On a problem à la Kummer-Vandiver for function fields
aa r X i v : . [ m a t h . N T ] F e b ON A PROBLEM `A LA KUMMER-VANDIVER FORFUNCTION FIELDS
BRUNO ANGL`ES AND LENNY TAELMAN
Abstract.
We use Artin-Schreier base change to construct coun-terexamples to a Kummer-Vandiver type question for functionfields. Introduction
Let p be a prime number and let F be the maximal real subfield of Q ( µ p ) . The famous Kummer-Vandiver conjecture asserts that Z p ⊗ Z Pic O F = { } . It has been verified for all p less than 163 , ,
856 [3]. However, heuris-tic arguments of Washington suggest that the number of counterexam-ples p up to X should grow as log log X , making it difficult to findeither counterexamples or convincing numerical evidence towards theconjecture.The second author has recently proven a function field analogue ofthe Herbrand-Ribet theorem, and formulated a version of the Kummer-Vandiver conjecture in this context [6]. In this note, which comple-ments [6], we construct counterexamples to this Kummer-Vandiverstatement.Let us now recall the statement of this analogue. Let A = F q [ T ] andlet C be the Carlitz module over A . This is the A -module scheme overSpec A whose underlying group scheme is the additive group G a,A , andon which A acts via the F q -algebra homomorphism φ : A → End G a,A : T T + τ where τ : G a,A → G a,A is the q -th power Frobenius endomorphism. Itis an example of a Drinfeld module, and in many ways it is an analogueof the multiplicative group in characteristic zero [4].Let P ∈ A be monic irreducible, let k = F q ( T ) be the fraction fieldof A and let K/k be the extension obtained by adjoining the P -torsionpoints of C . Then K/k is Galois and there is a canonical isomorphism ω P : Gal( K/k ) → ( A/P ) × , which one can think of as the mod P Teichm¨uller character. Let R bethe integral closure of A in K and put Y = Y P = Spec R .Let C [ P ] be the P -torsion subscheme of C and let C [ P ] D be itsCartier dual. Consider the flat cohomology group H P := H ( Y P, fl , C [ P ] D ) . This is an
A/P -vector space on which the Galois group Gal(
K/k ) acts,so it decomposes in isotypical components as H P = q deg P − M n =1 H P ( ω nP ) . In [6], it is shown that for n in the range 1 ≤ n < q deg P − q − H P ( ω n − P ) = { } if and only if B ( n ) ≡ P ) , where B ( n ) ∈ k is the n -th Bernoulli-Carlitz number. The Kummer-Vandiver problem can be stated as follows (see [6, Question 1]): Question 1. Is H P ( ω n − P ) = { } for n not divisible by q − ? The analogy with the classical Kummer-Vandiver conjecture is (im-plicitly) explained in [6, Remark 2]: using the Kummer sequence andflat duality it is shown that the classical Kummer-Vandiver conjectureis equivalent with the statement thatH ((Spec Z [ ζ ℓ ]) fl , µ D ℓ )( χ n − ℓ ) ? = 0 if n is odd , where ℓ is an odd prime, ζ ℓ a primitive ℓ -th root of unity and χ ℓ denotesthe mod ℓ cyclotomic character.In this paper we use Artin-Schreier change of variables and computercalculations to construct counterexamples to the above statement. Forexample, we use properties of the prime P = T − T + 1 ∈ F [ T ]to show that the prime Q = P ( T − T ) = T − T − T − T − T + 1satisfies H Q ( ω Q ) = 0. Note that 9840 = n − n = ( q deg Q − / Q is too high to allow for a direct compu-tation of H Q .In a forthcoming paper we compare the flat cohomology groups of [6]with the group of “units modulo circular units” introduced by Anderson[1], and show amongst other things that the Kummer-Vandiver problemof [6] is equivalent with Anderson’s Kummer-Vandiver conjecture [1, N A PROBLEM `A LA KUMMER-VANDIVER FOR FUNCTION FIELDS 3 § Acknowledgements.
The authors thank the referee for several sug-gestions and corrections that helped improve the paper.2.
Notation L -functions. Let
F/E be a finite abelian extension of functionfields of curves over F q . Assume that F q is algebraically closed in both E and F . Let χ : Gal( F/E ) → C × be a homomorphism, and let E χ ⊂ F be the fixed field of ker χ . We set L ( X, E, χ ) = Y v place of E (1 − χ ( v ) X deg v ) − ∈ C [[ X ]] , where χ ( v ) = χ (( v, E χ /E )) if v is unramified in E χ /E and χ ( v ) = 0otherwise. Here ( − , E χ /E ) denotes the global reciprocity map. Recallthat L ( X, E, χ ) is a rational function and that if χ = 1 then L ( X, E, χ )is a polynomial whose coefficients are algebraic integers.2.2.
The cyclotomic function fields.
Let p be a prime number. Let F q be a finite field having q elements, q = p s , where p is the charac-teristic of F q . Let A = F q [ T ] be the polynomial ring in one variable T and let k = F q ( T ) be its field of fractions. We denote the set ofmonic elements in A by A + . For n ≥ , we denote the set of elementsin A + of degree n by A n . We fix k , an algebraic closure of k. All finiteextensions of k considered in this note are assumed to be contained in k. We denote the unique place of k which is a pole of T by ∞ . Let P ∈ A be monic irreducible of degree d. We denote the P -thcyclotomic function field by K P (see [4], chapter 7). Recall that K P /k is the maximal abelian extension of k such that:(1) K P /k is unramified outside P and ∞ ,(2) K P /k is tamely ramified at P and ∞ ,(3) for every place v of K P above ∞ , the completion of K P at v isisomorphic to F q (( T ))( q − √− T ).The Galois group Gal( K P /k ) is canonically isomorphic with ( A/P A ) × and the subgroup F × q ⊂ ( A/P A ) × is both the inertia and the decom-position group of ∞ in K P /k .3. Cyclicity of divisor class groups
An Artin-Schreier extension and the function γ . Let i : A + → Z /p Z be the function that maps a polynomial T n + α T n − + · · · + α n BRUNO ANGL`ES AND LENNY TAELMAN to Tr F q / F p α ∈ Z /p Z . Observe that for all a, b ∈ A + we have i ( ab ) = i ( a ) + i ( b ).Let θ ∈ ¯ k be a root of X p − X = T . Then the extension ˜ k obtainedby adjoining θ to k is rational and we have ˜ k = F q ( θ ). The extensionramifies only at ∞ . The integral closure of A in ˜ k , which we denote by˜ A , is the polynomial ring F q [ θ ] in θ .We have an isomorphism of groups Z /p Z → Gal(˜ k/k ) given by n σ n = [ θ θ − n ] . Let ( − , e k/k ) be the Artin symbol for ideals, then for all a ∈ A + wehave [2, Lemma 2.1](1) ( aA, e k/k ) = σ i ( a ) . Let n be a positive integer. By [2, Lemma 3.2] for all m sufficientlylarge we have X a ∈ A m i ( a ) a n = 0 . We define γ ( n ) := X m ≥ X a ∈ A m i ( a ) a n ∈ A. Cyclotomic extensions of k and of ˜ k . Now, we fix a prime P in A of degree d such that i ( P ) = 0 . Set Q ( θ ) := P ( T ) ∈ e A . Note thatby (1) the polynomial Q ( θ ) ∈ F q [ θ ] is irreducible. Its degree is pd .Let K P be the P -th cyclotomic function field for A , with Galoisgroup ∆ = ( A/P A ) × , and let f K Q be the Q -th cyclotomic function fieldfor e A , with Galois group e ∆ = ( e A/Q e A ) × . By [2, Lemma 2.2] we have: K P ⊂ f K Q . Let L be the compositum of e k and K P inside f K Q . Then L is an abelianextension of k with Galois group Z /p Z × ∆. N A PROBLEM `A LA KUMMER-VANDIVER FOR FUNCTION FIELDS 5 e K Q LK P ˜ kk ∆ Z /p ZZ /p Z ∆The inclusion A/P A ⊂ e A/Q e A induces an injective homomorphism∆ → e ∆. On the other hand, we can identify the Galois group of L over˜ k with ∆, and obtain a surjective map e ∆ → ∆ , which is explicitly given by( e A/Q e A ) × → ( A/P A ) × : a Nm ˜ k/k a. Comparison of L -functions. Let W be the ring of Witt vectorsof A/QA and let W = W [ ζ p ], where ζ p is a primitive p -th root of unity.Let ω P : ∆ → W × and ω Q : e ∆ → W × be the Teichm¨uller characters.We will denote by f ω P the same character as ω P , but seen as a characteron Gal( L/ e k ). In particular, we have f ω P = ω qpd − qd − Q . Let n be an integer such that 1 ≤ n ≤ q d − . Then [2, Lemma 2.4](2) L ( X, L/ ˜ k, f ω P n ) = Y φ L ( X, L/k, φω nP ) , where φ runs over all characters of Gal(˜ k/k ) = Z /p Z .Observe that, if φ = 1 , then L ( X, φω nP ) is a polynomial of degree d (apply [2], Lemma 2.3 for both A and e A ). Furthermore, we have:(3) L ( X, L/k, ψω nP ) = d X m =0 X a ∈ A m ζ i ( a ) p ω P ( a ) n ! X m , where ψ : Z /p Z → W × is the character that maps 1 to ζ p . BRUNO ANGL`ES AND LENNY TAELMAN
Congruences.
