On a theory of the $b$-function in positive characteristic

We present a theory of the $b$-function (or Bernstein-Sato polynomial) in positive characteristic. Let $f$ be a non-constant polynomial with coefficients in a perfect field $k$ of characteristic $p>0.$ Its $b$-function $b_f$ is defined to be an ideal of the algebra of continuous $k$-valued functions on $\mathbb{Z}_p.$ The zero-locus of the $b$-function is thus naturally interpreted as a subset of $\mathbb{Z}_p,$ which we call the set of roots of $b_f.$ We prove that $b_f$ has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Musta\c{t}\u{a} and is in terms of $D$-modules, where $D$ is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of $b_f$ and relate it to the test ideals of $f.$

The goal of this note is to present a theory of the b-function in positive characteristic.
Let us first recall the classical complex theory, see for example [9,Chapter 6].
The classical theory. Let Q ∈ C[x 1 , . . . , x n ] be a nonconstant polynomial. We denote by D the ring D A n of polynomial differential operators on These are the main theorems of the classical theory. In order to present our results, we would like to rephrase the definition of the b-function, after [11].
For a commutative ring A and an element a ∈ A, we denote by A a the localization of A obtained by inverting a. Let A = C[x 1 , . . . , x n ] in what follows. Let . It naturally is a left D A n ×A 1 -module, where t is a coordinate of the extra A 1 . We then have: Proposition. There is an injective σ-linear morphism γ : Moreover γ induces an isomorphism Thus γ induces a D-module isomorphism: under which the action of s transports to that of −∂ t t. Hence we can reformulate the definition of the b-function as follows: Our definition of the b-function in positive characteristic is analogous.
Positive characteristic. Let k be a field of characteristic p > 0. Let f ∈ k[x 1 , . . . , x n ] =: R be a nonconstant polynomial. Then is the Grothendieck ring of differential oper- We note the following: It is thus natural to make the subsequent definition.
Definition. Let I be an ideal of k[−∂ We give two proofs of the existence of the b-function in positive characteristic. That is that the b-function has finitely many roots, see Corollaries 1 and 2. We also prove the following analogue of Kashiwara's Theorem: Theorem. (Corollary 2) The roots of the b-function of f are negative rational numbers, ≥ −1.
Remark. Note that the positive characteristic theory differs from its complex counterpart in that the roots of the b-function are always ≥ −1.
The framework of our construction is the theory of unit F -modules ( [10]) and the proofs are based on a relation to test ideals ( [6]). In fact, our main result is a positive characteristic parallel to [4,Theorem B]: The roots of the b-function of f are the opposites of the F -jumping exponents of f which are in (0, 1] ∩ Z (p) .
Contents. In the first section, after considering the algebra of higher Euler operators, which we call the algebra of binomial coefficients, we give the definition of the b-function and show that the D-module M f /tM f underlies a finitely generated unit F -module. This is achieved by approximating by the level of differential operators and using the relation of these approximations to test ideals. Key there is the discreetness of the set of F -jumping exponents.
In the second section, we carry out a finer study of the relation to test ideals and prove the announced theorems. They ultimately rely on elementary properties of p-adic and 1 p -adic expansions of rational numbers, which we gather in a preliminary subsection.
We give some examples in the third section.
Acknowledgements. This theory of the b-function in characteristic p sprang from my attempt to understand the work of Mircea Mustaţȃ [12]. I thank him for suggesting the problem as well as answering many questions about test ideals. I would also like to thank Roman Bezrukavnikov and Pavel Etingof for interesting discussions.
1. The construction 1.1. The algebra of binomial coefficients. Let k be a field of characteristic p > 0, and let D A 1 k be the full ring of differential operators on the affine line over k, with coordinate t. After [12], we set the higher Euler operators to be the global differential operators ∈ D A 1 k , ∀e ∈ N : t p e . Definition 1. Let k be a field of characteristic p. Then the algebra of binomial coefficients k[ s p 0 , s p 1 , s p 2 , · · · ] is the sub-k-algebra of D A 1 k generated by the higher Euler operators ν e , e ∈ N. In this algebra, ∀e ∈ N, we denote −ν e by the binomial coefficient symbol s p e . Thus k[ s p 0 , s p 1 , s p 2 , · · · ] is the quotient of the polynomial ring in infinitely many variables k[x e ; e ∈ N] by the relations, ∀e ∈ N, x p e = x e . The following remark is fundamental, and justifies the notation: are canonically identified to the p-adic integers.
Proof. Let π : Z → Z/pZ = F p be the quotient map and let s : To a p-adic integer α, we associate a maximal ideal m α of k[ s p 0 , s p 1 , s p 2 , · · · ]. Let α = e∈N α e p e , ∀e ∈ N, α e ∈ {0, 1, . . . , p − 1}, be the p-adic expansion. Define a surjective F p -algebra morphism, be the quotient map. Then K = k and we set α m := e∈N s(m( s p e ))p e . These maps are inverse to each other.
The roots of I are the p-adic integers corresponding by Remark 1 to the maximal ideals containing I.

