Poincaré series from Cryptology and Exceptional Towers Mike Fried, UCI and MSU-B 03/26/07

Transcription

1 Poincaré series from Cryptology and Exceptional Towers Mike Fried, UCI and MSU-B 03/26/07 Part 0: Exceptionality and fiber products Part I: Exceptional rational functions over number fields Part II: The exceptional tower T Z,Fq of any variety Z over F q Part III: Generalizing Exceptionality: Pr-exceptional covers and Davenport pairs Part IV: (Chow) motives from exceptional covers and Davenport pairs: Diophantine category of Poincare series over (Z, F q ) Part V: Comparing T P 1,F q with various subtowers: Generated by Serre s Open Image Theorem, CM part; By Serre s Open Image Theorem, GL part; By Wildly ramified polynomials. Typeset by FoilTEX 1

2 Part 0: Exceptionality and fiber products 1.a. Articles and Talks: Finite fields, Exceptional covers and motivic Poincare series An F q cover ϕ : X Z of absolutely irreducible normal varieties is exceptional if ϕ one-one on F q t points for infinitely many t. For a # field: ϕ has infinitely many exceptional residue class field reductions. We use the Davenport- Lewis name exceptional because, equivalently, a version of their geometric property holds for ϕ. Typeset by FoilTEX 2

3 Using fiber products Assume ϕ i : X i Z, i =1, 2, are two covers (of normal varieties) over K. The set theoretic fiber product has geometric points {(x 1,x 2 ) x i X i ( K),i=1, 2, ϕ 1 (x 1 )=ϕ 2 (x 2 )} : x X( Fq ) is a point in X with coordinates in Fq. Won t be normal at (x 1,x 2 ) if x 1 and x 2 both ramify over Z. The categorical fiber product here is normalization of the result: components are disjoint, normal varieties, X 1 Z X 2. Typeset by FoilTEX 3

4 Galois closure of a cover Denote X Z X minus the diagonal by XZ 2 \ Δ. XZ k \ Δ: kth iterate of the fiber product minus the fat diagonal; empty if k>n= deg(ϕ). Any K component ˆX of X n Z \ Δ is a K Galois closure of ϕ: unique up to K isomorphism of Galois covers of Z. S n action on XZ n \ Δ gives the Galois group def G( ˆX/Z) = Ĝϕ: subgroup fixing ˆX. Without ˆ, G ϕ, denotes absolute Galois closure. Typeset by FoilTEX 4

5 Part I: Exceptional rational functions over # fields Cyclic polynomials have the form x x n. RSA code scheme uses these. Fewer people know about Chebychev polynomials. Yet, these also have their cryptography use, as do compositions of these types. Proposition 1. If (n, p 1)=1, then we can use x n to scramble data into Z/p. Ifn is odd, there are infinitely many such primes p. Proof. Euler s Theorem: Powers of a single integer α fill out Z/p \{0} def = Z/p. Typeset by FoilTEX 5

6 Residue Primes that work for (odd) n Take p {k + m n m Z} where k satisfies: (k, n) =1(apply Dirichlet s Theorem); and (k 1,n)=1((p 1=k 1+m n, n) =1). Example: k =2works; other integers may too. Typeset by FoilTEX 6

7 Tchebychev polynomials of odd degree n T n ( 1 2 (x +1/x)) = 1 2 (xn +1/x n ), T n : {, ±1} {, ±1}. Proposition 2. If (n, 6)=1, then T n : Z/p Z/p maps one-one for infinitely many p. Exactly those primes p with (p 2 1,n)=1. Proof: Use finite fields F p 2 Z/p: F p 2 cyclic. Typeset by FoilTEX 7

8 2. Schur s Conjecture: Cryptography we recognize in modern algebra goes back to the middle of the 1800s. They used finite fields as the place to encode a message. Conjecture 3 (Schur 1921). Only compositions of cyclic, Tchebychev and degree 1 (x ax + b) give polynomials mapping 1-1 on Z/p for -ly many p. Problem 4. How to check if an f(x) is a composition of the correct polynomials? If so, how to check if it is 1-1 for of p (notation: 1 1 )? Typeset by FoilTEX 8