Assume that n is not divisible by q −
1. Then theBernoulli-Goss polynomial β ( n ) is defined as follows: β ( n ) = X m ≥ X a ∈ A m a n ∈ A. (The inner sum vanishes for all sufficiently large m .) Proposition 1.
Assume that n is not divisible by q − and that p isodd. Then the following are equivalent: (1) v p ( L (1 , L/k, ψω nP )) ≥ / ( p − , (2) β ( n ) and γ ( n ) are divisible by P .Proof. Using the congruence ζ ip ≡ i ( ζ p −
1) (mod ( ζ p − )we deduce from (3) the congruence L (1 , L/k, ψω nP ) ≡ L (1 , K P /k, ω nP ) + ( ζ p − d X m =0 X a ∈ A m i ( a ) ω P ( a ) n ! modulo ( ζ p − . Since L (1 , K P /k, ω nP ) ∈ W and p is odd, it followsthat L (1 , L/k, ψω nP ) vanishes modulo ( ζ p − if and only if both L (1 , K P /k, ω nP ) ≡ p )and d X m =0 X a ∈ A m i ( a ) ω P ( a ) n ≡ ζ p − . The first congruence holds if and only if P divides β ( n ) and the secondif and only if P divides γ ( n ). (cid:3) Divisor class groups.
Let E be a finite extension of k , withconstant field F q n . We have an exact sequence0 → F × q n → E × → Div E → Cl E → E is the group of degree 0 divisors on E and Cl E the groupof divisor classes of degree 0 of E . Since W is flat over Z this sequenceremains exact after tensoring with W , and since F × q n has no p -torsionwe obtain a short exact sequence(4) 0 → W ⊗ E × → W ⊗ Div E → C ( E ) → , where C ( E ) = W ⊗ Cl E . N A PROBLEM `A LA KUMMER-VANDIVER FOR FUNCTION FIELDS 7
Proposition 2.
Let
F/E be a finite Galois extension with Galois group G . Then there is a natural short exact sequence → C ( E ) → C ( F ) G → W ⊗ (Div F ) G Div E and (Div F ) G / Div E is generated by the ramified primes.Proof. By Hilbert 90 we have H ( G, F × ) = 0, and since W is flat over Z we have H ( G, W ⊗ F × ) = W ⊗ H ( G, F × ) = 0 . Taking G -invariants in the sequence (4) for F gives a short exact se-quence 0 → W ⊗ E × → W ⊗ (Div F ) G → C ( F ) G → . Comparing this with (4) for E gives the desired exact sequence. (cid:3) Corollary 1.
Assume that n is not divisible by q − . Then C ( K P )( ω − nP ) = C ( L )( f ω P − n ) Gal(
L/K P ) and C ( L )( f ω P − n ) = C ( f K Q )( ω − n ( q pd − / ( q d − Q ) . Proof.
