frac{13}{30}, \frac{17}{30}, \frac{19}{30}, \frac{23}{30}, \frac{29}{30} \right\}, \quad \mathbf{b} = \left\{ \frac{1}{5}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{4}{5}, 1 \right\}. $$

This proves in particular a beautiful observation due to Fernando Rodriguez-Villegas, namely that the generating function $\sum_{n \ge 0} u_n x^n$ of the sequence

$$ u_n := \frac{(30n)!n!}{(15n)!(10n)!(6n)!} $$

(used by Chebyshev in his work on estimates for the prime counting function) is algebraic.
### 3.3.2 Reading the $p$-curvature on the solutions

For differential equations of higher orders, the sparsity of the $p$-curvature no longer holds in general. However, we have the following result.

**Proposition 3.26.** Let

$$ \mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \cdots + b_1(x) \cdot \partial_x + b_0(x) $$

with $b_i(x) \in \mathbb{F}_p(x)$ and let $B(x)$ be the associated companion matrix (see Equation (23)). Let $f(x) \in \mathbb{F}_p(x)$ be a common denominator of the $b_i(x)$'s and

$$ d = \max(\deg f(x), \deg(f(x)b_0(x)), \dots, \deg(f(x)b_{n-1}(x))). $$

Let $B_p(x)$ be the matrix of the $p$-curvature of $\mathcal{L}$ defined by the recurrence (24). Then, the following holds.

(i) The matrix $B_p(x)$ has the form $\frac{1}{f(x)^p}C_p(x)$ where the entries of $C_p(x)$ are all polynomials of degree at most $dp$.

(ii) The matrix $B_p(x)$ is similar to a matrix with coefficients in $\mathbb{F}_p(x^p)$.

The last assertion implies that the trace, the determinant of $B_p(x)$ and, more generally, all the coefficients of its characteristic polynomial lie in $\mathbb{F}_p(x^p)$; they are then sparse rational functions. Combining with Proposition 3.26.(i), we deduce that their sizes is comparable to the size of the $b_i(x)$'s, although the size of the $p$-curvature itself is in general $p$ times larger.

In order to design fast algorithms for computing the $p$-curvature, it is useful to go beyond Cartier's lemma (see Proposition 3.18) and relate the $p$-curvature to the shape of solutions. Let

$$ \mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \cdots + b_1(x) \cdot \partial_x + b_0(x) $$

be a differential operator as before. In §3.2.1, we have seen that, when the $p$-curvature of $\mathcal{L}$ vanishes and the $b_i(x)$'s have no pole at 0, a fundamental system of solutions of $\mathcal{L}$ is explicitly given by

$$ \sum_{k=0}^{p-1} (-1)^k B_k(x) \frac{x^k}{k!} $$

where the $B_k(x)$'s are the matrices defined by the recurrence (24). In full generality, i.e. without assuming the vanishing of $B_p$, the idea is to consider the formal expansion

$$ \sum_{k=0}^{\infty} (-1)^k B_k(x) \frac{x^k}{k!}. $$

Of course, this does not make sense in $\mathbb{F}_p(x)$ because of the division by $k!$, but we shall see that it does make sense in a suitable ring. A natural idea to achieve this goal is to introduce divided powers, i.e. to consider the so-called ring of Hurwitz series, denoted by $\mathbb{F}_p[[x]]^{dp}$, whose elements are formal series of the form

$$ a_0 + a_1 \gamma_1(x) + a_2 \gamma_2(x) + \cdots + a_k \gamma_k(x) + \cdots. $$
Note that without the irreducibility assumption on $\mathcal{H}(a, b)$, the situation is much more subtle. For instance, ${}_2F_1([1/2, 1/3], [3/2]; x)$ is transcendental, while ${}_2F_1([3/2, 1/3], [1/2]; x)$ is algebraic. In a recent work by Fürnsinn and Yurkevich [52], a generalization of Theorem 3.23 is given, which allows to decide algebraicity/transcendence of arbitrary generalized hypergeometric functions (with potentially irrational parameters).

In addition to Theorem 3.23, Beukers and Heckman also drew up in [10] the list of generalized hypergeometric equations having a full basis of algebraic solutions, thus extending Schwarz's classification of algebraic ${}_2F_1$'s [76].

On the other hand, a calculation due to Katz in [57, §5.5] (also in [55, §6] for the specific case of ${}_2F_1$'s) shows that this list coincides with the list of generalized hypergeometric equations whose reductions modulo $p$ have a full basis of rational solutions for almost all primes $p$, in accordance with Grothendieck's conjecture.

### 3.3.2 State of the art on Grothendieck's conjecture

Besides for order-1 equations and for generalized hypergeometric equations, Grothendieck's conjecture has been proved in several particular cases.

On the one hand, for Picard-Fuchs differential equations (satisfied by periods of a family of smooth algebraic varieties), and more generally for certain direct factors, Grothendieck's conjecture was established by Katz [55]. As an application, Katz gave in [57, Theorem 5.5.3] a new proof of the aforementioned results of Beukers and Heckman [10] about the generalized hypergeometric equations. Katz [55, §1], and later André [5, §III], related the $p$-curvatures to the reduction modulo $p$ of the Kodaira-Spencer map. (See also Foucault [48] and Foucault and Toffin [49] for explicit computations for families of curves of genus 2 and 3.) As explained in [5, p. 108], this approach has a potential of delivering effective versions of Grothendieck's conjecture, similar to effective versions of Chebotarev's density theorem [63, 77]: the hope is to obtain, for instance for any Picard-Fuchs operator $\mathcal{L}$, an integer $N(\mathcal{L})$ such that the fact that $\mathcal{L}$ has a full basis of algebraic solutions can be read on the $p$-curvatures of $\mathcal{L}$ for the primes $p < N(\mathcal{L})$.

On the other hand, an arithmetic approach to Grothendieck's conjecture was introduced by the Chudnovsky brothers [35] who proved Grothendieck's conjecture for any rank one linear homogeneous differential equation over an algebraic curve [35, Theorem 8.1] (the case of first order equations over $\mathbb{P}^1$ had been proved by Honda in [53, §1], see Theorem 3.6). They also proved Grothendieck's conjecture for the class of Lamé equations [35, Theorem 7.2], of the form

$$p(x)y''(x) + \frac{1}{2}p'(x)y'(x) - (n(n+1)x + B) \cdot y(x) = 0$$

where $n \in \mathbb{N}$, $B \in \mathbb{Q}$ and $p(x) \in \mathbb{Q}[x]$ has degree 3. The arithmetic approach was extended by André to the case when the differential Galois group has a solvable neutral component [5] (see also [3], [4, Chap. VIII], [12, Thm. 2.9] and [29, Thm. 3.5]).

Katz [56] proposed a conjectural description of the differential Galois group in terms of $p$-curvatures and he proved that his conjecture is equivalent to the initial conjecture by Grothendieck.

Using the language of schemes and sheaves, Grothendieck's conjecture can be formulated more generally for differential equations over any algebraic smooth curve defined over a number field. In [5, Remark 7.1.4], André noticed that, using Belyi maps, one can reduce the
In the above expression, the $\gamma_k(x)$'s are just formal names without further additional meaning. Of course, they should be thought of as $\frac{x^k}{k!}$ but we cannot write this division because the denominator may vanish. The multiplication on $\mathbb{F}_p[[x]]^{\text{dp}}$ is governed by the rule

$$ \gamma_m(x) \cdot \gamma_n(x) = \binom{m+n}{m} \cdot \gamma_{n+m}(x) $$

for any nonnegative integers $m$ and $n$. Besides $k[[x]]^{\text{dp}}$ is equipped with a natural derivation that takes $\sum_k a_k \gamma_k(x)$ to $\sum_k a_{k+1} \gamma_k(x)$. We have to be careful however that $\mathbb{F}_p[[x]]^{\text{dp}}$ is not a domain (e.g. $\gamma_1(x)^p = 0$) and, because of that, we cannot consider its ring of fractions. But still, if the matrix $B(x)$ has polynomial coefficients, all the $B_k(x)$'s have the same property and we can consider their images $B_k^{\text{dp}}(x)$ is the ring $M_n(\mathbb{F}_p[[x]]^{\text{dp}})$. We then can form

$$ S^{\text{dp}}(x) = \sum_{k=0}^{\infty} (-1)^k B_k^{\text{dp}}(x) \cdot \gamma_k(x) \qquad (25) $$

obtaining this way a fundamental matrix of solutions of $\mathcal{L}$ over $\mathbb{F}_p[[x]]^{\text{dp}}$. This construction works actually more generally as soon as the entries of $B(x)$ have no pole at zero: in this case, we can expand them as series in $x$ in order to view them in $\mathbb{F}_p[[x]]^{\text{dp}}$. The precise relation between $S^{\text{dp}}(x)$ and the $p$-curvature is given by the next lemma.

**Lemma 3.27** (Bostan–Caruso–Schost [13]). We have the matrix relation:

$$ \frac{d^p S^{\text{dp}}(x)}{dx^p} = -B_p^{\text{dp}}(x) \cdot S^{\text{dp}}(x). $$

*Proof.* Set $M = \mathbb{F}_p[x]$ and let $\Delta : M^n \to M^n, Y \mapsto \frac{dY}{dx} + B(x)Y$. By definition of the $p$-curvature, $\Delta^p$ is the multiplication by $B_p(x)$.