1.2.
Definition of the b-function. Let k be a field of positive characteristic p > 0, X a smooth variety over k and f : X → A 1 a nonconstant function on X. The . In order to define it, we would like to introduce some notations.
Suppose that X is affine and let O(X) = R. Then is naturally a left D X×A 1 -module. Let δ be the class of 1 We obtain the corresponding notions for general X by gluing. One sees that tM f is a left sub-D X [ν e ; e ∈ N]-module of M f ([12, lemma 6.4]) and thus that the quotient ; e ∈ N]-module and in particular a k[ν e ; e ∈ N]-module i.e. a k[ s p 0 , s p 1 , s p 2 , · · · ]-module. We then set: Definition 3. Let k be a field of positive characteristic p, X a smooth variety over k and f : X → A 1 a nonconstant function on X. The b-function b f of f is the annihilator of N f in k[ s p 0 , s p 1 , s p 2 , · · · ]. 1.3. On the Frobenius structure of N f . Let us show that the D X -module N f is a unit F -module. In order to do so, we consider the ring of differential operators as the inductive limit of the rings of differential operators of bounded level, and similarly for the modules of the theory.
Let l be a non negative integer and let D (l) X ⊂ D X be the sheaf of rings of differential operators with divided powers of level at most p l+1 − 1. We set One has that tM It induces an isomorphism: Hence the higher Euler operators act on the right-hand side by transport of structure. The actions are as follow: If the base p expansion of n is n = 0≤e<l+1 a e p e , then ∀e < l + 1, ν e acts on D (l) Proof. The first part of the statement is [12, Proposition 6.1.], the second is [12, Corollary 6.5.] and the last is [12, Remark 5.7.].
We would like to express the N (l) f as Frobenius pullbacks of a coherent sheaf. This coherent sheaf is expressed in terms of the test ideals of f.
Let λ ∈ R ≥0 and let τ (f λ ) ⊂ O X be the test ideal of exponent λ of f. The test ideals form a decreasing sequence of ideals ⊂ O X . A F -jumping exponent of f is a positive real number λ ∈ R ≥0 , such that, ∀ǫ > 0, τ (f λ−ǫ ) = τ (f λ ), see [12, Section 3] for definitions. They satisfy the following finiteness theorem: Thus, by Proposition 1, ∀l ∈ N, For l large enough, this direct sum is concentrated at the F -jumping exponents of f.
Indeed, for ǫ > 0 very small, let gr λ∈(0,1] (τ (f λ )) be the direct sum over the F -jumping exponents of f in (0, 1] : The sum is finite since by (1) of Theorem 1, there are only finitely many F -jumping exponents of f in (0, 1]. One has, Proposition 2. Let l ∈ N be large enough so that ∀n, 0 ≤ n < p l+1 , each interval ( n p l+1 , n+1 p l+1 ] contains at most one F -jumping exponent of f. Then there is a canonical isomorphism of left D (l) Proof. The proof is clear.
We can now prove the Theorem 2. Let k be a field of positive characteristic p, X a smooth variety over k and f : X → A 1 a nonconstant function on X. Then the left D X -module N f is a finitely generated unit F -module.
Proof. By Proposition 2, the left D X -module N f is expressed as an inductive limit of Frobenius pullbacks of a coherent O X -module, Hence it is a finitely generated unit F -module.

The relation to test ideals
In order to understand the finer properties of the b-function and its roots, we want to study further its relationship to test ideals.

Preliminaries on p-adic and 1
p -adic expansions. We will express the padic expansion of the roots of the b-function of f in terms of the 1 p -adic expansion of the F -jumping exponents of f. Let us first introduce some definitions.

Definition 4. The 1
p -adic expansion of a positive real number r is its unique base p expansion which does not have infinitely many consecutive zero coefficients, r = −b≤n r n ( 1 p ) n . Let us focus our attention on properties of the 1 p -adic expansion of rational numbers in (0, 1]. We say that the 1 p -adic expansion of a number r is periodic if ∃l ≥ 0, ∃d ≥ 1 and {b 1 , . . . , b l , a 1 , . . . , a d } ⊂ {0, . . . , p − 1} such that If l = 0, this means that the 1 p -adic expansion is of the form: r = a 1 p + · · · + a d p d + a 1 p d+1 + · · · + a d p 2d + a 1 p 2d+1 + . . . It is then called strictly periodic. The minima for l and d are called the length of the preperiod of r and the length of the period of r, respectively.
In the following lemma, we characterize the periodicity properties of 1 p -adic expansions.

