Linear Differential Equations with Algebraic Solutions
Hélène Esnault, Freie Universität Berlin

Sackler Lectures, Nov. 2016

Hélène Esnault, Freie Universität Berlin

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Acknowledgements
Thank you tothe mathematicians of Tel Aviv university for the kind invitation to deliver
the Sackler Distinguished Lectures;Hélène Esnault, Freie Universität Berlin

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Acknowledgements
Thank you tothe mathematicians of Tel Aviv university for the kind invitation to deliver
the Sackler Distinguished Lectures;Mark Kisin for constructive remarks and comments on the slides, Yves
André for historical references.

Hélène Esnault, Freie Universität Berlin

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Table of ContentsHélène Esnault, Freie Universität Berlin

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Roots of unity and rational numbers
Question
Let a be a complex number, a 6= 0. How do we recognize whether a is aroot of unity, that is satisfies an equation of the formHélène Esnault, Freie Universität Berlin

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Roots of unity and rational numbers
Question
Let a be a complex number, a 6= 0. How do we recognize whether a is aroot of unity, that is satisfies an equation of the form
an = 1for some natural number n?

Hélène Esnault, Freie Universität Berlin

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Roots of unity and rational numbers
Question
Let a be a complex number, a 6= 0. How do we recognize whether a is aroot of unity, that is satisfies an equation of the form
an = 1for some natural number n?

Answer√Well, write a = exp(2π −1b) for some complex number b. Then bshould be a rational number.

Hélène Esnault, Freie Universität Berlin

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Roots of unity and rational numbers
Question
Let a be a complex number, a 6= 0. How do we recognize whether a is aroot of unity, that is satisfies an equation of the form
an = 1for some natural number n?

Answer√Well, write a = exp(2π −1b) for some complex number b. Then bshould be a rational number.

Too elementary!
Let us ask further, what makes the complex number a a root of unity, or
equivalently the complex number b a rational number?
Hélène Esnault, Freie Universität Berlin

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Kronecker’s analytic answer
TheoremAmong the complex numbers a which satisfy an equation of the form
an + c1 an−1 + . . . + cn = 0 for some integers ci ∈ Z, the roots of unity arethose for which all the complex solutions have absolute value 1.

Hélène Esnault, Freie Universität Berlin

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Kronecker’s analytic answer
TheoremAmong the complex numbers a which satisfy an equation of the form
an + c1 an−1 + . . . + cn = 0 for some integers ci ∈ Z, the roots of unity arethose for which all the complex solutions have absolute value 1.
We could equivalently say: If a ∈ OK , the ring of integers of a number field K ,then a ∈ µ∞ , the group of roots of unity, if and only if for all complex
embeddings ι : K ,→ C, |ι(a)| = 1.

Hélène Esnault, Freie Universität Berlin

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Kronecker’s analytic answer
TheoremAmong the complex numbers a which satisfy an equation of the form
an + c1 an−1 + . . . + cn = 0 for some integers ci ∈ Z, the roots of unity arethose for which all the complex solutions have absolute value 1.
We could equivalently say: If a ∈ OK , the ring of integers of a number field K ,then a ∈ µ∞ , the group of roots of unity, if and only if for all complex
embeddings ι : K ,→ C, |ι(a)| = 1.

One needs ci ∈ Z
If a is algebraic, but not integral, the theorem does not hold. For example,
we know√there are more Weil numbers of weight 0 than roots of unity, e.g.
a = 3+45 −1 , a is algebraic of degree 2, has norm 1 with respect to the two
embeddings, but is not a root of unity.
Hélène Esnault, Freie Universität Berlin

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Kronecker’s arithmetic answer
TheoremAmong the complex numbers b which satisfy an equation of the formb m + q1 b n−1 + . . . + qm = 0 for some rational numbers qi ∈ Q, therational numbers are those for which for all prime numbers p not dividing
the denominators of the qi , the mod p reduction of b satisfies(b mod p)p = (b mod p).