Now consider the endomorphism $\Delta^{\text{dp}}$ of $M^{\text{dp}} = \mathbb{F}_p[[x]]^{\text{dp}} \otimes_{\mathbb{F}_p[x]} M$ defined by

$$ \Delta^{\text{dp}} = \frac{d}{dx} \otimes \text{id}_M + 1 \otimes \Delta_M. $$

One checks that it satisfies the Leibniz rule: for $f \in \mathbb{F}_p[[x]]^{\text{dp}}$ and $m \in M$, we have

$$ \Delta^{\text{dp}}(f \otimes m) = \frac{df}{dx} \otimes m + f \otimes \Delta(m). $$

Hence, raising it to the $p$-th power, we obtain

$$ (\Delta^{\text{dp}})^p (Y^{\text{dp}}) = \frac{d^p Y^{\text{dp}}}{dx^p} + B_p^{\text{dp}}(x) \cdot Y^{\text{dp}} $$

for all vector $Y^{\text{dp}} \in M^{\text{dp}}$. The equality of the lemma follows given that the columns of $S^{\text{dp}}(x)$ maps to 0 under $\Delta^{\text{dp}}$. $\square$
general case to that of the curve $X = \mathbb{P}^1\setminus\{0, 1, \infty\}$. In our setting, this means that one can safely assume that the differential operator $\mathcal{L}$ has only singularities at 0, 1 and $\infty$. Under this additional assumption, Tang [83] proves that if all<fnref n="3" /> the $p$-curvatures of $\mathcal{L}$ vanish, then $\mathcal{L}$ has a full basis of *rational* solutions. Although this latter result differs from Grothendieck's in the hypotheses (which are stronger) and the conclusion (which is also stronger), it is closely related.

We also point out the work of Bost in [12] giving an algebraicity criterion for leaves of algebraic foliations defined over number fields. For additional details, we refer to [29]. We mention the work of van der Put in [85] concerned with inhomogeneous equations of order 1. Other special cases of the conjecture have been proven recently, see [44, 79, 71]. Last but not least, an analogue of Grothendieck's conjecture for $q$-difference equations was conjectured by Bézivin [11, §5] and proved by Di Vizio in [37].

## 3.4 A formal parallel with Kronecker's theorem

It is instructive to observe that Grothendieck's conjecture appears to be, in some sense, a differential version of Kronecker's theorem we have already encountered earlier (see Theorem 3.7). Indeed, Kronecker's theorem can be reformulated as follows.

**Theorem 3.25.** For a separable polynomial $L \in \mathbb{Q}[x]$, the following conditions are equivalent:

(1) all the roots of $L$ are in $\mathbb{Q}$;

(2) for almost all primes $p$, all the roots of $L \bmod p$ are in $\mathbb{F}_p$;

(3) for almost all primes $p$, we have $X^p \equiv X \pmod L, p$.

It is striking that the three conditions of Theorem 3.25 are formal analogues of the conditions of Conjecture 3.17, at least if we admit that algebraic solutions in the differential case correspond to rational solutions in the algebraic case. Besides, the fact that the condition $X^p \equiv X \pmod L, p$ translates to $\partial_x^p \equiv 0 \pmod{\mathcal{L}, p}$, i.e. that the right-hand side shifts from $X$ to 0, is explained by the fact that the classical Frobenius map behaves “multiplicatively” (it belongs naturally to some Galois group) while the $p$-curvature behaves “additively” (it belongs naturally to some Lie algebra).

In the classical setting, Kronecker's theorem is obtained as a corollary of Chebotarev's density theorem, which is itself proved by means of Artin's $L$-functions. Unfortunately, similar tools do not seem to be available so far in the differential context; developing them might then sound as an exciting project.

As mentioned above, Honda proved that the Grothendieck conjecture for first order differential equations is equivalent to Kronecker's theorem. In [35, §4], the Chudnovsky brothers gave an elementary (although “extravagant”) proof of these equivalent statements; their approach is based on Hermite's explicit Hermite-Padé approximants to binomial functions. More precisely, they proved that if $y'(x) = \frac{x}{\alpha}y(x)$ has zero $p$-curvature for almost all primes $p$, then for all primes ideals $\mathfrak{P}$ of $\mathbb{Q}(\alpha)$ all the binomial coefficients $\binom{\alpha}{n}$ are $\mathfrak{P}$-integral for all $n$. From there, it is shown that Hermite-Padé approximants to $1, x^\alpha, \dots, x^{(m-1)\alpha}$ at $x=1$ with weights $(N, \dots, N)$ are trivial for large $m$ and $N$. This in turn implies that $1, x^\alpha, \dots, x^{(m-1)\alpha}$ are linearly dependent over $\mathbb{Q}(x)$, that is $x^\alpha$ is an algebraic function, which is equivalent to $\alpha \in \mathbb{Q}$.

<footnote n="3">³When $\mathcal{L}$ does not reduce properly at a prime $p$, the $p$-curvature of $\mathcal{L}_p$ is a priori not defined; however Tang manages to give an alternative definition of the vanishing of the $p$-curvature, see [83, Definition 2.1.7].</footnote>
### 3.3.3 Application to algorithmics

Given that the matrix $S^{dp}(x)$ is invertible, it follows from Lemma 3.27 that one can deduce the value of $B_p^{dp}(x)$ from that of $S^{dp}(x)$ which, in turn, can be computed using the techniques of [15]. However, at this point, we have not solved entirely the question of the computation of the p-curvature because the knowledge of $B_p^{dp}(x)$ is not enough to recover $B_p$. Precisely, letting $\mathbb{F}_p(x)_0$ denote the subring of $\mathbb{F}_p(x)$ consisting of functions with no pole at 0, the natural map $\delta_0 : \mathbb{F}_p(x)_0 \to \mathbb{F}_p([x])^{dp}$ is not injective; its kernel is the ideal generated by $x^p$.

To tackle this issue, the idea is shift around any other base point $a \in \mathbb{F}_p$. Doing so, we get a new differential ring homomorphism $\delta_a : \mathbb{F}_p(x)_a \to k([x-a])^{dp}$ and, reusing the same techniques, we end up with a fast algorithm that computes the p-curvature $B_p$ modulo $(x-a)^p$. Since we have moreover at our disposal a priori bounds on the size of the p-curvature (see Proposition 3.26), one can pick enough elements $a$ in $\mathbb{F}_p$ (or in a finite extension of $\mathbb{F}_p$, if needed), compute the p-curvature modulo $(x-a)^p$ for all those points $a$ and reconstruct the complete matrix $B_p(x)$ using the Chinese Remainder Theorem. Implementing this strategy, we end up with the following theorem.

**Theorem 3.28 (Bostan–Caruso–Schost [13]).** There exists an algorithm that takes as input a differential operator

$$ \mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \dots + b_1(x) \cdot \partial_x + b_0(x) $$

over $\mathbb{F}_p(x)$ and outputs its p-curvature for a cost of $O(\mathrm{deg}(dn^\omega p))$ operations in $\mathbb{F}_p$ with

$$ d = \max(\mathrm{deg} f(x), \mathrm{deg}(f(x)b_0(x)), \dots, \mathrm{deg}(f(x)b_{n-1}(x))) $$

where $f(x)$ is a common denominator of the $b_i(x)$'s.

Before commenting on the above result, we need to explain some notation. Firstly, the notation $O(\omega)$ means that we are hiding logarithmic factors. Secondly, the exponent $\omega$ refers to what we usually call a feasible exponent for the matrix multiplication. It simply means that we suppose that we are given an algorithm that computes the product of two square matrices of size $n$ with at most $O(n^\omega)$ operations in the base field. The naive method for multiplying matrices (the one we have all learnt in our first course of linear algebra) indicates that we can take $\omega = 3$. However, it turns out that better algorithms exist. For example, Strassen's algorithm [66] results in $\omega = \log_2 7 \approx 2.8$. Nowadays, the best known value for $\omega$ is about 2.37188 and the corresponding algorithm is due to Duan, Wu and Zhou [33]. It is a widely open conjecture if one can take $\omega = 2 + \varepsilon$ for all $\varepsilon > 0$.

The announced complexity $O(\mathrm{deg}(dn^\omega p))$ should be compared to the size of the output, i.e. the number of scalars in $\mathbb{F}_p$ needed to write down completely the p-curvature. By Proposition 3.26, $B_p$ if an $n \times n$ matrix whose entries are rational functions with numerators and denominators of degree at most $dp$; in practice, this bound is in general sharp. Therefore, the size of the output is about $dn^2p$. As a consequence, the algorithm behind Theorem 3.28 would be quasi-optimal (i.e. optimal up to constant and logarithmic factors) if $\omega$ were equal to 2. Even if this limit cannot be attained, this comparison underlines the good performances of the algorithm. In practice, using it makes it possible to compute p-curvatures of operators of order and degree 20 in a few seconds when $p < 100$ and in about half an hour when $p = 12007$.
# 4 About the computation of the p-curvature

After Cartier's lemma (Theorem 3.19) and Grothendieck's conjecture (Conjecture 3.17), it is clear that the p-curvature is an invariant of primary importance of linear differential equations in characteristic p. In this section, we outline some algorithms for computing it (or other quantities associated to it) efficiently.

## 4.1 Operators of order 1

In the case of differential equations of order 1 of the form (22):

$$y' + b(x)y = 0$$

we have seen in Theorem 3.12 that the p-curvature is explicitly given by the formula

$$b_p(x) = b^{(p-1)}(x) + b(x)^p.$$

Furthermore, computing explicitly the latter is quite an easy task which directly reduces to writing down the partial fraction decomposition of $b(x)$. Indeed, we have seen that if $b(x)$ decomposes as

$$b(x) = P(x) + \sum_{i=1}^{m} \sum_{j=1}^{r_i} \frac{\alpha_{i,j}}{(x - a_i)^j}$$

then

$$b_p(x) = P^{(p-1)}(x) + P(x)^p - \sum_{i=1}^{m} \sum_{\substack{1 \le j \le r_i \\ j \equiv 1 \pmod p}} \frac{\alpha_{i,j}}{(x - a_i)^{j+p-1}} + \sum_{i=1}^{m} \sum_{j=1}^{r_i} \frac{\alpha_{i,j}^p}{(x - a_i)^{pj}}.$$

Importantly for algorithmic purposes, we observe that the size of $b_p(x)$ is roughly the same as the size of the input $b(x)$, although the degree of (the numerator and the denominator of) the former is p times larger the degree of the latter. This apparent contradiction is explained by the fact that $b_p(x)$ is actually a function of $x^p$; it is then a sparse rational function, in the sense that a large proportion of its coefficients vanish.