9 Points toward proving Schur s conjecture: Step 1: If f = f 1 f 2 (f i F q [x]), then f is 1 1 if and only f 1 and f 2 are 1 1. Subtle reduction: If f decomposes over C then it decomposes over Q (not automatic for rational functions). So, to prove Schur s conjecture we consider f indecomposable over K. Step 2: Consider 1 1 f with f : Z/p Z/p 1-1. Then, the polynomial expression f(x) f(y) ( ) ϕ(x, y) = =0 x y has no solutions (x 0,y 0 ) Z/p Z/p, x 0 y 0. Typeset by FoilTEX 9

10 Cover characterization of exceptionality Proposition 5 (Weil). If ϕ(x, y) has u absolutely irreducible factors (over F p ), then (*) has at least u p + A p solutions (some A constant in p). Corollary 6. If f is 1 1, then ϕ(x, y) mod p has no absolutely irreducible factors (for p large). Proposition 7. [DL63] [Mc67] [Fr74] [Fr05] [GLTZ07]: General F q cover of normal varieties: ϕ : X Z exceptional over F q t XZ 2 \ Δ has no F qt abs. irred. components. Typeset by FoilTEX 10

11 For 1 1 f : P 1 x P 1 z, the groups Ĝf and G f Consider f(x) z =0with z a variable. Find n solutions x 1,...,x n in some algebraic closure F of Q(z): f(x i ) = z; they generate a field Q(x 1,...,x n,z) def = L f. Then, Ĝ f = G(L f /Q(z)). Proposition 8. Then, G f S n is primitive, not doubly transitive, and contains an n-cycle. Example 9. Assume n>2 is prime. The group D n (Dihedral of degree n) with generators g 1 = (1n)(2 n 1) ( n 1 n ) g 2 = (2n)(3 n 1) ( n+1 n ) is primitive,not double transitive, has an n-cycle. Typeset by FoilTEX 11

12 Why primitive with an n-cycle? With f(x) =x n +a 1 x n 1 + +a n (exceptionality allows monic). Solve for x from f(x) =z. Solution: x 1 = z 1/n + b 0 + b 1 z 1/n + b 2 z 2/n +. Substitute e 2πi k n z n 1 z 1/n for n-cycle in G f. Let G f (x 1 ) be the subgroup of G f fixing x 1. Primitive means no proper group H with G f (x 1 ) < H<G f. Galois correspondence: Such an H would mean a field L = Q(w) with Q(z) <L< Q(x 1 ). So, w = f 2 (x 1 ), and z = f 1 (w). Contrary to indecomposable f: f 1 (f 2 (x 1 )) = z. Typeset by FoilTEX 12

13 Concluding Schur s Conjecture Why G f is not doubly transitive: Equivalent to ϕ(x, y) (XZ 2 \ Δ) has at least two factors over Q (from no abs. irred. factors over Q). Get Schur s conjecture if 1 1 and indecomposable f is variable change of cyclic or Chebychev polynomial. Chebychev case: variable change, (z,x) (az + b, a x + b ) (a, b, a,b K), allows f(±u) =±u with u 2 = a K. def Then, with l u : x ux, f = l u T n l u 1 = T n,a : u n 1 T n,a is what a large literature calls a Dickson polynomial [LMT93]. Typeset by FoilTEX 13

14 All exceptional prime degree rational f Step 1: Show G f is a cyclic or dihedral group. Proposition 10 (Famous Group Results). If n is a prime, then (Burnside): G f {( u v) u (Z/n),v Z/n } def = Z/n s (Z/n). 0 1 For n not prime there is no such G f : Schur. Step 2: Show G f dihedral (resp. cyclic) polynomial f is Chebychev (resp. cyclic) after changing variables. Best part: Monodromy method solves many other problems (Schur s conjecture the easiest). Typeset by FoilTEX 14