Since
L/K P is unramified away from the primes above ∞ , wehave that (Div L ) G / Div K P is generated by the primes above ∞ . Let S be the set of primes of L above ∞ and W S the free W -module withbasis S . Because n is not divisible by q − W S ( f ω P − n ) = 0 , hence the first claim follows from the Proposition. For the secondclaim, use that f K Q /L is unramified away from Q and the primes above ∞ , and that Q is totally ramified. (cid:3) Theorem 1.
Assume p = 2 . Let P ∈ A be monic irreducible of degree d , and such that i ( P ) = 0 . Let n be an integer such that ≤ n ≤ q d − ,not divisible by q − and such that β ( n ) and γ ( n ) are divisible by P .Then C ( f K Q )( ω − n ( q pd − / ( q d − Q ) is not cyclic.Proof. Set U = C ( L )( f ω P − n ) = C ( f K Q )( ω − n ( q pd − / ( q d − Q )and assume that U is W -cyclic, but that β ( n ) and γ ( n ) are divisibleby P .Since β ( n ) is divisible by P , it follows that U Gal(
L/K P ) = C ( K P )( ω − nP )is nonzero, and in particular that U is nonzero. Let x ∈ U be agenerator, so that U = W x . Let g be a generator of Gal( L/K P ) . BRUNO ANGL`ES AND LENNY TAELMAN
We have gx = wx for some w ∈ W × . This implies that w p x = x ,and it follows that w p − ≡ p ) and w ≡ p ) . Since v p (1 + w + · · · + w p − ) = 1 we find pU = (1 + w + · · · + w p − ) U ⊂ U Gal(
L/K P ) and therefore the length of U/U
Gal(
L/K P ) is at most 1.On the other hand, by [5] and Corollary 1, we have that the lengthof U/U
Gal(
L/K P ) equals v p ( L (1 , L/ ˜ k, f ω P n )) − v p ( L (1 , K P /k, ω nP ))and by (2) this equals( p − v p ( L (1 , L/k, ψω nP )) . From Proposition 1 we deduce that the length of
U/U
Gal(
L/K P ) is atleast 2, a contradiction. (cid:3) Kummer-Vandiver If P ∈ A is monic irreducible we write Y P for the spectrum of theintegral closure of A in K P . Theorem 2.
Assume p = 2 . Let P ∈ A be monic irreducible of degree d and such that i ( P ) = 0 . Let n be an integer such that (1) β ( n ) is divisible by P if n is not divisible by q − ; (2) γ ( n ) is divisible by P .Let Q ( T ) = P ( T p − T ) and N = n ( q pd − / ( q d − . Then Q isirreducible in A and H ( Y Q, fl , C [ Q ] D )( ω − N − Q ) = { } . Proof.
We split the proof in cases depending on the divisibility of n and dn by q −
1. Note that N is divisible by q − nd isdivisible by q − Case 1.
Assume that n is divisible by q −
1. This case is treated in[2]. By [2, Proposition 2.6], we get( W ⊗ Pic Y Q )( ω − NQ ) = { } . Without loss of generality we may assume that 1 ≤ n < q d −
1. Thenby the work of Okada ([7], see also [4, § B ( q pd − − N ) ≡ Q ( T )) . By Theorem 1 of [6] (the “Herbrand-Ribet theorem”) we concludeH ( Y Q, fl , C [ Q ] D )( ω − N − ) = { } . N A PROBLEM `A LA KUMMER-VANDIVER FOR FUNCTION FIELDS 9
Case 2.
Now assume that n is not divisible by q − dn is. Thenby Theorem 1 the module C ( K Q )( ω Q − N ) is not cyclic, and so we musthave: ( W ⊗ Pic Y Q )( ω Q − N ) = { } . We conclude with the same argument as in case 1.
Case 3.
Assume that nd is not divisible by q −
1. As in the previouscase, we find that ( W ⊗ Pic Y Q )( ω Q − N ) = { } . Now we conclude we a different argument. By the above non vanishing,and by exact sequence (2) of [6] we find that the space of Cartier-invariant ω − N -typical differential forms A/Q ⊗ F q Γ( Y, Ω Y ) c=1 ( ω − N )is at least two-dimensional. With the exact sequence of Theorem 2 of loc. cit. one concludes thatH ( Y Q, fl , C [ Q ] D )( ω − N − ) = { } . (Using the fact that the ω − N -part of the last module of the exactsequence of Theorem 2 is 1-dimensional. Note that the same argumentis used in the proof of Theorem 1 of loc. cit. , see [6, § (cid:3) An example.
Let q = 3. One can verify that with P = T − T +1and n = 13 the conditions of Theorem 2 are satisfied. Indeed, one has β (13) = − T − T − T + 1and γ (13) = − T − T + T − T + T + T − , both of which are divisible by P = T − T + 1 in F [ T ]. Also, notethat n is not divisible by q −
1. Using Theorem 2 we thus find that theprime Q ( T ) = P ( T − T ) = T − T − T − T − T + 1satisfies H ( Y Q, fl , C [ Q ] D )( ω − N − ) = { } where we have N = n q pd − q d − . This is the counterexample to the analogue of the Kummer-Vandiverconjecture stated at the end of the introduction (we have − ≡ q pd − Characteristic p = 2We now assume that p = 2. With some minor changes, the abovearguments still work, but the result is weaker.We keep the notations of section 3. If P ( T ) is a prime in A of degree d such that i ( P ) = 0, then Q ( θ ) = P ( T ) is a prime of degree 2 d in e A = F q [ θ ] , where θ − θ = T . Set again L = e kK P ⊂ f K Q . We have thefollowing version of Theorem 1.
Theorem 3.
Assume p = 2 . Let P ∈ A be monic irreducible of degree d , and such that i ( P ) = 0 . Let n be an integer such that ≤ n ≤ q d − , not divisible by q − and such that γ ( n ) is divisible by P and L (1 , K P /k, ω P ) is divisible by . Then C ( K Q )( ω − n ( q pd − / ( q d − Q ) is notcyclic. Note that β ( n ) is divisible by P if and only if L (1 , K P /k, ω P ) isdivisible by 2, so the hypothesis are stronger than those in Theorem 1. Proof of Theorem 3.
The proof is almost identical to that of Theorem1. Let ψ be the unique non-trivial character of G = Gal( e k/k ) . Proposition 1 does no longer hold, since we no longer have that ( ζ p − divides p . However, if γ ( n ) is divisible by P and if L (1 , K P /k, ω P )is divisible by 4 (instead of 2), we can still conclude v p ( L (1 , L/k, ψω nP )) ≥ . Denote the length of U = C ( L )( e ω − nP ) by N . We have N = v p ( L (1 , L/k, ψω nP )) + v p ( L (1 , L/K, ω nP )) ≥ . Let g be the nontrivial element of Gal( L/K P ). Then the length of U g equals v p ( L (1 , L/K, ω nP )), which by hypothesis is at least 2.Suppose that U is a cyclic W -module, and let x ∈ U be a generator,so that U = W x . There is a w ∈ W × so that gx = wx . We thenhave that w − N . We find that w − N − but not by 2 N (since U g = U .) It follows that (1 + g ) U = 2 U ,and as in the proof of Theorem 1 we conclude that U/U g has length atmost 1, contradicting our hypothesis. We conclude that U cannot be W -cyclic. (cid:3) Using this, we get the following variation of Theorem 2 in character-istic 2, with the same proof.
Theorem 4.
Assume p = 2 . Let P ∈ A be monic irreducible of degree d and such that i ( P ) = 0 . Let n be an integer such (1) L (1 , ω nP ) is divisible by if n is not divisible by q − ; (2) γ ( n ) is divisible by P . N A PROBLEM `A LA KUMMER-VANDIVER FOR FUNCTION FIELDS 11
Let Q ( T ) = P ( T − T ) and N = n ( q d + 1) . Then Q is irreducible in A and H ( Y Q, fl , C [ Q ] D )( ω − N − ) = { } . An example.
Let q = 4 and F = F ( α ). Then P = T + α T + T + αT + α with n = 341 = (4 − / (4 −
1) satisfies the hypothesis,leading to a counterexample Q ∈ F [ T ] to Question 1.6. Heuristics
This section contains no mathematical theorems, but only crudeheuristic arguments and numerical observations. Our main goal is toconvince the reader that one could a priori expect to construct manycounterexamples using the above base change strategy.The arguments are specific to odd q so we assume throughout thesection that q is odd.We argue that one could expect that Theorem 2 yields at least cX /p (log X ) − counter-examples of residue cardinality at most X toQuestion 1 (for some constant c > X counter-examples predicted by Washington’s heuristics.In fact we will only consider counter-examples of a particular form.Assume that q is odd. Note that by Theorem 2, if we are given(1) an integer m with 1 ≤ m < q − P ∈ A of degree d ,such that(1) q − md ;(2) β ( m ( q d − / ( q − ≡ γ ( m ( q d − / ( q − ≡ P );then Q ( T ) = P ( T p − T ) is monic irreducible of degree pd andH ( Y Q, fl , C [ Q ] D )( ω m ( q pd − / ( q − ) = { } , giving a counterexample to Question 1.The reason to restrict to n of the form m ( q d − / ( q −
1) lies in thefollowing trivial observation:
Lemma 1.
Let P ∈ A be irreducible of degree d . If n is a multiple of ( q d − / ( q − then β ( n ) and γ ( n ) modulo P lie inside F q ⊂ A/P . (cid:3) So we may expect that β ( n ) and γ ( n ) are much more likely to vanishmodulo a prime P of degree d if n is of the form m ( q d − / ( q − q − md . We make the followinghypotheses on a “random” monic irreducible P of degree d :(1) i ( P ) is non-zero with probability ( p − /p ;(2) β ( m ( q d − / ( q − P with probability 1 /q ; (3) γ ( m ( q d − / ( q − P with probability 1 /q ;(4) the above probabilities are independent of each other, and in-dependent of the vanishing of i ( P ).The first hypothesis is essentially an instance of the Chebotarev den-sity theorem, the second and the third are motivated by Lemma 1, andthe fourth is nothing more than wishful thinking. To some extent onecan verify these statements experimentally. In Table 1 we reproducesome numerical data regarding these hypotheses. Note that in the ex-ample of Table 1 β seems to have a slight bias towards vanishing, wehave no explanation for this bias.Finally, we show that under the above hypothesis, for some c > X sufficiently large there are at least cX /p (log X ) − primes of residue cardinality at most X that contradict Question 1.Indeed, for all X sufficiently large there is a positive integer d with(log q X /p ) − < d ≤ log q X /p and d not divisible by q −
1. Taking m = 1, we should find p − p · q · q · q d d ≥ c X /p log X monic irreducibles P of degree d satisfying the conditions (with c > X ). Each of these leads to a counter-example Q ofresidue cardinality at most X . d β ( n ) ≡ P ) γ ( n ) ≡ P ) β ( n ) ≡ γ ( n ) ≡ P )9 428 318 14211 395 344 13713 416 332 147 Table 1.
The number of P satisfying various congru-ences, out of random samples of 1000 primes P in F [ t ]with i ( P ) = 0, of degrees 9, 11 and 13. Again n =( q d − /
2. Note that every P counted in the rightmostcolumn gives rise to a prime Q which gives a counterex-ample to Question 1. References [1] G. Anderson, Log-Algebraicity of Twisted A -Harmonic Series and Special Val-ues of L -Series in Characteristic p, J. Number Theory (1996), 165–209.[2] B. Angl`es. On L -functions of cyclotomic function fields. J. of Number Theory (2006), 247–269.
N A PROBLEM `A LA KUMMER-VANDIVER FOR FUNCTION FIELDS 13 [3] J.P. Buhler and D. Harvey. Irregular primes to 163 million.
Math. Comp. (2011), 2435–2444.[4] D. Goss. Basic Structures of Function Field Arithmetic.
Springer, Berlin, 1996.[5] D. Goss, W. Sinnott. Class groups of function fields.
Duke Math. J. (1985),507–516.[6] L. Taelman. A Herbrand-Ribet theorem for function fields. To appear in Invent.Math. [7] S. Okada. Kummer’s theorem for function fields.
J. Number Theory (1991),212–215. Universit´e de Caen, CNRS UMR 6139, Campus II, Boulevard Mar´echalJuin, B.P. 5186, 14032 Caen Cedex, France.
E-mail address : [email protected] Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RALeiden, The Netherlands.
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