**Remark 4.1.** Another option for computing explicitly the p-curvature of differential operators of order 1 is presented in [23, Thm. 2]; it avoids the computation of partial fraction decomposition and the factorization of the denominator of $b(x)$. Let us briefly sketch it with the differential operator

$$\partial_x - \frac{1}{x^2 + 1}$$

of Example 3.4. We write $b(x) = -1/(x^2 + 1)$ and assume $p > 2$ for simplicity. In order to compute $b_p(x)$, we expand $b(x)$ in power series:

$$b(x) = - \sum_{n=0}^{\infty} (-1)^n x^{2n}.$$
### 3.3.4 Similarity class and characteristic polynomial

We have seen previously (after Proposition 3.26) that, although the size of the p-curvature grows linearly with respect to p, its characteristic polynomial has roughly the same size as the input operator $\mathcal{L}$ even when p gets large. For this reason, one might hope to be able to compute the characteristic polynomial faster than the p-curvature itself.

The main observation for achieving this is a refinement of Lemma 3.27 which asserts that $B_p^{dp}(x) = B_p(x) \mod x^p$ is not only equal to

$$-(S^{dp}(x))^{-1} \cdot \frac{d^p S^{dp}(x)}{dx^p}$$

but it is further similar to the value at $x = 0$ (i.e. the reduction modulo $x$) of the latter product. On the other hand, evaluating this reduction can be done with standard algorithmic techniques (based on a so-called “baby step / giant step” approach) in time proportional to $\sqrt{p}$. Based on this, we obtain the next theorem.

**Theorem 3.29 (Bostan–Caruso–Schost [14]).** There exists an algorithm that takes as input a differential operator

$$\mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \dots + b_1(x) \cdot \partial_x + b_0(x)$$

over $\mathbb{F}_p(x)$ and outputs the invariant factors of its p-curvature for a cost of

$$O \sim (d^{\omega+\frac{3}{2}} n^{\omega+1} \sqrt{p})$$

operations in $\mathbb{F}_p$ where $d$ is defined as in Theorem 3.28.

We notice that the knowledge of the invariant factors is finer than that of the characteristic polynomial since the latter is the product of the formers. Furthermore, knowing the invariant factors, one can decide whether the p-curvature vanishes or not, whereas the characteristic polynomial only gives information about its nilpotency.

On the complexity side, we notice that the cost of the algorithm of Theorem 3.29 is worse with respect to the parameters $d$ and $n$ but better with respect to $p$. It is then interesting for small operators but large characteristic.

Finally, we mention that, if we are only interested by the characteristic polynomial of the p-curvature, faster algorithms (based on different techniques) exist.

**Theorem 3.30 (Bostan–Caruso–Schost [12]).** There exists an algorithm that takes as input a differential operator

$$\mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \dots + b_1(x) \cdot \partial_x + b_0(x)$$

over $\mathbb{F}_p(x)$ and outputs the characteristic polynomial of its p-curvature for a cost of

$$O \sim ((d+n)^\omega \min(d, n) \sqrt{p} + (d+n)^{\omega+1} \min(d, n))$$

operations in $\mathbb{F}_p$ where $d$ is defined as in Theorem 3.28.
The $(p-1)$-st derivative of $x^{2n}$ is 0 when $2n \neq -1 \pmod p$, and it is $-x^{2n-p+2}$ otherwise thanks to Wilson's theorem. Writing $2n = p-1 + pk$ and noticing that $k$ has to be even, $k=2\ell$, we end up with

$$b^{(p-1)}(x) = - \sum_{\ell=0}^{\infty} (-1)^{\ell-\frac{p-1}{2}} x^{2\ell p}$$

On the other hand, it is clear that $b(x)^p = -\sum_{n=0}^{\infty} (-1)^n x^{2np}$. Adding both sums, we find

$$b_p(x) = - \sum_{n=0}^{\infty} (-1)^n \cdot \left(1 - (-1)^{\frac{p-1}{2}}\right) \cdot x^{2np}.$$

When $p \equiv 1 \pmod 4$, the exponent $\frac{p-1}{2}$ is even and the term in the parenthesis vanishes. Therefore $b_p(x) = 0$ in this case. On the contrary, when $p \equiv 3 \pmod 4$, we have

$$b_p(x) = -2 \cdot \sum_{n=0}^{\infty} (-1)^n \cdot x^{2np} = -\frac{2}{1+x^{2p}} = -\frac{2}{(1+x^2)^p}.$$

and we recover the result of Example 3.14.

The same idea applies actually to any differential operator $\mathcal{L} = \partial_x - b(x)$. Indeed, as already noticed its $p$-curvature is a rational function in $x^p$. Besides, it is of the form $f(x^p)/\mathrm{denom}(b(x))^p$, where $f(x)$ is a polynomial of degree at most $d = \deg(b(x))$. Hence, it is enough to determine the power series expansion of $(b(x)^{(p-1)})^{1/p}$ at precision $x^d$, starting from the power series expansion of $b(x)$ at the same precision $d$. If $b(x) = \sum_{n \ge 0} u_n x^n$, then by Wilson's theorem we have $(b(x)^{(p-1)})^{1/p} = - \sum_{n \ge 1} u_{np-1} x^{n-1}$, and hence it is enough to be able to compute the terms $u_{p-1}, \dots, u_{dp-1}$. Since $\bar{b}(x)$ is a rational function, the sequence $(u_n)_{n \ge 0}$ satisfies a linear recursion of order at most $d$, with coefficients in $\mathbb{F}_p$ (given by the coefficients of $\mathrm{denom}(b)$). As the $N$-th term of such a linear recurrence with constant coefficients can be computed using $O(d \log(d) \log(N))$ operations in $\mathbb{F}_p$ using the technique of binary powering combined with fast polynomial multiplication in $\mathbb{F}_p[x]$, we conclude that the $p$-curvature $b_p(x)$ can be computed by an algorithm that uses $O(d^2 \log(d) \log(p))$ operations in $\mathbb{F}_p$.

Note that the reasoning above shows that the $p$-curvature $b_p(x)$ of $\mathcal{L} = \partial_x - b(x)$ is zero if and only if the following infinite systems of congruences holds

$$u_n \equiv u_{(n+1)p-1} \pmod p \quad \text{for all } n \ge 0.$$

## 4.2 Reading the $p$-curvature on the solutions

For differential equations of higher orders, the sparsity of the $p$-curvature no longer holds in general. However, we have the following result.

**Proposition 4.2.** Let

$$\mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \dots + b_1(x) \cdot \partial_x + b_0(x)$$

with $b_i(x) \in \mathbb{F}_p(x)$ and let $B(x)$ be the associated companion matrix (see equation (27)). Let $f(x) \in \mathbb{F}_p[x]$ be a common denominator of the $b_i(x)$'s and

$$d = \max(\deg f(x), \deg(f(x)b_0(x)), \dots, \deg(f(x)b_{n-1}(x))).$$
In practice, this algorithm performs quite well and allows for computing the characteristic polynomial in less than one hour for the parameters $d = n = 20$ and $p = 120 011$.

Recently, Pagès proved an “average version” of this theorem which, roughly speaking, states that, starting with a differential operator over $\mathbb{Q}(x)$, one can compute all its $p$-curvatures (up to some given bound) in average time proportional to $\log p$.

**Theorem 3.31** (Pagès [57]). There exists an algorithm that takes as input a differential operator

$$ \mathcal{L} = b_n(x)\partial_x^n + b_{n-1}(x)\partial_x^{n-1} + \cdots + b_1(x)\partial_x + b_0(x) $$

with $b_i(x) \in \mathbb{Z}[x]$ and outputs the characteristic polynomial of all the $p$-curvatures of $\mathcal{L}$ mod $p$ for $p \le N$ for a cost of

$$ O^(((d+n)^\omega(d+m) + (d+n)^3) \cdot Nd) $$

operations on bits where $d$ is the maximal degree of the $b_i(x)$'s and $m$ is the maximal bitsize of an integer appearing as the coefficient of one of the $b_i(x)$'s.

## 3.4 Algebraicity and integrality

The theoretical developments we carried out in §3.3.2 have also interesting consequences in characteristic 0. Let

$$ \mathcal{L} = \partial_x^n + b_{n-1}(x)\partial_x^{n-1} + \cdots + b_1(x)\partial_x + b_0(x) $$

be a differential operator over $\mathbb{Q}(x)$ and set:

$$ S(x) = \sum_{k=0}^{\infty} (-1)^k B_k(x) \cdot \frac{x^k}{k!} \qquad (26) $$

where the $B_k(x)$'s are defined, as previously, by Equation (24). The matrix $S(x)$ has entries in $\mathbb{Q}[[x]]$ and we write $S(x) = \sum_{i=0}^{\infty} S_i x^i$ where the $S_i$ are matrices over $\mathbb{Q}$. We emphasize that $S_i$ is not equal to $\frac{(-1)^i}{i!} B_i(x)$ because the latter has in general coefficients in $\mathbb{Q}(x)$.

### 3.4.1 Growth of denominators and $p$-curvatures

We would like to reduce everything modulo a prime number $p$ in order to compare $S(x)$ with the matrix $S^{\mathrm{dp}}(x)$ we have studied earlier; however, this operation requires some care because the $B_k(x)$'s may exhibit denominators. In order to do it properly, we introduce new rings. For any subring $R \subset \mathbb{Q}$, we set:

$$ R(x) = \left\{ \frac{P}{Q} \quad \text{with} \quad P, Q \in R[x] \text{ and } Q \text{ monic} \right\}, $$

$$ R(x)_0 = \left\{ \frac{P}{Q} \quad \text{with} \quad P, Q \in R[x], Q \text{ monic and } Q(0) \neq 0 \right\}, $$

$$ R[[x]]^{\mathrm{dp}} = \left\{ \sum_{i=0}^{\infty} a_i \frac{x^i}{i!} \quad \text{with} \quad a_i \in R \text{ for all } i \right\}. $$
Let $B_p(x)$ be the matrix of the $p$-curvature of $\mathcal{L}$ defined by the recurrence (28). Then, the following holds.

(i) The matrix $B_p(x)$ has the form $\frac{1}{f(x)^p}C_p(x)$ where the entries of $C_p(x)$ are all polynomials of degree at most $dp$.

(ii) The matrix $B_p(x)$ is similar to a matrix with coefficients in $\mathbb{F}_p(x^p)$.