Lemma 1.
(1) The 1 p -adic expansion of a positive real number r is periodic if and only if r is rational.
(2) The 1 p -adic expansion of a rational number r ∈ (0, 1] is strictly periodic if and only if r ∈ Z (p) .
Proof. The proof is straightforward and left to the reader.
To each rational number r ∈ (0, 1] ∩ Z (p) correspond finitely many conjugated rational numbers ∈ Z (p) : Definition 5. Let r be a positive rational number. By Lemma 1, its 1 p -adic expansion is periodic. Let a 1 a 2 · · · a d be its period. The 1 p -conjugates of r are the d positive rational numbers whose 1 p -adic expansion is obtained by replacing the period by its cyclic permutations {a 1 a 2 · · · a d , a d a 1 a 2 · · · , . . . , a 2 · · · a d a 1 }.
Let us now point out another property of strictly periodic 1 p -adic expansions. Lemma 2. Let s be a sequence s : N 0 → {0, . . . , p − 1}, and let Λ ⊂ (0, 1] ∩ Q be a finite set. Suppose that ∀N >> 0, ∃λ ∈ Λ such that the 1 p -adic expansion of λ starts by Proof. Let us, ∀l ≥ 1, denote by s l the number s l := s(l) p + s(l−1) p 2 + · · · + s(1) p l and let us say that the 1 p -adic expansion of a number r starts by s l , if r = s(l) p + s(l − 1) p 2 + · · · + s(1) p l + . . . By the finiteness of Λ, ∃λ ∈ Λ such that λ starts by s l , for arbitrary large l. Also, ∃L ≥ 0 such that, ∀l ≥ L, if α ∈ Λ starts with s l , then it is of this type. Let us show that the 1 p -adic expansion of such a λ is strictly periodic. By Lemma 1.1, the 1 p -adic expansion of λ is periodic. Let l be the length of the preperiod of λ. Choose n ≥ l and N ≥ n + l such that λ starts by s n and s N . That is, . . Thus, by the uniqueness of the 1 p -adic expansion of λ, the second equality implies that (b 1 , . . . , b l ) = (s(n), . . . , s(n − l + 1)). This combines with the third equality to imply that (b 1 , . . . , b l ) is a subsegment of (a 1 , a 2 , . . . , a d , a 1 , . . . ). Thus the 1 p -adic expansion of λ is strictly periodic.
Since the 1 p -adic expansion of λ is strictly periodic, each of the µ ∈ Λ∩Q starting by s n for arbitrary large n is a 1 p -conjugate of λ. Thus µ has a strictly periodic Proposition 4. Let d be the lcm of the lengths of the periods of the 1 p -adic expansions of the F -jumping exponents of f in (0, 1] ∩ Z (p) . Then ∃e > 0, a multiple of d, such that ∀l ∈ N as in Proposition 3 and ∀λ F -jumping exponent of f in (0, 1], the e-th iterate of the structure maps: induces a morphism: Moreover, this morphism vanishes if λ ∈ Z (p) .
Proof. By Proposition 3 and the description of the structure maps in Proposition 1, ∀i > 0, the i-th iterate of the structure maps sends for the unique m such that m p l+1 is the truncated 1 p -adic expansion of λ. By Proposition 3 and Lemma 2, the finiteness of the number of F -jumping exponents of f implies that, ∃N > 0 independent of l, such that, ∀λ a F -jumping exponent of f in (0, 1] but not in Z (p) , the above map for i ≥ N vanishes. Indeed, there is no n as above such that n p l+i+1 is the truncation of the 1 p -adic expansion of a F -jumping exponent of f.
Suppose that λ ∈ Z (p) . Then it is straightforward to see that for i a positive multiple of d, ∃!n, 0 ≤ n < p l+i+1 and n = m mod p l+1 such that n p l+i+1 is the truncation of the 1 p -adic expansion of a F -jumping exponent of f in (0, 1] ∩ Z (p) . Furthermore, that F -jumping exponent is λ.
Thus if e ′ ≥ N is a multiple of d, then e = e ′ + d fulfills the proposition.
We can now prove the Theorem 3. There is an integer e > 0 such that, ∀l ∈ N as in Proposition 3, the natural injection gr λ∈(0,1]∩Z (p) (τ (f λ )) → gr λ∈(0,1] (τ (f λ )) induces an isomorphism of unit F e -modules: Hence, by Lemma 3,λ l is the opposite of a 1 p -conjugate of λ. Thus by Remark 2,λ l is the opposite of a F -jumping exponent of f in (0, 1] ∩ Z (p) , and all their opposites are obtained this way. This concludes the proof of the theorem.

Remark 3.
Since there are only finitely many F -jumping exponents of f in (0, 1] by Theorem 1, this reproves Corollary 1.
The following is a direct consequence of Theorem 5: