On v-adic aspects

of Function Field arithmetic

Jing Yu

National Center for Theoretical Sciences

TAIWAN

July 2004, at Clermont-Ferrand, France8-th International Conference on p-adic Functional Analysis

Typeset by AMS-TEX

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The group Gm and the exponential function ez.

C ez

−−−−→ Gm(C) = C×

n(·)y

y(·)n

C ez

−−−−→ Gm(C) = C×

ez =∞∑

n=0

zn

n!

Entire function of finite order satisfying algebraicdifferential equation.

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Analytic Consequences: Hermite-Lindemann The-orem, Lindemann-Weierstrass Theorem, Baker’s The-orem...

Take any prime number p, the series ez also definesanalytic function of a p-adic variable, no longerentire, only as function on the open disc normalizedradius p−1/(p−1).

Mahler 1930. p-adic Hermite-Lindemann Theo-rem. Also p-adic Baker’s Theorem. However p-adicLindemann-Weierstrass remains open:

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Conjecture: If α1, . . . , αn are algebraic numberslinearly independent overQ, with |αi|p < p−1/(p−1),then eα1 , . . . , eαn are algebraically independent overQ.

Arithmetic of function fields.Polynomial ring Fq[t] playing the role of Z.

L. Carlitz 1935, V.G. Drinfeld, 1973Fq[t]-action on Ga.

Example: Carlitz module.φC : Fq[t] → EndFq (Ga),

φC(t) : x 7−→ tx + xq .

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Let K = Fq(t) be fixed algebraic closure of Fq(t).

1973 Drinfeld: t-modules defined over K of rankr are:

φ(t) : x 7→ tx + g1xq + · · ·+ gr−1x

qr−1+ ∆xqr

where gi,∆ ∈ K, ∆ 6= 0.

Let C∞ be a fixed algebraic closure of Fq((1/t)),with K embedded in it. Have the exponential func-tion associated to a given t-action:

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C∞eφ−−−−→ Ga(C∞) = C∞

t(·)y

yφ(t)

C∞eφ−−−−→ Ga(C∞) = C∞

Equation: eφ(tz) = φ(t)(eφ(z)).

Its zero set Lφ ⊂ C∞ is a discrete free Fq[t]-submoduleof rank r(Fq[t]-“lattice”):

0 → Lφ → C∞eφ−−−−→ C∞ → 0.

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Facts: The exponential functions are Fq-linear en-tire functions of finite order which linearize theFq[t]-actions, with nice Taylor coefficients from fi-nite extensions of Fq(t). They also behave likefunctions satisfying algebraic differential equations.

Example: the Carlitz exponential is given by

eC(z) =∞∑

h=0

zqh

Dh,

where D0 = 1, and for i ≥ 1

Di = (tqi − tq

i−1) · · · (tqi − t).

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Moreover its zero set LC = Fq[t] π̃, with

π̃ = (t− tq)1

q−1

∞∏

i=1

(1− tq

i − t

tqi+1 − t

).

Non-archimedean analytic consequences. analoguesof Hermite-Lindemann Theorem, Lindemann-WeierstrassTheorem, Baker’s Theorem...

Theorem (J. Yu 1986). Let α 6= 0 ∈ C∞. Then atleast one of the two elements α, eφ(α) is transcen-dental. (Here transcendental means these elementsof C∞ are not in K = Fq(t).)

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Now take any “finite”prime v, let kv be the comple-tion of k = Fq(t) at v, and let Cv be the completionof a fixed algebraic closure of kv. Also fix embed-ding of K in Cv. We normalize additive valuationon Cv so that dv(α) = − ordv(α) deg v for α ∈ k.

The series eC(z) thus defines v-adic analytic func-tion on the open disc D ⊂ Cv of normalized radius

−deg v

qdeg v − 1.

We have v-adic Hermite-Lindemann Theorem. Alsov-adic Baker’s Theorem:

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Theorem (J.Yu 1992) Let β1, · · · , βn in D be lin-early independent over k, such that eC(β1), · · · , eC(βn)are all falls in K. Then 1, β1, · · · , βn must be lin-early independent over K.

Here eC(z) can be replaced by any eφ(z), the sameD works provided the Drinfeld modules φ in ques-tion are defined over the valuation ring Ov.

Point: The analysis come up is always non-archimedean.However the chosen place ∞ plays a role distinctfrom the other (finite) places v. For function fieldarithmetic, the starting place ∞ can be changed.

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Applications to arithmetic.Carlitz zeta values(1935), n ≥ 1,

ζC(n) =∑

a∈Fq [t]a monic

1an

∈ Fq

(( 1

t )).

Through “analytic continuation”, Goss 1980 alsodefines ζC(n) ∈ k for integers n ≤ 0. Identify vwith the corresponding monic irreducible polyno-mial in Fq[t], and let (analogue of Kubota-Leopoldt)

ζv(n) = (1− v−n)ζC(n), n ≤ 0,

ζv(n) = limi→∞

ζv(n− (qdeg v − 1)qi), n > 0.

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These v-adic zeta values ζv(n) ∈ kv for n ∈ Z.Theorem (J. Yu) Let n ≥ 0 be an integer. Ifn 6≡ (mod q − 1) then ζv(n) is transcendental.Otherwise ζv(n) = 0.

Note q − 1 is the number of signs in Fq[t].