The first assertion of the proposition follows easily from the induction formula (28) (see [23, Lemma 1]). The assertion (ii) is more subtle and requires the construction of a differential extension of $\mathbb{F}_p(x)$ over which $\mathcal{L}$ has a full basis of solutions. We refer to [41, Proposition 2.1.2] for a detailed proof.

Besides, we notice that (ii) implies that the trace, the determinant of $B_p(x)$ and, more generally, all the coefficients of its characteristic polynomial lie in $\mathbb{F}_p(x^p)$. This latter statement can be also seen as a consequence of the following easy lemma: the determinant of a linear mapping $\mathbb{F}_p(x)^n \to \mathbb{F}_p(x)^n$ commuting with $\nabla: F \mapsto F' + B(x)F$ has zero derivative. This alternative proof has the advantage to avoid going to an extension. We derive from what precedes that the sizes of the coefficients of the characteristic polynomial of $B_p(x)$ are comparable to the sizes of the $b_i(x)$'s, although the size of the $p$-curvature itself is in general $p$ times larger.

In order to design fast algorithms for computing the $p$-curvature, it is useful to go beyond Cartier's lemma (see Theorem 3.19) and relate the $p$-curvature to the shape of solutions. Let

$$ \mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \dots + b_1(x) \cdot \partial_x + b_0(x) $$

be a differential operator as before. In §3.2, we have seen that, when the $p$-curvature of $\mathcal{L}$ vanishes and the $b_i(x)$'s have no pole at 0, a fundamental system of solutions of $\mathcal{L}$ is explicitly given by

$$ \sum_{k=0}^{p-1} (-1)^k B_k(x) \frac{x^k}{k!} $$

where the $B_k(x)$'s are the matrices defined by the recurrence (28). In full generality, i.e. without assuming the vanishing of $B_p$, the idea is to consider the formal expansion

$$ \sum_{k=0}^{\infty} (-1)^k B_k(x) \frac{x^k}{k!} $$

Of course, this does not make sense in $\mathbb{F}_p(x)$ because of the division by $k!$, but we shall see that it does make sense in a suitable ring. A natural idea to achieve this goal is to introduce divided powers, i.e. to consider the ring of *Hurwitz series*, denoted by $\mathbb{F}_p[[x]]^{\text{dp}}$, whose elements are formal series of the form

$$ a_0 + a_1 \gamma_1(x) + a_2 \gamma_2(x) + \dots + a_k \gamma_k(x) + \dots $$

In the above expression, the $\gamma_k(x)$'s are formal names without further additional meaning. Of course, they should be thought of as $\frac{x^k}{k!}$ but we cannot write this division because the denominator may vanish. The multiplication on $\mathbb{F}_p[[x]]^{\text{dp}}$ is governed by the rule

$$ \gamma_m(x) \cdot \gamma_n(x) = \binom{m+n}{m} \cdot \gamma_{n+m}(x) $$
One has the following chain of inclusions $R[x] \subset R(x)_0 \subset R(x)$ together with an injective morphism of rings $R(x)_0 \hookrightarrow R[[x]]$. Besides, all these maps commute with the derivation $\frac{d}{dx}$ and, if $p$ is a prime number which is noninvertible in $R$, they are compatible with the reduction modulo $p$; in particular, we have the following commutative diagram:

<figure><img src="image_2.png" /></figure>

where all the arrows are homomorphisms of rings and commute with the derivation. We now assume that the entries of $B(x)$ have no pole at 0 and choose $R$ in such a way that they all belong to $R(x)_0$ (one can always take $R = \mathbb{Z}[\frac{1}{D}]$ for $D$ large enough). All the $B_m(x)$'s then assume coefficients in $R(x)_0$ as well and the matrix $S(x)$ is defined over $R[[x]]^{\text{dp}}$. Besides, the image of $S(x)$ modulo $p$ is the matrix $S^{\text{dp}}(x)$ modulo $p$ associated to the differential system $Y' + (B(x) \text{ mod } p) \cdot Y = 0$ in characteristic $p$. Lemma 3.27 then lefts us with the congruence:

$$S_p \equiv \frac{B_p(x)}{p!} \pmod{x}, \quad \text{i.e.} \quad S_p = \frac{B_p(0)}{p!}. \qquad (27)$$

Hence the $p$-curvatures (which, we recall, are the matrices $B_p(x) \text{ mod } p$) are directly related to the coefficients appearing in a fundamental system of solutions. In particular, the vanishing of $B_p(0)$ modulo $p$ is equivalent to the fact that the denominator of $S_p$ is coprime with $p$. Many variations on this theme are possible; a beautiful example is given by the next theorem.

**Theorem 3.32** (see Proposition 5.3.3 in [4]). Let

$$\mathcal{L} = \partial_x^n + b_{n-1}(x) \partial_x^{n-1} + \dots + b_1(x) \partial_x + b_0(x)$$

be a differential operator over $\mathbb{Q}(x)$ and $D$ be a positive integer. We assume that $\mathcal{L}$ admits $n$ solutions $Y_1, \dots, Y_r$ which have coordinates in $\mathbb{Z}[\frac{1}{D}][[x]]$ and are linearly independent over $\mathbb{Q}$. Then almost all the $p$-curvatures of $\mathcal{L}$ vanish.

**Remark 3.33.** Under Grothendieck conjecture, Theorem 3.32 can be elegantly rephrased as follows: if a differential system admits a basis of solutions in $\mathbb{Z}[\frac{1}{D}][[x]]$ for some positive integer $D$ (i.e., a basis of globally bounded solutions), then it also admits a basis of algebraic solutions. This is known as *Bézivin's conjecture*; it was formulated by Bézivin in [9, p. 299] and proved by him for $q$-differential equations [9, Thm. 7-1]. It is widely open whether Bézivin's conjecture is more difficult than Grothendieck's conjecture; at any rate, it appears that for the time being the only cases for which Bézivin's conjecture is proven are those for which Grothendieck's conjecture is known to be true.

**Example 3.34.** We illustrate Theorem 3.32 with the differential equation

$$y' = \frac{1}{x^2 + 1} y$$

already considered in Example 3.4. Over the rationals, the solutions are all proportional to the fundamental solution

$$y_0(x) = \exp(\arctan(x)) = \sum_{n=0}^{\infty} c_n x^n$$
for any nonnegative integers $m$ and $n$. Besides $\mathbb{F}_p[[x]]^{\text{dp}}$ is equipped with a natural derivation that takes $\sum_k a_k \gamma_k(x)$ to $\sum_k a_{k+1} \gamma_k(x)$. We have to be careful however that $\mathbb{F}_p[[x]]^{\text{dp}}$ is not a domain (e.g. $\gamma_1(x)^p = 0$) and, because of that, we cannot consider its ring of fractions. But still, if the matrix $B(x)$ has polynomial coefficients, all the $B_k(x)$'s have the same property and we can consider their images $B_k^{\text{dp}}(x)$ is the ring $M_n(\mathbb{F}_p[[x]]^{\text{dp}})$. We then can form

$$S^{\text{dp}}(x) = \sum_{k=0}^{\infty} (-1)^k B_k^{\text{dp}}(x) \cdot \gamma_k(x) \qquad (29)$$

obtaining this way a fundamental matrix of solutions of $\mathcal{L}$ over $\mathbb{F}_p[[x]]^{\text{dp}}$. This construction works actually more generally as soon as the entries of $B(x)$ have no pole at zero: in this case, we can expand them as series in $x$ in order to view them in $\mathbb{F}_p[[x]]^{\text{dp}}$. The precise relation between $S^{\text{dp}}(x)$ and the $p$-curvature is given by the next lemma.

**Lemma 4.3** (Bostan–Caruso–Schost [15]). We have the matrix relation:

$$\frac{d^p S^{\text{dp}}(x)}{dx^p} = -B_p^{\text{dp}}(x) \cdot S^{\text{dp}}(x).$$

*Proof.* Set $M = \mathbb{F}_p[x]$ and let $\Delta : M^n \to M^n, Y \mapsto \frac{dY}{dx} + B(x)Y$. By definition of the $p$-curvature, $\Delta^p$ is the multiplication by $B_p(x)$.

Now consider the endomorphism $\Delta^{\text{dp}}$ of $M^{\text{dp}} = \mathbb{F}_p[[x]]^{\text{dp}} \otimes_{\mathbb{F}_p[x]} M$ defined by

$$\Delta^{\text{dp}} = \frac{d}{dx} \otimes \text{id}_M + 1 \otimes \Delta_M.$$

One checks that it satisfies the Leibniz rule: for $f \in \mathbb{F}_p[[x]]^{\text{dp}}$ and $m \in M$, we have

$$\Delta^{\text{dp}}(f \otimes m) = \frac{df}{dx} \otimes m + f \otimes \Delta(m).$$

Hence, raising it to the $p$-th power, we obtain

$$(\Delta^{\text{dp}})^p (Y^{\text{dp}}) = \frac{d^p Y^{\text{dp}}}{dx^p} + B_p^{\text{dp}}(x) \cdot Y^{\text{dp}}$$

for all vectors $Y^{\text{dp}} \in M^{\text{dp}}$. The equality of the lemma follows given that the columns of $S^{\text{dp}}(x)$ map to 0 under $\Delta^{\text{dp}}$. $\square$

**Remark 4.4.** Hurwitz series provide a framework in which the Picard–Lindelöf (or, Cauchy-Lipschitz) theorem holds for linear differential equations in characteristic $p$. The recent article [51] provides another construction allowing this theorem, which has the advantage of yielding an integral domain but which, in return, does not seem so directly linked to the $p$-curvature.

## 4.3 Application to algorithmics

Given that the matrix $S^{\text{dp}}(x)$ is invertible, it follows from Lemma 4.3 that one can deduce the value of $B_p^{\text{dp}}(x)$ from that of $S^{\text{dp}}(x)$ which, in turn, can be computed using the techniques
where the $c_n$'s are rational numbers. We would like to find bounds on denominators of the $c_n$'s. More precisely, we fix a prime number $p \neq 2$ (for simplicity) and ask whether the denominators of the $c_n$'s are all coprime with $p$. For this, we use Dwork's criterion (see for instance [61, p. 409]) which asserts that the previous property holds if and only if

$$ \arctan(x^p) - p \cdot \arctan(x) \in p\mathbb{Z}_{(p)}[[x]] $$

where we recall that $\mathbb{Z}_{(p)}$ is the subring of $\mathbb{Q}$ consisting of fractions $\frac{a}{b}$ with $b$ coprime with $p$. We have:

$$ \arctan(x^p) - p \cdot \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{(2n+1)p} - p \cdot \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{(2n+1)}. \quad (28) $$

Clearly, when $2n + 1$ is coprime with $p$, the coefficient $p \cdot \frac{(-1)^n}{2n+1}$ is divisible by $p$. Therefore, we can only retain in the second sum of Eq. (28) the terms for which $2n \equiv -1 \pmod p$, i.e. $2n = p - 1 + 2\ell p$ with $\ell \in \mathbb{N}$. We thus get:

$$ \begin{aligned} \arctan(x^p) - p \cdot \arctan(x) &\equiv \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{(2n+1)p} - \sum_{\ell=0}^{\infty} \frac{(-1)^{\ell-\frac{p-1}{2}}}{2\ell+1} x^{(2\ell+1)p} \\ &= \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} \cdot \left(1 - (-1)^{\frac{p-1}{2}}\right) \cdot x^{(2n+1)p} \pmod{p\mathbb{Z}_{(p)}[[x]]}, \end{aligned} $$

hence $\arctan(x^p) - p \cdot \arctan(x)$ is divisible by $p$ when $p \equiv 1 \pmod 4$ and is not otherwise. In conclusion, the denominators of the $c_n$'s are all coprime with $p$ (that is, $\exp(\arctan(x))$ can be reduced modulo $p$) if and only if $p \equiv 1 \pmod 4$.

**Remark 3.35.** Note that the sequence $(T_n)_{n \ge 0}$ defined by $T_n = n! \cdot c_n$, satisfies the linear recurrence $T_{n+2} = T_{n+1} - n(n+1)T_n$ with $T_0 = T_1 = 1$, hence its terms are all integer numbers. Kelinsky proved in [50, Thm. 3] that for any prime $p \neq 2$, the term $T_p$ is congruent to 0 modulo $p$ if $p \equiv 1 \pmod 4$ (and to 2 if $p \equiv 3 \pmod 4$). The computation in Example 3.34 shows that a much stronger property holds: if $p \equiv 1 \pmod 4$, then $T_n$ is congruent to 0 modulo $p$ for all $n \ge p$, in other terms the generating function $\sum_{n \ge 0} T_n x^n$ is a polynomial modulo $p$.

In the same orbit, we mention two other theorems which are not directly related to our discussion but highlights other intricate relations between algebraicity and integrality.

**Theorem 3.36 (Conjectured by Ogus [56], proved by André [3]).** Let $f(x) \in \mathbb{Z}[[x]]$ such that $f'(x)$ is algebraic over $\mathbb{Q}(x)$. Then $f(x)$ is algebraic over $\mathbb{Q}(x)$.

**Theorem 3.37 (Conjectured by Katz [45], proved by the Chudnovsky-Chudnovsky [30]).** Let $f(x) \in \mathbb{Z}[[x]]$ such that $f'(x)/f(x)$ is algebraic over $\mathbb{Q}(x)$. Then $f(x)$ is algebraic over $\mathbb{Q}(x)$.

### 3.4.2 An analytic perspective on the $p$-curvature

All what precedes indicates that the vanishing properties of the $p$-curvatures tend to control the growth of the denominators of the coefficients of the fundamental system of solutions $S(x)$. Typically, after Eq. (27), we have seen that $B_p(0) \equiv 0 \pmod p$ is equivalent to the fact that $p$ does not divide the smallest common denominator of the entries of $S_p$.
of [17]. However, at this point, we have not solved entirely the question of the computation of the p-curvature for two reasons. Firstly, the previous reasoning assumes implicitly that the coefficients $b_i(x)$'s have no pole at 0. Secondly, and more importantly, the knowledge of $B_p^{dp}(x)$ is not enough to recover $B_p$. Precisely, letting $\mathbb{F}_p(x)_0$ denote the subring of $\mathbb{F}_p(x)$ consisting of functions with no pole at 0, the natural map $\delta_0 : \mathbb{F}_p(x)_0 \to \mathbb{F}_p[[x]]^{dp}$ is not injective; its kernel is the ideal generated by $x^p$.

To tackle these issues, the idea is to shift around any other base point $a \in \mathbb{F}_p$. Doing so, we get a new differential ring homomorphism $\delta_a : \mathbb{F}_p(x)_a \to \mathbb{F}_p[[x-a]]^{dp}$ and, reusing the same techniques, we end up with a fast algorithm that computes the p-curvature $B_p$ modulo $(x-a)^p$. Since we have moreover at our disposal a priori bounds on the size of the p-curvature (see Proposition 4.2), one can pick enough elements $a$ in $\mathbb{F}_p$ (or in a finite extension of $\mathbb{F}_p$, if needed), compute the p-curvature modulo $(x-a)^p$ for all those points $a$ and reconstruct the complete matrix $B_p(x)$ using the Chinese Remainder Theorem. Implementing this strategy, we end up with the following theorem.

**Theorem 4.5 (Bostan–Caruso–Schost [15]).** There exists an algorithm that takes as input a differential operator

$$ \mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \cdots + b_1(x) \cdot \partial_x + b_0(x) $$

over $\mathbb{F}_p(x)$ and outputs its p-curvature for a cost of $O(\log^2 p)$ operations in $\mathbb{F}_p$ with

$$ d = \max \left( \deg f(x), \deg(f(x)b_0(x)), \dots, \deg(f(x)b_{n-1}(x)) \right) $$

where $f(x)$ is a common denominator of the $b_i(x)$'s.

Before commenting on the above result, we need to explain some notation. Firstly, the notation $O(\cdot)$ means that we are hiding logarithmic factors. Secondly, the exponent $\omega$ refers to what we usually call a feasible exponent for the matrix multiplication. It simply means that we suppose that we are given an algorithm that computes the product of two square matrices of size $n$ using at most $O(n^\omega)$ operations in the base field. The naive method for multiplying matrices (the one we have all learnt in our first course of linear algebra) indicates that we can take $\omega = 3$. However, it turns out that better algorithms exist. For example, Strassen's algorithm [81] results in $\omega = \log_2 7 \approx 2.8$. Nowadays, the best known value for $\omega$ is about 2.37188 and the corresponding algorithm is due to Duan, Wu and Zhou [39]. It is a widely open conjecture if one can take $\omega = 2 + \varepsilon$ for all $\varepsilon > 0$.

The announced complexity $O(\log^2 p)$ should be compared to the size of the output, i.e. the number of scalars in $\mathbb{F}_p$ needed to write down completely the p-curvature. By Proposition 4.2, $B_p$ if an $n \times n$ matrix whose entries are rational functions with numerators and denominators of degree at most $dp$; in practice, this bound is in general sharp. Therefore, the size of the output is about $dn^2p$. As a consequence, the algorithm behind Theorem 4.5 would be quasi-optimal (i.e. optimal up to constant and logarithmic factors) if $\omega$ were equal to 2. Even if this limit cannot be attained, this comparison underlines the good performances of the algorithm. In practice, using it makes it possible to compute p-curvatures of operators of order and degree 20 in a few seconds when $p < 100$ and in about half an hour when $p = 12007$.
It is convenient to reformulate this class of properties in terms of *p*-adic valuation and *p*-adic numbers. We recall that the *p*-adic valuation of an integer *n*, denoted by $v_p(n)$, is the greatest integer *v* such that $p^v$ divides *n*. We then define the *p*-adic valuation of a rational number *x* = $\frac{a}{b}$ by setting $v_p(x) = v_p(a) - v_p(b)$. Having a denominator coprime with *p* is then equivalent to having nonnegative *p*-adic valuation. If *M* is a matrix over $\mathbb{Q}$, we define $v_p(M)$ as the minimum of the *p*-valuations of its entries.

Recall that, for all nonnegative integer $i$, we have:

$$v_p(i!) = \sum_{n=1}^{\infty} \lfloor \frac{i}{p^n} \rfloor \le \frac{i}{p-1}$$

(where $\lfloor \cdot \rfloor$ is the floor function). Hence, if $S(x)$ is defined over $\mathbb{Z}[\frac{1}{p}][\lfloor x \rfloor]$ and $p$ is a prime number which does not divide $N$, we deduce from the very first definition of $S(x)$ (see Eq. (26)) that $v_p(S_i) \ge -v_p(i!) \ge \frac{-i}{p-1}$ for all $i$. On the other hand, the property we have recalled above indicates that $B_p(0) \equiv 0 \pmod{p}$ if and only if $v_p(S_p) \ge 0$. It turns out actually that the vanishing of the *p*-curvature implies a general lower bound on the *p*-adic valuation of the $S_i$'s.