15 Step 2 cont: Apply Riemann s Existence Theorem. For g S n, ind(g) def = n # of disjoint cycles in g (including length 1). If f : C x { } C z { }, with branch points z 1,...,z r = r elements g 1,...,g r G f (branch cycles) with these properties: G f = g 1,...,g r 1 (generation); r i=1 g i =1(product-one); and 2(n 1) = r i=1 ind(g i) (genus 0). Typeset by FoilTEX 15

16 Finish Polynomial case def g r = g is an n-cycle; and n 1= r 1 i=1 ind(g i) (genus 0). Proposition 11. Combine with g 1,...,g r 1,g Z/n s (Z/n). Polynomial Result: {g 1,...,g r 1 } = {g 1,g 2 } as in Ex. 9 modulo conjugation in S n, g =(12... n) 1 ;or r =2and g 1 =(12... n). Tchebychev/cyclic polynomial branch cycles. Typeset by FoilTEX 16

17 Dominant rational (not polynomial) function case Branch cycles are (g 1,g 2,g 3,g 4 ), g i s conjugate to ) Z/n s {±1}. Most new functions from ( Weierstrass -functions through this diagram: C {±w} { } mod {±1} f C {±z} { } mod {±1} mod L z /L w Z/n C w /L w C z /L z. Here L w L z both generated over Z by two linearly independent (over R) complex numbers. Typeset by FoilTEX 17

18 Part II: Exceptional tower T Z,Fq of variety Z over F q Extension of constants series Let ˆK ϕ (k) be the minimal def. field of (geom.) K components of XZ k \ Δ, 1 k n: ker(ĝϕ G( ˆK ϕ (n)/k)) = G ϕ. Each ˆK ϕ (k)/k is Galois: kth ext. of constants field: G( ˆK ϕ (k)/k) permutes geom. components of X k Y \ Δ. Denote perm. rep. by T ϕ,k. Typeset by FoilTEX 18

19 Characterize exceptional There is a natural sequence of quotients G( ˆX/Y ) G( ˆK ϕ (n)/k) G( ˆK ϕ (k)/k) G( ˆK ϕ (1)/K). G( ˆK(1)/K) is trivial iff all K components of X are absolutely irreducible. Theorem 12. For K a finite field, G( ˆK ϕ (2)/K) having no fixed points under T ϕ,2 characterizes ϕ being exceptional ([Fr74], [Fr05], [GLTZ07]). Typeset by FoilTEX 19

20 The tower T Z,Fq and its cryptology potential Morphisms (X, ϕ) T Z,Fq to (X,ϕ ) T Z,Fq are covers ψ : X X with ϕ = ϕ ψ. Partially order T Z,Fq by (X, ϕ) > (X,ϕ ) if there is an (F q ) morphism ψ from (X, ϕ) to (X,ϕ ). Then ψ induces: a homomorphism G( ˆX ϕ /X ϕ ) to G( ˆX ϕ /X ϕ ); and canonical map from cosets of G( ˆX ϕ /X ϕ ) in G( ˆX ϕ /Z) to the corresponding cosets for X. Note: (X, ψ) is automatically in T X,F q. Typeset by FoilTEX 20

21 Forming the exceptional tower Nub of an exceptional tower of (Z, F q ): unique minimal exceptional cover X the fiber product dominating exceptional covers ϕ i : X i Z, i =1, 2. Note: Everything depends on F q. For (X, ϕ) T Z,Fq denote cosets of G( ˆX ϕ /X ϕ ) in G( ˆX ϕ /Z)=Ĝϕ by V ϕ ; coset of 1 by v ϕ and the rep. of Ĝϕ on these cosets by T ϕ : Ĝϕ S Vϕ. Write G( ˆK ϕi (2)/F q ) as Z/d(ϕ i ), i =1, 2. Typeset by FoilTEX 21