This is based on v-adic analogue of Baker’s the-ory extended to higher dimensional Drinfeld mod-ules. Also rely on Anderson-Thakur’s work ex-pressing zeta values in terms of v-adic logarithmsof algebraic points on tensor power of Carlitz mod-ules.

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We may compre the situation with the classi-cal result on the non-vanishing of abelian p-adicregulators by a p-adic version of Baker’s theory(Leopoldt’s conjecture, A. Brumer). It amountsto the values of p-adic L-functions at n = 1.

Drinfeld module φ defined over a finite extensionL/k has the property that all coefficients of φ(t)are in Ov ∩ L, we can consider its reduction, i.e.reading the coefficients modulo v.

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Given any φ over L of rank r. We may change φto φ′ = b◦φ◦b−1 with suitable b ∈ L× if necessary,so that reduction at v makes sense. If there existsφ′ whose reduction is a Drinfeld module of rank r1

over the residue field L(v), φ is said to have stablereduction at v. In case r1 = r, φ is said to havegood reduction at v.

All Drinfeld modules have stable reduction at vafter finite extension of the coefficient field if nec-essary. Moreover, those Drinfeld modules φ nothaving potentially good stable reductions at v areparametrized by v-adic lattices in Cv.

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Analogues of Tate’s uniformization for elliptic curves.Let Cp be the completion of a fixed algebraic clo-sure of Qp. There is one-to-one correspondence be-tween isomorphism classes of elliptic curves E de-fined over Cp not having good reduction and rankone multiplicative lattices inside C×p .

Let φ(Cv) = (Ga(Cv), φ). By a lattice (v-adic)Λ ⊂ φ(Cv) we mean a discrete finitely generatedfree A-submodule.

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Theorem (V.G. Drinfeld) There is one-to-one cor-respondence between isomorphism classes of pairs(φ, Λ), φ is Drinfeld module of rank r1 over Cv hav-ing good reduction, Λ is lattice of rank r2 in φ(Cv),and isomorphism classes of Drinfeld modules overCv of rank r1 + r2 having reduction rank r1.

This is in contrast to the∞-adic uniformization.According to which there is a correspondence be-tween lattices of rank r inside C∞ and Drinfeldmodules of rank r over C∞.

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Given a lattice Λ ⊂ C∞, one considers entirefunction on C∞

eΛ(z) = z∏

0 6=ω∈Λ

(1− z

ω).

The Drinfeld module φ defined over C∞ correspond-ing to Λ satisfies

eΛ(az) = φ(a)(eΛ(z)),

for a ∈ Fq[t]. This entire function has an expansionof the form eΛ(z) =

∑∞i=0 aiz

qi

. If φ is defined overK, then all the Taylor coefficients ai are in K.

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Let v be a finite place of k. Given a pair (φ, Λ),where φ is a Drinfeld module over Cv having goodreduction and Λ ⊂ φ(Cv) is a v-adic lattice. Onealso associates entire function on Cv

eΛ,φ(z) = z∏

0 6=ω∈Λ

(1− z

ω).

The Drinfeld module ψ defined over Cv corre-sponding to Λ satisfies

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eΛ,φ(φ(a)(z)) = ψ(a)(eΛ,φ(z)), a ∈ Fq[t]

CveΛ,φ−−−−→ Cv

φ(t)

yyψ(t)

CveΛ,φ−−−−→ Cv

This entire function also has an expansion of theform eΛ,φ(z) =

∑∞i=0 biz

qi

. If both φ and ψ aredefined over K, then all the Taylor coefficients bi

are in K.

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Theorem (J. Yu). Let ψ and φ be Drinfeld mod-ules defined over K ⊂ Cv, ψ has rank r1 + r2 andreduction rank r1, φ has rank r1 and good reduc-tion. Suppose Λ is a lattice of rank r2 inside φ(Cv)such that (φ, Λ) is the pair corresponding to ψ un-der Drinfeld correspondence. Let α 6= 0 ∈ Cv.Then at least one of the two elements α, eΛ,φ(α) istranscendental.

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We are interested in Drinfeld module ψ definedover global field L ⊂ K. Embedding L into C∞ byσ, we obtain ∞-adic function eψσ (z) as analytictool. On the other hand, at those “finite”places ofL where ψ has potentially bad reductions, we alsohave a finite number of v-adic functions of the formeΛ,φ(z) which tell important information about theDrinfeld module ψ. All these entire functions areactually defined over L, one may change the place,studying them w-adically with w 6= ∞ or v.

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Drinfeld modules of rank 2 over L ⊂ K ⊂ Cv

are given by

ψ(t) = tτ0 + gτ + ∆τ2, g, ∆ ∈ L, ∆ 6= 0.

The isomorphism class of ψ over K is deter-mined by the invariant j(ψ) = gq+1/∆.

Suppose ψ has reduction rank one. Then ψ cor-respondes to a pair (φ, Λ), where φ is a Drinfeldmodule of rank one, and Λ ∈ φ(Cv) is a v-adiclattice of rank one.

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As all Drinfeld modules of rank one over Cv areisomorphic, we may take φ = φC , the Carlitz mod-ule. The rank one v-adic lattices in φC(Cv) aregiven by Λ = φC(Fq[t])(wΛ) with dv(wΛ) > 0 inCv. Set j(ψ) = j(wΛ), we then have the followinganalogue of Manin’s conjecture

Theorem. Suppose w ∈ K ⊂ Cv with dv(w) > 0.Then j(w) /∈ K.