**Proposition 3.38.** Let

$$\mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \cdots + b_1(x) \cdot \partial_x + b_0(x)$$

be a differential operator over $\mathbb{Q}(x)$ and let $S$ be the matrix defined by Eq. (26). If the reduction of $\mathcal{L}$ modulo $p$ is well-defined and if the *p*-curvature of $\mathcal{L}$ mod $p$ vanishes, then for all $i \ge 0$:

$$v_p(S_i) \ge -v_p([i/p]!) \ge \frac{-i}{p(p-1)}. \qquad (29)$$

**Proposition 3.38** can be further rephrased in more analytic terms using *p*-adic numbers. We recall briefly that the field of *p*-adic numbers is the completion of $\mathbb{Q}$ for the *p*-adic norm $\|\cdot\|_p$ defined by $\|x\|_p = p^{-v_p(x)}$. Set $\omega = p^{-1/(p-1)}$. Without any assumption on the *p*-curvature, we have seen that $v_p(S_i) \ge \frac{-i}{p-1}$, that is $\|S_i\| \le \omega^{-i}$. This upper bound indicates that the (*p*-adic) radius of convergence of the series $S(x) = \sum_{i=0}^{\infty} S_ix^i$ is at least $\omega$. On the contrary, when the *p*-curvature vanishes, Proposition 3.38 tells us that $\|S_i\|_p \le \omega^{i/p}$ for all $i$. Hence, the radius of convergence of $S$ is now at least $\omega^{1/p} > \omega$. The *p*-curvature then measures some analytic properties of the solutions of our differential system in the *p*-adic world. A classical result in the theory of *p*-adic differential equations [49, Theorem 10.4.2], refining the so-called *Frobenius antecedent theorem* of Christol and Dwork [29, Thm. 5.4] (see also [48, Theorem 6.15]), asserts that when the radius of convergence of a fundamental system of solutions is strictly greater than $\omega$, the corresponding differential system $Y'(x) + B(x)Y(x) = 0$ is equivalent, up to a base change, to a system of the form $Y'(x^p) + C(x^p)Y(x^p) = 0$ where the entries of $C(x)$ are *p*-adic analytic functions converging on the closed unit disk. The differential system:

$$(\Sigma_1) : Y' + C(x)Y = 0$$

is called a *Frobenius antecedent* of $(\Sigma)$. The aforementioned convergence conditions allows for reducing $(\Sigma_1)$ modulo $p$, thus obtaining a new differential system on $\mathbb{F}_p(x)$. The latter has a well-defined *p*-curvature and if this second *p*-curvature persists to vanish, one eventually deduces that the radius of convergence of $S$ is at least $\omega^{1/p^2}$. When this occurs, one can continue this process and find a second Frobenius antecedent $(\Sigma_2)$ of $(\Sigma)$. If its *p*-curvature vanishes, the radius of convergence of $S$ will be at least $\omega^{1/p^3}$ and so on and so forth.
## 4.4 Similarity class and characteristic polynomial

We have seen previously (after Proposition 4.2) that, although the size of the p-curvature grows linearly with respect to p, its characteristic polynomial has roughly the same size as the input operator $\mathcal{L}$ even when p gets large. For this reason, one might hope to be able to compute the characteristic polynomial faster than the p-curvature itself.

The main observation for achieving this is a refinement of Lemma 4.3 which asserts that $B_p^{dp}(x) = B_p(x) \mod x^p$ is not only equal to

$$-(S^{dp}(x))^{-1} \cdot \frac{d^p S^{dp}(x)}{dx^p}$$

but it is further similar to the value at $x = 0$ (i.e. the reduction modulo $x$) of the latter product. On the other hand, evaluating this reduction can be done with standard algorithmic techniques (based on a “baby step / giant step” approach) in time proportional to $\sqrt{p}$. Based on this, we obtain the next theorem.

**Theorem 4.6** (Bostan–Caruso–Schost [16]). There exists an algorithm that takes as input a differential operator

$$\mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \cdots + b_1(x) \cdot \partial_x + b_0(x)$$

over $\mathbb{F}_p(x)$ and outputs the invariant factors of its p-curvature for a cost of

$$O \sim (d^{\omega+\frac{3}{2}} n^{\omega+1} \sqrt{p})$$

operations in $\mathbb{F}_p$ where $d$ is defined as in Theorem 4.5.

We notice that the knowledge of the invariant factors is finer than that of the characteristic polynomial since the latter is the product of the formers. Furthermore, knowing the invariant factors, one can decide whether the p-curvature vanishes or not, whereas the characteristic polynomial only gives information about its nilpotency.

On the complexity side, we notice that the cost of the algorithm of Theorem 4.6 is worse with respect to the parameters $d$ and $n$ but better with respect to $p$. It is then interesting for small operators but large characteristic.

Finally, we mention that, if we are only interested by the characteristic polynomial of the p-curvature, faster algorithms (based on different techniques) exist.

**Theorem 4.7** (Bostan–Caruso–Schost [14]). There exists an algorithm that takes as input a differential operator

$$\mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \cdots + b_1(x) \cdot \partial_x + b_0(x)$$

over $\mathbb{F}_p(x)$ and outputs the characteristic polynomial of its p-curvature for a cost of

$$O \sim ((d+n)^\omega \min(d, n) \sqrt{p} + (d+n)^{\omega+1} \min(d, n))$$

operations in $\mathbb{F}_p$ where $d$ is defined as in Theorem 4.5.
### 3.4.3 Towards Berkovich geometry

The connexion we have outlined above between p-curvature and p-adic analysis is quite nice but it is restricted to a fixed prime number p. In the perspective of the Grothendieck conjecture, we would like however to let p vary and study interactions between different primes.

An object capable to reflect these interactions is the *Berkovich line over $\mathbb{Z}$*, which was anticipated by Berkovich himself in [5] and then developed by Poineau [59]. By definition, it is the space $M(\mathbb{Z}[x])$ consisting of all *bounded multiplicative semi-norms $\| \cdot \| : \mathbb{Z}[x] \to \mathbb{R}$*. By definition, a semi-norm is a norm except that we authorize nonzero elements to have norm zero. It is said multiplicative if $\|fg\| = \|f\| \cdot \|g\|$ for all $f, g \in \mathbb{Z}[x]$ and bounded when

$$ \|a_0 + a_1 x + \dots + a_n x^n\| \le \max (|a_0|, |a_1|, \dots, |a_n|), $$

where $|a_i|$ denotes the usual absolute value of $a_i$. Of course, the Berkovich line $M(\mathbb{Z}[x])$ is not only a set but is endowed with a rich additional geometrical structure: a topology, a structural sheaf, etc. Besides, after Poineau's work, we have at our disposal a whole panel of powerful tools (inspired by modern algebraic geometry) to work with it.

Describing entirely the space $M(\mathbb{Z}[x])$ is not obvious, but it is not difficult to exhibit elements in it. Take $K = \mathbb{R}$ or $\mathbb{Q}_p$ (the field of p-adic numbers) for some prime number p and write $\| \cdot \|_K$ for the standard absolute value of K. Pick in addition an element $a \in K$ of norm at most 1 and a nonnegative real number r. Polynomials in $\mathbb{Z}[x]$ then define (real or p-adic) analytic functions on the closed ball $B(a, r)$ of center a and radius r. For $f \in \mathbb{Z}[x]$, we can then consider the sup norm on this domain:

$$ \|f\|_{a,r} = \sup_{x \in B(a,r)} \|f(x)\|_K. $$

One checks that it is an element of $M(\mathbb{Z}[x])$. Moreover, at least in the p-adic case, the completion of $\mathbb{Z}[x]$ with respect to this norm is the ring of p-adic analytic functions on $B(a, r)$; we shall denote it by $A_{a,r}$ in what follows. Another nice observation is that the notion of “ball of center a and radius r” has a well-defined meaning in the Berkovich geometry. Indeed, let $M(A_{a,r})$ be the Berkovich space associated to the ring $A_{a,r}$, i.e. the set of bounded multiplicative semi-norms on $A_{a,r}$. Restricting a semi-norm from $A_{a,r}$ to $\mathbb{Z}[x]$ lefts us with an injective map:

$$ M(A_{a,r}) \hookrightarrow M(\mathbb{Z}[x]) $$

whose image, denoted by $U_{a,r}$, is an open subset (for the Berkovich topology) in $M(\mathbb{Z}[x])$. In addition, we observe that any analytic function $f$ on $B(a, r)$, that is any element $f \in A_{a,r}$, induces a function on $U_{a,r}$: to each semi-norm $\| \cdot \| \in M(A_{a,r})$, we associate $\|f\|$. For this reason, it is natural to think at $U_{a,r}$ as the Berkovich incarnation of the ball of center a and radius r.

Coming back to our topic, let us consider a differential system $(\Sigma) : Y' + A(x)Y = 0$ over $\mathbb{Z}[x]$. By what we have seen previously, for almost all prime numbers p and all $a \in \mathbb{Z}_p$, the system $(\Sigma)$ admits a full basis of solutions in $A_{a,\omega}$ where we recall that we have set $\omega = p^{-1/(p-1)}$. In the Berkovich language, these functions give rise to new functions defined on $U_{a,\omega}$. Putting them together, we conclude that $(\Sigma)$ always admits a basis of solutions on a certain open subspace $U_0 \subset M(\mathbb{Z}[x])$. Now, the assumption that almost all the p-curvatures vanish shows that those solutions extend automatically to a larger subspace $U_1 \subset M(\mathbb{Z}[x])$. On the other
In practice, this algorithm performs quite well and allows for computing the characteristic polynomial in less than one hour for the parameters $d = n = 20$ and $p = 120 011$.

Recently, Pagès proved an “average version” of this theorem which, roughly speaking, states that, starting with a differential operator over $\mathbb{Q}(x)$, one can compute all its $p$-curvatures (up to some given bound) in average time proportional to $\log p$.

**Theorem 4.8** (Pagès [70]). There exists an algorithm that takes as input a differential operator

$$ \mathcal{L} = b_n(x) \cdot \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \dots + b_1(x) \cdot \partial_x + b_0(x) $$

with $b_i(x) \in \mathbb{Z}[x]$ and outputs the characteristic polynomial of all the $p$-curvatures of $\mathcal{L}$ mod $p$ for $p \le N$ for a cost of

$$ O^\sim(((d+n)^\omega(d+m) + (d+n)^3) \cdot Nd) $$

operations on bits, where $d$ is the maximal degree of the $b_i(x)$'s and $m$ is the maximal bitsize of an integer appearing as the coefficient of one of the $b_i(x)$'s.

# 5 Algebraicity and integrality

The theoretical developments we carried out in §4.2 have also interesting consequences in characteristic 0, as they allow to relate algebraicity of solutions with the growth of denominators of the coefficients in their Taylor expansions. The aim of this section is to appetize the reader with some charming results and perspectives in this direction.