Hélène Esnault, Freie Universität Berlin

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Kronecker’s arithmetic answer
TheoremAmong the complex numbers b which satisfy an equation of the formb m + q1 b n−1 + . . . + qm = 0 for some rational numbers qi ∈ Q, therational numbers are those for which for all prime numbers p not dividing
the denominators of the qi , the mod p reduction of b satisfies(b mod p)p = (b mod p).
We could equivalently say: If b ∈ K , a number field, then it lies in the ring OK ,Sof integers except at finitely many places S; thenb ∈ Q ⊂ K , the prime field of K
if and only if for almost all prime numbers p,(b mod pOK ,S ) ∈ Fp ⊂ OK ,S /pOK ,S , the prime field of the residue ring .
Hélène Esnault, Freie Universität Berlin

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A Linear Differential Equation
Let us show that the Question above is equivalent to a question on rank
one linear differential equations on the affine line minus a point.

Hélène Esnault, Freie Universität Berlin

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A Linear Differential Equation
Let us show that the Question above is equivalent to a question on rank
one linear differential equations on the affine line minus a point.
• Let X := C \ {0} be the complex affine line minus the origin, withparameter t.

Hélène Esnault, Freie Universität Berlin

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A Linear Differential Equation
Let us show that the Question above is equivalent to a question on rank
one linear differential equations on the affine line minus a point.
• Let X := C \ {0} be the complex affine line minus the origin, withparameter t.
• Linear differential equation(?) df = b · f ·Hélène Esnault, Freie Universität Berlin

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A Linear Differential Equation
Let us show that the Question above is equivalent to a question on rank
one linear differential equations on the affine line minus a point.
• Let X := C \ {0} be the complex affine line minus the origin, withparameter t.
• Linear differential equation(?) df = b · f ·dt.
t• Solve in f , an analytic multivalued function on X .

Hélène Esnault, Freie Universität Berlin

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A Linear Differential Equation• Solution: f = λ · t b , for some λ ∈ C; 1-dimensional vector space over C
near any point.

Hélène Esnault, Freie Universität Berlin

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A Linear Differential Equation• Solution: f = λ · t b , for some λ ∈ C; 1-dimensional vector space over C
near any point.
√• Monodromy: γ : exp(2π −1θ), θ ∈ [0, 1]; follow solution f aroundt = 1 along√ the loop γ, so from θ = 0 to θ = 1. It transforms f toexp(2π −1b)f = af . So the local system is defined by:ρ : Z · γ = π1 (X ) → C× , γ 7→ a.

Hélène Esnault, Freie Universität Berlin

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A Linear Differential Equation• Solution: f = λ · t b , for some λ ∈ C; 1-dimensional vector space over C
near any point.
√• Monodromy: γ : exp(2π −1θ), θ ∈ [0, 1]; follow solution f aroundt = 1 along√ the loop γ, so from θ = 0 to θ = 1. It transforms f toexp(2π −1b)f = af . So the local system is defined by:ρ : Z · γ = π1 (X ) → C× , γ 7→ a.
• The datum of the local system ρ is equivalent to the datum of the linear
differential equation (?): this is an instance of the Riemann-Hilbertcorrespondence, here valid as the solutions have moderate growth atinfinity, or equivalently, (?) is regular singular.

Hélène Esnault, Freie Universität Berlin

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Kronecker’s theorems expressed with linear differential
equations: analytic side
• Then the solution f is algebraic over the field of rational functions C(t)if and only if the monodromy a is a root of unity (”if” is clear, ”only if”
discussed later).

Hélène Esnault, Freie Universität Berlin

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Kronecker’s theorems expressed with linear differential
equations: analytic side
• Then the solution f is algebraic over the field of rational functions C(t)if and only if the monodromy a is a root of unity (”if” is clear, ”only if”
discussed later).
• Note: to say |a| = 1 is to say that the monodromy is unitary, i.e.
π1 (X ) → U(1) = S 1 ⊂ C× = GL(1, C). And to say a is integral is to saythat a ∈ GL(1, OK ), where K is a number field and OK is its ring ofintegers (called ‘number ring’).