Throughout this section, we let

$$ \mathcal{L} = \partial_x^n + b_{n-1}(x) \cdot \partial_x^{n-1} + \dots + b_1(x) \cdot \partial_x + b_0(x) $$

be a differential operator over $\mathbb{Q}(x)$ and assume that $b_i(x)$ has no pole at 0 for all $i$. We set:

$$ S(x) = \sum_{k=0}^{\infty} (-1)^k B_k(x) \cdot \frac{x^k}{k!} \qquad (30) $$

where the $B_k(x)$'s are defined by equation (28). The matrix $S(x)$ has entries in $\mathbb{Q}[[x]]$ and we write $S(x) = \sum_{i=0}^{\infty} S_i x^i$ where the $S_i$ are matrices over $\mathbb{Q}$. We emphasize that $S_i$ is not equal to $\frac{(-1)^i}{i!} B_i(x)$ because the latter has in general coefficients in $\mathbb{Q}(x)$.

## 5.1 Growth of denominators and $p$-curvatures

We would like to compare $S(x)$ with the matrix $S^{\mathrm{dp}}(x)$ introduced in equation (29) and, for this, to reduce everything $(\mathcal{L}, S(x), \text{etc.})$ modulo a prime number $p$. However, this operation requires some care because the $B_k(x)$'s may exhibit denominators. In order to do it properly, we introduce new rings. For any subring $R \subset \mathbb{Q}$, we set:

$$ R(x) = \left\{ \frac{P}{Q} \text{ with } P, Q \in R[x] \text{ and } Q \text{ monic} \right\}, $$

$$ R(x)_0 = \left\{ \frac{P}{Q} \text{ with } P, Q \in R[x], Q \text{ monic and } Q(0) \neq 0 \right\}, $$

$$ R[[x]]^{\mathrm{dp}} = \left\{ \sum_{i=0}^{\infty} a_i \frac{x^i}{i!} \text{ with } a_i \in R \text{ for all } i \right\}. $$
hand, in the Berkovich language, proving the Grothendieck conjecture amounts to showing
that there exist a full basis of solution on an étale covering of $M(\mathbb{Z}[x])$. Of course, these
remarks do not give any proof of the Grothendieck conjecture because $U_1$ itself is certainly
not an étale covering of $M(\mathbb{Z}[x])$. However, we think that this point of view has the potential
to lead to new interesting developments towards the Grothendieck conjecture in the future.

**Acknowledgements.** We are very grateful to Herwig Hauser for the initial idea of this sur-
vey, and for his constant support along the various phases of the project. Our warm thanks go
to Florian Fürnsinn and Sergey Yurkevich for their careful reading and helpful comments. This
work has been partially supported by the French grants CLAP-CLAP (ANR-18-CE40-0026) and
DeRerumNatura (ANR-19-CE40-0018), and by the French–Austrian project EAGLES (ANR-22-
CE91-0007 & FWF I6130-N).

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One has the following chain of inclusions $R[x] \subset R(x)_0 \subset R(x)$ together with an injective morphism of rings $R(x)_0 \hookrightarrow R[[x]]$. Besides, all these maps commute with the derivation $\frac{d}{dx}$ and, if $p$ is a prime number which is noninvertible in $R$, they are compatible with the reduction modulo $p$; in particular, we have the following commutative diagram:

<figure><img src="image_4.png" /></figure>

where all the arrows are homomorphisms of rings and commute with the derivation. We now assume that the entries of $B(x)$ have no pole at 0 and choose $R$ in such a way that they all belong to $R(x)_0$ (one can always take $R = \mathbb{Z}[\frac{1}{D}]$ for $D$ large enough). All the $B_m(x)$'s then take coefficients in $R(x)_0$ as well and the matrix $S(x)$ is defined over $R[[x]]^{\text{dp}}$. Besides, the image of $S(x)$ modulo $p$ is the matrix $S^{\text{dp}}(x)$ modulo $p$ associated to the differential system $Y' + (B(x) \text{ mod } p) \cdot Y = 0$ in characteristic $p$. Lemma 4.3 then leaves us with the congruence:

$$S_p \equiv \frac{B_p(x)}{p!} \pmod{x}, \quad \text{i.e.} \quad S_p = \frac{B_p(0)}{p!}. \qquad (31)$$

Hence the $p$-curvatures (which, we recall, are the matrices $B_p(x) \text{ mod } p$) are directly related to the coefficients appearing in a fundamental system of solutions. In particular, the vanishing of $B_p(0)$ modulo $p$ is equivalent to the fact that the denominator of $S_p$ is coprime with $p$. Many variations on this theme are possible; a beautiful example is given by the following theorem.

**Theorem 5.1** (see Proposition 5.3.3 in [5]). Let

$$\mathcal{L} = \partial_x^n + b_{n-1}(x) \partial_x^{n-1} + \dots + b_1(x) \partial_x + b_0(x)$$

be a differential operator over $\mathbb{Q}(x)$ and $D$ be a positive integer. We assume that $\mathcal{L}$ admits $n$ solutions $Y_1, \dots, Y_n$ which have coordinates in $\mathbb{Z}[\frac{1}{D}][[x]]$ and are linearly independent over $\mathbb{Q}$. Then almost all the $p$-curvatures of $\mathcal{L}$ vanish.

**Remark 5.2.** Under Grothendieck's conjecture, Theorem 5.1 can be elegantly rephrased as follows: if a differential system admits a basis of solutions in $\mathbb{Z}[\frac{1}{D}][[x]]$ for some positive integer $D$ (i.e., a basis of globally bounded solutions), then it also admits a basis of algebraic solutions. This is known as *Bézivin's conjecture*; it was formulated by Bézivin in [11, p. 299] and proved by him for $q$-differential equations [11, Thm. 7-1]; see also Conjecture 6.3 in [30]. It is widely open whether Bézivin's conjecture is more difficult than Grothendieck's conjecture; at any rate, it appears that for the time being the only cases for which Bézivin's conjecture is proven are those for which Grothendieck's conjecture is known to be true.

**Example 5.3.** We illustrate Theorem 5.1 with the differential equation

$$y' = \frac{1}{x^2 + 1} y$$

already considered in Example 3.4. Over the rationals, the solutions are all proportional to the fundamental solution

$$y_0(x) = \exp(\arctan(x)) = \sum_{n=0}^{\infty} c_n x^n$$
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where the $c_n$'s are rational numbers. We would like to find bounds on denominators of the $c_n$'s. More precisely, we fix a prime number $p \neq 2$ (for simplicity) and ask whether the denominators of the $c_n$'s are all coprime with $p$. For this, we use the following important result due to Dwork.

**Theorem 5.4** (Dwork's criterion, [75, p. 409]). Given a prime $p$ and $f(x) \in x\mathbb{Q}[[x]]$, we have $\exp(f(x)) \in 1 + x\mathbb{Z}_{(p)}[[x]]$ if and only if $f(x^p) - pf(x) \in px\mathbb{Z}_{(p)}[[x]]$, where $\mathbb{Z}_{(p)}$ is the subring of $\mathbb{Q}$ consisting of fractions $\frac{a}{b}$ with $b$ coprime with $p$.

According to Dwork's criterion, the denominators of the $c_n$'s are all coprime with $p$ if and only if

$$ \arctan(x^p) - p \cdot \arctan(x) \in p\mathbb{Z}_{(p)}[[x]]. $$

We have:

$$ \arctan(x^p) - p \cdot \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{(2n+1)p} - p \cdot \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{(2n+1)}. \quad (32) $$

Clearly, when $2n + 1$ is coprime with $p$, the coefficient $p \cdot \frac{(-1)^n}{2n+1}$ is divisible by $p$. Therefore, we can only retain in the second sum of equation (32) the terms for which $2n \equiv -1 \pmod p$, i.e. $2n = p - 1 + 2\ell p$ with $\ell \in \mathbb{N}$. We thus get:

$$ \begin{aligned} \arctan(x^p) - p \cdot \arctan(x) &\equiv \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{(2n+1)p} - \sum_{\ell=0}^{\infty} \frac{(-1)^{\ell-\frac{p-1}{2}}}{2\ell+1} x^{(2\ell+1)p} \\ &= \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} \cdot \left(1 - (-1)^{\frac{p-1}{2}}\right) \cdot x^{(2n+1)p} \pmod{p\mathbb{Z}_{(p)}[[x]]}, \end{aligned} $$

hence $\arctan(x^p) - p \cdot \arctan(x)$ is divisible by $p$ when $p \equiv 1 \pmod 4$ and is not otherwise. In conclusion, the denominators of the $c_n$'s are all coprime with $p$ (that is, $\exp(\arctan(x))$ can be reduced modulo $p$) if and only if $p \equiv 1 \pmod 4$.

**Remark 5.5.** Note that the sequence $(T_n)_{n \geq 0}$ defined by $T_n = n! \cdot c_n$, satisfies the linear recurrence $T_{n+2} = T_{n+1} - n(n+1)T_n$ with $T_0 = T_1 = 1$, hence its terms are all integer numbers. Kelinsky proved in [61, Thm. 3] that for any prime $p \neq 2$, the term $T_p$ is congruent to 0 modulo $p$ if $p \equiv 1 \pmod 4$ (and to 2 if $p \equiv 3 \pmod 4$). The computation in Example 5.3 provides a new proof of this statement: if $p \equiv 1 \pmod 4$, then $T_n$ is congruent to 0 modulo $p$ for all $n \geq p$, in other terms the generating function $\sum_{n \geq 0} T_n x^n$ is a polynomial modulo $p$.

In the same orbit, we mention two other theorems which are not directly related to our discussion but highlights other intricate relations between algebraicity and integrality.

**Theorem 5.6** (Conjectured by Ogus [69], proved by André [4]). Let $f(x) \in \mathbb{Z}[[x]]$ such that $f'(x)$ is algebraic over $\mathbb{Q}(x)$. Then $f(x)$ is algebraic over $\mathbb{Q}(x)$.

**Theorem 5.7** (Conjectured by Katz [55], proved by the Chudnovsky-Chudnovsky [35]). Let $f(x) \in \mathbb{Z}[[x]]$ such that $f'(x)/f(x)$ is algebraic over $\mathbb{Q}(x)$. Then $f(x)$ is algebraic over $\mathbb{Q}(x)$.
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## 5.2 An analytic perspective on the p-curvature

All that precedes indicates that the vanishing properties of the p-curvatures tend to control the growth of the denominators of the coefficients of the fundamental system of solutions $S(x)$. Typically, after equation (31), we have seen that $B_p(0) \equiv 0 \pmod{p}$ is equivalent to the fact that $p$ does not divide the smallest common denominator of the entries of $S_p$.

It is convenient to reformulate this class of properties in terms of p-adic valuation and p-adic numbers. We recall that the p-adic valuation of an integer $n$, denoted by $v_p(n)$, is the greatest integer $v$ such that $p^v$ divides $n$. We then define the p-adic valuation of a rational number $x = \frac{a}{b}$ by setting $v_p(x) = v_p(a) - v_p(b)$. Having a denominator coprime with $p$ is then equivalent to having nonnegative p-adic valuation. If $M$ is a matrix over $\mathbb{Q}$, we define $v_p(M)$ as the minimum of the p-valuations of its entries.

Recall that, for all nonnegative integer $i$, we have:

$$v_p(i!) = \sum_{n=1}^{\infty} \lfloor \frac{i}{p^n} \rfloor \le \frac{i}{p-1}$$

(where $\lfloor \cdot \rfloor$ is the floor function). Hence, if $S(x)$ is defined over $\mathbb{Z}[\frac{1}{p}][x]$ and $p$ is a prime number which does not divide $N$, we deduce from the very first definition of $S(x)$ (see equation (30)) that $v_p(S_i) \ge -v_p(i!) \ge \frac{-i}{p-1}$ for all $i$. On the other hand, the property we have recalled above indicates that $B_p(0) \equiv 0 \pmod{p}$ if and only if $v_p(S_p) \ge 0$. It turns out actually that the vanishing of the p-curvature implies a general lower bound on the p-adic valuation of the $S_i$'s.

**Proposition 5.8.** Let

$$\mathcal{L} = \partial_x^n + b_{n-1}(x) \partial_x^{n-1} + \cdots + b_1(x) \partial_x + b_0(x)$$

be a differential operator over $\mathbb{Q}(x)$ and let $S$ be the matrix defined by equation (30). If the reduction of $\mathcal{L}$ modulo $p$ is well-defined and if the p-curvature of $\mathcal{L}$ mod $p$ vanishes, then for all $i \ge 0$:

$$v_p(S_i) \ge -v_p(\lfloor i/p \rfloor!) \ge \frac{-i}{p(p-1)}. \qquad (33)$$

Proposition 5.8 can be further rephrased in more analytic terms using p-adic numbers. We recall briefly that the field of p-adic numbers is the completion of $\mathbb{Q}$ for the p-adic norm $\| \cdot \|_p$ defined by $\|x\|_p = p^{-v_p(x)}$. Set $\omega = p^{-1/(p-1)}$. Without any assumption on the p-curvature, we have seen that $v_p(S_i) \ge \frac{-i}{p-1}$, that is $\|S_i\|_p \le \omega^{-i}$. This upper bound indicates that the (p-adic) radius of convergence of the series $S(x) = \sum_{i=0}^{\infty} S_i x^i$ is at least $\omega$. On the contrary, when the p-curvature vanishes, Proposition 5.8 tells us that $\|S_i\|_p \le \omega^{-i/p}$ for all $i$. Hence, the radius of convergence of $S$ is now at least $\omega^{1/p} > \omega$. The p-curvature then measures some analytic properties of the solutions of our differential system in the p-adic world. A classical result in the theory of p-adic differential equations [60, Theorem 10.4.2], refining the Frobenius antecedent theorem of Christol and Dwork [34, Thm. 5.4] (see also [59, Theorem 6.15]), asserts that when the radius of convergence of a fundamental system of solutions is strictly greater than $\omega$, the corresponding differential system $Y'(x) + B(x)Y(x) = 0$ is equivalent, up to a base change, to a system of the form $Y'(x^p) + C(x^p)Y(x^p) = 0$ where the entries of $C(x)$ are p-adic analytic functions converging on the closed unit disk. The differential system:

$$(\Sigma_1) : Y' + C(x)Y = 0$$
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is called a *Frobenius antecedent* of $(\Sigma)$. The aforementioned convergence conditions allow for reducing $(\Sigma_1)$ modulo $p$, thus obtaining a new differential system on $\mathbb{F}_p(x)$. The latter has a well-defined $p$-curvature and if this second $p$-curvature persists to vanish, one eventually deduces that the radius of convergence of $S$ is at least $\omega^{1/p^2}$. When this occurs, one can continue this process and find a second Frobenius antecedent $(\Sigma_2)$ of $(\Sigma)$. If its $p$-curvature vanishes, the radius of convergence of $S$ will be at least $\omega^{1/p^3}$ and so on and so forth.

In the perspective of the Grothendieck conjecture, we would like to let $p$ vary and understand how the aforementioned convergence properties interact. A promising object, which looks capable to reflect these interactions is the *Berkovich line over $\mathbb{Z}$*, which was anticipated by Berkovich himself in [6] and then developed by Poineau [72]. By definition, it is the space $\mathcal{M}(\mathbb{Z}[x])$ consisting of all *bounded multiplicative semi-norms* $\|\cdot\| : \mathbb{Z}[x] \to \mathbb{R}$. A semi-norm is a norm except that we authorize nonzero elements to have norm zero. It is said multiplicative if $\|fg\| = \|f\| \cdot \|g\|$ for all $f, g \in \mathbb{Z}[x]$ and bounded when

$$ \|a_0 + a_1 x + \dots + a_n x^n\| \le \max (|a_0|, |a_1|, \dots, |a_n|), $$

where $|a_i|$ denotes the usual absolute value of $a_i$. Of course, the Berkovich line $\mathcal{M}(\mathbb{Z}[x])$ is not only a set but is endowed with a rich additional geometrical structure: a topology, a structural sheaf, etc. Besides, after Poineau's work, we have at our disposal a whole panel of powerful tools (inspired by modern algebraic geometry) to work with it.

Describing entirely the space $\mathcal{M}(\mathbb{Z}[x])$ is not obvious, but it is not difficult to exhibit elements in it. Take $K = \mathbb{R}$ or $\mathbb{Q}_p$ (the field of $p$-adic numbers) for some prime number $p$ and write $\|\cdot\|_K$ for the standard absolute value of $K$. Pick in addition an element $a \in K$ of norm at most 1 and a nonnegative real number $r$. Polynomials in $\mathbb{Z}[x]$ then define (real or $p$-adic) analytic functions on the closed ball $B(a, r)$ of center $a$ and radius $r$. For $f \in \mathbb{Z}[x]$, we can then consider the sup norm on this domain:

$$ \|f\|_{a,r} = \sup_{x \in B(a,r)} \|f(x)\|_K. $$

One checks that it is an element of $\mathcal{M}(\mathbb{Z}[x])$. Moreover, at least in the $p$-adic case, the completion of $\mathbb{Z}[x]$ with respect to this norm is the ring of $p$-adic analytic functions on $B(a, r)$; we shall denote it by $\mathcal{A}_{a,r}$ in what follows. Another nice observation is that the notion of “ball of center $a$ and radius $r$” has a well-defined meaning in the Berkovich geometry. Indeed, let $\mathcal{M}(\mathcal{A}_{a,r})$ be the Berkovich space associated to the ring $\mathcal{A}_{a,r}$, i.e. the set of bounded multiplicative semi-norms on $\mathcal{A}_{a,r}$. Restricting a semi-norm from $\mathcal{A}_{a,r}$ to $\mathbb{Z}[x]$ leaves us with an injective map:

$$ \mathcal{M}(\mathcal{A}_{a,r}) \hookrightarrow \mathcal{M}(\mathbb{Z}[x]) $$

whose image, denoted by $U_{a,r}$, is an open subset (for the Berkovich topology) in $\mathcal{M}(\mathbb{Z}[x])$. In addition, we observe that any analytic function $f$ on $B(a, r)$, that is any element $f \in \mathcal{A}_{a,r}$, induces a function on $U_{a,r}$: to each semi-norm $\|\cdot\| \in \mathcal{M}(\mathcal{A}_{a,r})$, we associate $\|f\|$. For this reason, it is natural to think at $U_{a,r}$ as the Berkovich incarnation of the ball of center $a$ and radius $r$.

Coming back to our topic, let us consider a differential system $(\Sigma) : Y' + A(x)Y = 0$ over $\mathbb{Z}[x]$. By what we have seen previously, for almost all prime numbers $p$ and all $a \in \mathbb{Z}_p$, the system $(\Sigma)$ admits a full basis of solutions in $\mathcal{A}_{a,\omega}$ where we recall that we have set $\omega = p^{-1/(p-1)}$. In the Berkovich language, these functions give rise to new functions defined on $U_{a,\omega}$. Putting
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them together, we conclude that $(\Sigma)$ always admits a basis of solutions on a certain open subspace $U_0 \subset \mathcal{M}(\mathbb{Z}[x])$. Now, the assumption that almost all the $p$-curvatures vanish shows that those solutions extend automatically to a larger subspace $U_1 \subset \mathcal{M}(\mathbb{Z}[x])$. On the other hand, in the Berkovich language, proving the Grothendieck conjecture amounts to showing that there exist a full basis of solution on an étale covering of $\mathcal{M}(\mathbb{Z}[x])$. Of course, these remarks do not give any proof of the Grothendieck conjecture because $U_1$ itself is certainly not an étale covering of $\mathcal{M}(\mathbb{Z}[x])$. However, we think that this point of view has the potential to lead to new interesting developments towards the Grothendieck conjecture in the future.

**Acknowledgements.** We are very grateful to Herwig Hauser for the initial idea of this survey, for his constant support along the various phases of the project, and for the marvelous workshops he organized in Lisbon these last years, from which the three authors benefited a lot (supported by the Austrian Science Fund FWF, project P-34765). Our warm thanks go to Florian Fürnsinn, Mark van Hoeij and Sergey Yurkevich for their careful reading and helpful comments. This work has been partially supported by the French grants CLap-CLap (ANR-18-CE40-0026) and DeRerumNatura (ANR-19-CE40-0018), and by the French–Austrian project EAGLES (ANR-22-CE91-0007 & FWF I6130-N).

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