Hélène Esnault, Freie Universität Berlin

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Kronecker’s theorems expressed with linear differential
equations: analytic side
• Then the solution f is algebraic over the field of rational functions C(t)if and only if the monodromy a is a root of unity (”if” is clear, ”only if”
discussed later).
• Note: to say |a| = 1 is to say that the monodromy is unitary, i.e.
π1 (X ) → U(1) = S 1 ⊂ C× = GL(1, C). And to say a is integral is to saythat a ∈ GL(1, OK ), where K is a number field and OK is its ring ofintegers (called ‘number ring’).

Analytic characterization
The linear differential equation (?) has finite monodromy if and only if themonodromy is both unitary and defined over a number ring.

Hélène Esnault, Freie Universität Berlin

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Kronecker’s theorems expressed with linear differential
equations: arithmetic side
• Then the solution f is algebraic over the field of rational functions C(t)if and only if b is a rational number (again ”if” clear and ”only if”
discussed later).

Hélène Esnault, Freie Universität Berlin

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Kronecker’s theorems expressed with linear differential
equations: arithmetic side
• Then the solution f is algebraic over the field of rational functions C(t)if and only if b is a rational number (again ”if” clear and ”only if”
discussed later).
P• Note: solution (f mod p) is a finite expansion f = i λi t i for λi in(OK ,S mod p) satisfying
X
X
bλ0
+(
bλi t i−1 )(iλi t i−1 ) =t
i6=0Hélène Esnault, Freie Universität Berlin

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Kronecker’s theorems expressed with linear differential
equations: arithmetic side
• Then the solution f is algebraic over the field of rational functions C(t)if and only if b is a rational number (again ”if” clear and ”only if”
discussed later).
P• Note: solution (f mod p) is a finite expansion f = i λi t i for λi in(OK ,S mod p) satisfying
X
X
bλ0
+(
bλi t i−1 )(iλi t i−1 ) =t
i6=0• Compute: f solution if and only if b = 0, or λ0 = 0 and b = i ∈ (Z modp) \ {0} for some i.

Hélène Esnault, Freie Universität Berlin

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Kronecker’s theorems expressed with linear differential
equations: arithmetic side
• We conclude: b ∈ Q if and only if the solution (f mod p) exists foralmost all p.

Hélène Esnault, Freie Universität Berlin

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Kronecker’s theorems expressed with linear differential
equations: arithmetic side
• We conclude: b ∈ Q if and only if the solution (f mod p) exists foralmost all p.
• Discussion ”only if”: b ∈ Q is equivalent to f = t b (and thus to f = λt b , ∀λ ∈ C)
being algebraic over C(t).

Hélène Esnault, Freie Universität Berlin

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Kronecker’s theorems expressed with linear differential
equations: arithmetic side
• We conclude: b ∈ Q if and only if the solution (f mod p) exists foralmost all p.
• Discussion ”only if”: b ∈ Q is equivalent to f = t b (and thus to f = λt b , ∀λ ∈ C)
being algebraic over C(t).
n, m, n ∈ Q, (n, m) = 1, the minimal equation of• Easy direction: if b = mmn
f is (X − t ) ∈ C(t)[X ] as this is an irreducible polynomial. Vice-versa:
an algebraic equation (t b )N + ϕ1 (t)(t b )N−1 + . . . + ϕ0 (t) = 0 withϕi (t) ∈ C(t) implies that b ∈ Q.

Hélène Esnault, Freie Universität Berlin

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Kronecker’s theorems expressed with linear differential
equations: arithmetic side
• We conclude: b ∈ Q if and only if the solution (f mod p) exists foralmost all p.
• Discussion ”only if”: b ∈ Q is equivalent to f = t b (and thus to f = λt b , ∀λ ∈ C)
being algebraic over C(t).
n, m, n ∈ Q, (n, m) = 1, the minimal equation of• Easy direction: if b = mmn
f is (X − t ) ∈ C(t)[X ] as this is an irreducible polynomial. Vice-versa:
an algebraic equation (t b )N + ϕ1 (t)(t b )N−1 + . . . + ϕ0 (t) = 0 withϕi (t) ∈ C(t) implies that b ∈ Q.

Arithmetic characterization
The linear differential equation (?) has algebraic solutions if and only if it
is defined over a number field K , and for almost all p, it has a full set ofsolutions modulo p.
Hélène Esnault, Freie Universität Berlin

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Generalization of the analytic theorem•
QWe can reinterpret Kronecker’s analytic theorem as follows: writing(ι) : K ,→ VC for an r -dimensional vector space over C, where r equals
the number of real embeddings plus twice the number of complex
embeddings, and OK as a lattice VZ ⊂ VC , then×OK
⊂ GL(VZ ) ⊂ GL(VC ). The condition |ι(a)| = 1 ∀ι saysa ∈ diag(S 1 × . . . × S 1 ) ∩ GL(r , VZ ) ⊂ GL(VC ).

Hélène Esnault, Freie Universität Berlin

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Generalization of the analytic theorem•
QWe can reinterpret Kronecker’s analytic theorem as follows: writing(ι) : K ,→ VC for an r -dimensional vector space over C, where r equals
the number of real embeddings plus twice the number of complex
embeddings, and OK as a lattice VZ ⊂ VC , then×OK
⊂ GL(VZ ) ⊂ GL(VC ). The condition |ι(a)| = 1 ∀ι saysa ∈ diag(S 1 × . . . × S 1 ) ∩ GL(r , VZ ) ⊂ GL(VC ).
• In GL(VC ) endowed with the complex topology, the subgroupdiag(S 1 × . . . × S 1 ) is compact, while the subgroup GL(r , VZ ) is discrete.

Hélène Esnault, Freie Universität Berlin

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Generalization of the analytic theorem•
QWe can reinterpret Kronecker’s analytic theorem as follows: writing(ι) : K ,→ VC for an r -dimensional vector space over C, where r equals
the number of real embeddings plus twice the number of complex
embeddings, and OK as a lattice VZ ⊂ VC , then×OK
⊂ GL(VZ ) ⊂ GL(VC ). The condition |ι(a)| = 1 ∀ι saysa ∈ diag(S 1 × . . . × S 1 ) ∩ GL(r , VZ ) ⊂ GL(VC ).
• In GL(VC ) endowed with the complex topology, the subgroupdiag(S 1 × . . . × S 1 ) is compact, while the subgroup GL(r , VZ ) is discrete.
• Thus diag(S 1 × . . . × S 1 ) ∩ GL(r , VZ ) is finite.

Hélène Esnault, Freie Universität Berlin

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Generalization of the analytic theorem•
QWe can reinterpret Kronecker’s analytic theorem as follows: writing(ι) : K ,→ VC for an r -dimensional vector space over C, where r equals
the number of real embeddings plus twice the number of complex
embeddings, and OK as a lattice VZ ⊂ VC , then×OK
⊂ GL(VZ ) ⊂ GL(VC ). The condition |ι(a)| = 1 ∀ι saysa ∈ diag(S 1 × . . . × S 1 ) ∩ GL(r , VZ ) ⊂ GL(VC ).
• In GL(VC ) endowed with the complex topology, the subgroupdiag(S 1 × . . . × S 1 ) is compact, while the subgroup GL(r , VZ ) is discrete.
• Thus diag(S 1 × . . . × S 1 ) ∩ GL(r , VZ ) is finite.
• Thus a is a root of unity.

Hélène Esnault, Freie Universität Berlin

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Generalization of the analytic theorem
Theorem (Nobody’s theorem, it is obvious: same topologicalargument)
Let X be a complex manifold, ρ : π1 (X ) → GL(r , C) be a local system.
Then it has finite monodromy if it is definable over Z and is unitary.

Hélène Esnault, Freie Universität Berlin

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Generalization of the analytic theorem
Theorem (Nobody’s theorem, it is obvious: same topologicalargument)
Let X be a complex manifold, ρ : π1 (X ) → GL(r , C) be a local system.
Then it has finite monodromy if it is definable over Z and is unitary.
• Later on, we shall interpret unitary local systems as polarizable variationsof Hodge structure of weight 0. We keep this in mind.

Hélène Esnault, Freie Universität Berlin

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Restricted formulation of Grothendieck’s p-curvature
conjecture
Grothendieck’s p-curvature conjecture, restricted formulation
Let X be a smooth variety defined over a number field K . Then a systemof linear differential equations M has algebraic solutions if and only if it
has a full set of solutions modulo p for almost all p.

Hélène Esnault, Freie Universität Berlin

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Restricted formulation of Grothendieck’s p-curvature
conjecture
Grothendieck’s p-curvature conjecture, restricted formulation
Let X be a smooth variety defined over a number field K . Then a systemof linear differential equations M has algebraic solutions if and only if it
has a full set of solutions modulo p for almost all p.
• This is the precise generalization of the ’Arithmetic Formulation’ of
Kronecker’s theorem translated for (X , M) = (Gm , (?)).

Hélène Esnault, Freie Universität Berlin

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Restricted formulation of Grothendieck’s p-curvature
conjecture
Grothendieck’s p-curvature conjecture, restricted formulation
Let X be a smooth variety defined over a number field K . Then a systemof linear differential equations M has algebraic solutions if and only if it
has a full set of solutions modulo p for almost all p.
• This is the precise generalization of the ’Arithmetic Formulation’ of
Kronecker’s theorem translated for (X , M) = (Gm , (?)).

Small historical remark by Yves AndréThe problem of characterizing which linear differential equations havealgebraic solutions goes back to Fuchs (1875), followed by Schwarz,
Jordan, Poincaré...Landau ”Eine Anwendung des Eisensteinschen Satzes
auf die Theorie der Gausschen Differentialgleichung” (Crelle (1904)).
Hélène Esnault, Freie Universität Berlin

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More precise formulation• We should be more precise.

Hélène Esnault, Freie Universität Berlin

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More precise formulation• We should be more precise.
• To say that (M mod p) has a full set of solutions means the following.

Hélène Esnault, Freie Universität Berlin

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More precise formulation• We should be more precise.
• To say that (M mod p) has a full set of solutions means the following.
• Locally on U Zariski open in X with local coordinates (x1 , . . . , xn ), M isdefined byrX(??) ∂xs fi =aij fj ,j=1

for s = 1, . . . , n, i, j = 1, . . . , r , aij regular functions. Then there are r
r -vectors of functions g j := (g1j , . . . , grj ), j = 1, . . . , r in (O(U) mod p)
such that any r -vector with entries in (O(U) mod p) is a linear
combination of the vectors g j with coefficients in (O(U) mod p).

Hélène Esnault, Freie Universität Berlin

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More precise formulation• To say that (X , M) is defined over a number field means X is defined byalgebraic equations with coefficients in a number field, K say, and (??) has
coefficients aij ∈ O(UK ).

Hélène Esnault, Freie Universität Berlin

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More complete formulation of Grothendieck’s p-curvature
conjecture
We wish to be free of the assumption that (X , M) be defined over anumber field. It goes as follows. If not, that is if (X , M) is only definedover C, then it is indeed defined over a Z-algebra of finite type, R say,which contains all the coefficients of the equations defining X , soaij ∈ O(UR ) in (??). One writes (X , M) = (X , M)R ⊗R C. For all
specializations R → K , where K is a number field, (called K -valued
points), (X , M) ⊗R K is defined over the number field K . For
specializations R → κ, where κ is a finite field, (called closed points),(X , M) ⊗R κ is defined over a finite field.

Hélène Esnault, Freie Universität Berlin

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More complete formulation of Grothendieck’s p-curvature
conjecture
Grothendieck’s p-curvature conjecture, more general formulation
Let X be a smooth complex variety. Then a system of linear differential
equations M has algebraic solutions if and only if M ⊗R κ as a full set ofsolutions for all closed points κ of a ring of definition R.

Hélène Esnault, Freie Universität Berlin

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Easy direction
Theorem (Easy direction)
If M has algebraic solutions, then M ⊗R κ has a full set of solutions for all
closed points κ of a ring of definition R.

Hélène Esnault, Freie Universität Berlin

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Easy direction
Theorem (Easy direction)
If M has algebraic solutions, then M ⊗R κ has a full set of solutions for all
closed points κ of a ring of definition R.

Proof.
• There is an easy argument which relies on the notion of p-curvature
which we carefully avoided up to now (even though the conjecture carries
this name).

Hélène Esnault, Freie Universität Berlin

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Easy direction
Theorem (Easy direction)
If M has algebraic solutions, then M ⊗R κ has a full set of solutions for all
closed points κ of a ring of definition R.

Proof.
• There is an easy argument which relies on the notion of p-curvature
which we carefully avoided up to now (even though the conjecture carries
this name).
• One has to show that when we take the mod p reduction of a finite étale
cover Y → X which trivializes the differential equation MY , then thesolutions of the differential equation on (Y mod p) come from (X mod p).
The solutions on (Y mod p) form a bundle on the Frobenius twist of (Ymod p), and because MY comes from M, this bundle comes from the
Frobenius twist of (X mod p). This bundle is the sheaf of solutions.
Hélène Esnault, Freie Universität Berlin

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Two examples of global problems and theorems
understood via localization/specialization, and local-global
considerations
Theorem of Minkowski-Hasse
Minkowski: if a quadratic form defined over Q has a real zero and zeros inall the p-adic completions Qp of Q, then it has a zero on Q. (Hasse moregenerally: replace Q by a number field, and R and Qp by the completionat all places).

Differences
Minkowski-Hasse consider all the completions, Grothendieck considers
almost all non-archimedian specializations. Also, while a p-adic zero yields
a zero (mod p), it is true vice-versa only if the quadric has good reduction.

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Two examples of global problems and theorems understood
via localization/reduction, and local-global considerations
Theorem of Deligne-Sullivan
On X complex manifold, the C ∞ -vector bundle associated to a local
system ρ : π1top (X ) → GL(r , A), where A is a finite type over Z, is trivial ifρ ⊗A κi , i = 1, 2, are trivial for two closed points withchar(κ1 ) 6= char(κ2 ).

Differences
We have here specializations, not localizations, but of the monodromy, notof the underlying algebraic vector bundle; in addition, one needs only two
specializations.

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Two other classical examples
Additive version
Let XS be smooth over an open S of a number ring, ω be a regulardifferential form. Then ω is d-exact if and only if it is mod p for almost allp.

Hélène Esnault, Freie Universität Berlin

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Two other classical examples
Additive version
Let XS be smooth over an open S of a number ring, ω be a regulardifferential form. Then ω is d-exact if and only if it is mod p for almost allp.

Multiplicative version
Let XS be smooth an open S of a number ring, ω be a regular differential
form. Then ω is dlog-exact if and only if it is mod p for almost all p.

Hélène Esnault, Freie Universität Berlin

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Two other classical examples
Additive version
Let XS be smooth over an open S of a number ring, ω be a regulardifferential form. Then ω is d-exact if and only if it is mod p for almost allp.

Multiplicative version
Let XS be smooth an open S of a number ring, ω be a regular differential
form. Then ω is dlog-exact if and only if it is mod p for almost all p.
We shall see later that those last two statements are true as a corollary of
one of the rare instances where one has a solution to the conjecture.

Hélène Esnault, Freie Universität Berlin

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Number field condition
Theorem of André-Hrushovski
The more general formulation of Grothendieck’s p-curvature conjecture
reduces to the restricted formulation. More precisely: Let R be a ring offinite type over Q. If MR ⊗R K has a full set of algebraic solutions for all
points with values in number fields K , then M itself has a full set ofalgebraic solutions. So one may assume that (X , M) is defined over anumber field.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201623 / 34

Number field condition
Theorem of André-Hrushovski
The more general formulation of Grothendieck’s p-curvature conjecture
reduces to the restricted formulation. More precisely: Let R be a ring offinite type over Q. If MR ⊗R K has a full set of algebraic solutions for all
points with values in number fields K , then M itself has a full set ofalgebraic solutions. So one may assume that (X , M) is defined over anumber field.

Hrushovski: model theory
I’m incompetent to comment.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201623 / 34

Number field condition
André: pure algebraic geometry and group theory• Camille Jordan’s theorem, proved in fact in connection with Fuchs’
problem (1878): any finite subgroup of GL(r , C) has a normal abeliansubgroup of index bounded in term of r , thus independently of thecomplex linear representations associated to the various complex
embeddings of K for the specializations M ⊗ K . .

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201624 / 34

Number field condition
André: pure algebraic geometry and group theory• Camille Jordan’s theorem, proved in fact in connection with Fuchs’
problem (1878): any finite subgroup of GL(r , C) has a normal abeliansubgroup of index bounded in term of r , thus independently of thecomplex linear representations associated to the various complex
embeddings of K for the specializations M ⊗ K . .
• This reduces the problem to the case where the monodromies are all
finite abelian groups.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201624 / 34

Number field condition
André: pure algebraic geometry and group theory• Camille Jordan’s theorem, proved in fact in connection with Fuchs’
problem (1878): any finite subgroup of GL(r , C) has a normal abeliansubgroup of index bounded in term of r , thus independently of thecomplex linear representations associated to the various complex
embeddings of K for the specializations M ⊗ K . .
• This reduces the problem to the case where the monodromies are all
finite abelian groups.
• One uses algebraic geometry and the representability of the Pic-functor
to conclude.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201624 / 34

First Katz’ theorem: conditions at infinity
Theorem (Katz)
Let X be a smooth variety defined over a number field, let M be a systemof linear differential equations which has a full set of algebraic solutions
mod p for almost all p. Then after replacing X by a finite cover, M
extends to a good compactification. In other words, one may assume that
X is projective.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201625 / 34

Lefschetz’ and Belyi’s theorems• The Lefschetz theorem reduces the problem to X being a smoothprojective curve, defined over a number field.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 2016

26 / 34

Lefschetz’ and Belyi’s theorems• The Lefschetz theorem reduces the problem to X being a smoothprojective curve, defined over a number field.
• Belyi’s theorem according to which smooth projective curves definedover a number field ramify over the projective line P1 in three (rational)
points reduces the problem to X being P1 \ {0, 1, ∞}, defined over anumber field.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 2016

26 / 34Résumé
To summarize:
Grothendieck’s p-curvature conjecture predicts:(i) If X is a smooth projective curve defined over a number field, (Mmod p) has a full set of solutions for almost all p; or(ii) if X = P1 \ {0, 1, ∞} over a number field, (M mod p) has a full set ofsolutions for almost all p;then M has a full set of algebraic solutions.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201627 / 34Résumé
To summarize:
Grothendieck’s p-curvature conjecture predicts:(i) If X is a smooth projective curve defined over a number field, (Mmod p) has a full set of solutions for almost all p; or(ii) if X = P1 \ {0, 1, ∞} over a number field, (M mod p) has a full set ofsolutions for almost all p;then M has a full set of algebraic solutions.

Comment(i) suggests perhaps the use of techniques of global algebraic geometry;(ii) suggests perhaps the use of p-adic analysis.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201627 / 34

Second Katz’ theorem: Gauß-Manin connections and theuse of Nobody’s analytic theorem
Theorem (Katz)
Gauß-Manin connections verify Grothendieck’s p-curvature conjecture.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201628 / 34

Second Katz’ theorem: Gauß-Manin connections and theuse of Nobody’s analytic theorem
Theorem (Katz)
Gauß-Manin connections verify Grothendieck’s p-curvature conjecture.

Method of proof• By definition, the monodromy is integral, i.e. lies in GL(r , Z), and insome U(p, q) (up to conjugacy), as a variation of Hodge structure. Show:in fact it is unitary, i.e. in U(r , C), and apply Nobody’s theorem.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201628 / 34

Second Katz’ theorem: Gauß-Manin connections and theuse of Nobody’s analytic theorem
Theorem (Katz)
Gauß-Manin connections verify Grothendieck’s p-curvature conjecture.

Method of proof• By definition, the monodromy is integral, i.e. lies in GL(r , Z), and insome U(p, q) (up to conjugacy), as a variation of Hodge structure. Show:in fact it is unitary, i.e. in U(r , C), and apply Nobody’s theorem.
• To this: use geometry. Show “full sets of solutions mod p” implies(F -filtration mod p) stable under the connection, thus stable to start with.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201628 / 34

Theorems of Honda, Chudnovsky-Chudnovsky, André:
p-adic analysis
Theorem (Honda, Chudnovsky-Chudnovsky, André)Grothendieck’s p-curvature conjecture is true for M being a successive
extension of rank 1 connections.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201629 / 34

Theorems of Honda, Chudnovsky-Chudnovsky, André:
p-adic analysis
Theorem (Honda, Chudnovsky-Chudnovsky, André)Grothendieck’s p-curvature conjecture is true for M being a successive
extension of rank 1 connections.

Additive and multiplicative examples fall under this theorem
For any system: to have a full set of algebraic solutions is a property one
checks locally Zariski on X . Then the multiplicative example is precisely
the case of a rank 1 system of differential equations, while the additive
example if precisely the case of an extension of the trivial rank 1 system ofdifferential equations by itself.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201629 / 34

Theorems of Honda, Chudnovsky-Chudnovsky, André:
p-adic analysis
Method of proof• Honda-Chudnovsky2 : Dwork’s criterion for a p-adic function to be arational function;

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201630 / 34

Theorems of Honda, Chudnovsky-Chudnovsky, André:
p-adic analysis
Method of proof• Honda-Chudnovsky2 : Dwork’s criterion for a p-adic function to be arational function;• Extension by André: criterion for a p-adic function to be algebraic.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201630 / 34

Theorems of Honda, Chudnovsky-Chudnovsky, André:
p-adic analysis
Method of proof• Honda-Chudnovsky2 : Dwork’s criterion for a p-adic function to be arational function;• Extension by André: criterion for a p-adic function to be algebraic.
• Tannakian considerations.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201630 / 34

End of general knowledge/understanding
This is essentially all we know on Grothendieck’s p-curvature
conjecture, without reinforcing the assumptions.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201631 / 34

End of general knowledge/understanding
This is essentially all we know on Grothendieck’s p-curvature
conjecture, without reinforcing the assumptions.
The résumé suggested to try to use algebraic geometry and start with X
being a smooth projective curve. This is what we shall do in the next
lecture, based on joint work with Mark Kisin.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201631 / 34

Outlook• A generalisation: M with algebraic solutions is a special instance of an
M which is a Gauß-Manin connection, that is of an M which comes fromthe variation of Betti cohomology of a fibration Y → X over C. The caseof algebraic solutions is equivalent to Y → X finite.
• The generalization of the p-curvature conjecture is to say that
Gauß-Manin connections and their summands are characterized by theirp-curvatures as in the previous case, except that here one requires the M
on (X mod p) to be filtered so that the graded pieces have a full set ofsolutions. Again the ’easy’ direction is known (and due to Deligne): if M
is a summand of a Gauß-Manin, then mod p for almost all p, M is filteredwith graded having a full set of solutions.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201632 / 34

SimpsonA very recent development is based on Simpson’s earlier conjecture: alocal system which is rigid (i.e. isolated in its moduli) should be asummand of a Gauß-Manin connection. He proved this with Corlette infull generality for SL(2)-local, and with various restrictions with Langer for
SL(3)-local systems.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201633 / 34

Outlook: a conjecture
Conjecture
Let (X , M) be a system of linear differential equations on X smooth over
C. Assume M is simple, has quasi-nilpotent monodromies at infinity and isrigid. Then there is S affine over Z and a model (XS , MS ) such that for alls ∈ S, Ms is filtered and grMs has a full set of solutions.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201634 / 34