22 Why X 1 Z X 2 has exactly one abs. irred. comp. Do 1 2, suppose none! Let F q t 0 contain coefficients of all absolutely irred. X 1 Z X 2 comps. Then, if (t, t 0 )=1, X 1 Z X 2 has no abs. irr. com. over F q t. Normality = X 1 Z X 2 (F q t)=. D-L criterion allows assuming ϕ i s are étale. Then, t (Z/d(ϕ i )), i =1, 2, = ϕ i is 1-1 and onto (over F q t), i =1, 2. For t large, z Z(F q t) = x i X i (F q t) z, i =1, 2. So (x 1,x 2 ) X 1 Z X 2 (F q t). Typeset by FoilTEX 22

23 T Z,Fq is a very rigid category Proposition 13. In T Z,Fq there is at most one (F q ) morphism between any two objects. So, ϕ : X Z has no F q automorphisms: Cen SV ϕ (Ĝϕ) ={1}. Then, {(Ĝϕ,T ϕ,v ϕ )} (X,ϕ) TZ,F canonically defines q a compatible system of permutation representations; it has a projective limit (ĜZ,T Z ). Value of the Tower: It now makes sense to form the subtower generated by special exceptional covers: The minimal tower including all covers in the set. Examples: Tamely ramified subtower; Schur-Dickson subtower of T P 1 z,f q ; Subtower generated by CM (or GL 2 ) covers from Serre s OIT (Part V). Typeset by FoilTEX 23

24 Exceptional scrambling For any t let T Z,Fq (t) be those covers with t in their exceptionality set. Cryptology starts by encoding a message into a set. For t large our message encodes in F q t. Then, select (X, ϕ) T Z,Fq (t). Embed our message as x 0 X(F q t). Use ϕ as a one-one function to pass x 0 to ϕ(x 0 )=z 0 Z(F q t) for publication. You and everyone else who can understand message x 0 can see z 0 below it. To find out what is x 0 from z 0, need an inverting function ϕt 1 : Z(F q t) X(F q t). Typeset by FoilTEX 24

25 Inverting the scrambling map Question 14 (Periods). With X = P 1 x and Z = P 1 z, identify them to regard ϕ on F q t as ϕ t, permuting F q t { }. Label the order of ϕ t as m ϕ,t = m t. Then, ϕ m t 1 t inverts ϕ t. How does m ϕ,t vary, for genus 0 exceptional ϕ, ast varies? Standard RSA inverts x x n by inverting the nth power map on F q t (mult. by n on Z/(q t 1) Euler s Theorem). Works for all covers in the Schur Sub-Tower of (P 1 y, F q ) generated by x n s and T n s. (For T n s, invert mult. by n onz/(q 2t 1).) Typeset by FoilTEX 25

26 Part III: pr-exceptional covers and Davenport pairs Definition 15. ϕ : X Z is p(ossibly)r(educible)-exceptional: ϕ : X(F q t) Z(F q t) surjective for -ly many t. Then, ϕ is exceptional iff X is abs. irreducible. allow X to have no abs. irred. comps. We even Form ˆX Z (with its canonical rep. T ϕ ), the Galois closure with group Ĝϕ, and get an extension of constants field with G(ˆFϕ /F q )=Z/ ˆd(ϕ). Typeset by FoilTEX 26

27 D-L generalization; pr-exceptional characterization For t Z/ ˆd(ϕ): def Ĝ ϕ,t = {g Ĝϕ restricts to t Z/ ˆd(ϕ)}. Exceptionality set E ϕ of a pr-exceptional cover: {t Z/ ˆd(ϕ) g Ĝϕ,t fixes 1 letter of T ϕ }. pr-exceptional correspondences: W X 1 X 2 with projections W X i s pr-exceptional. Exceptional correspondence between X 1 and X 2 = X 1 (F q t) = X 2 (F q t) for -ly many t. If X 2 = P 1 z, then t=1 (a n q t +1for -ly many t. def = X 1 (F q t) )u t has a n = Typeset by FoilTEX 27

28 A zoo of high genus except. correspondences between P 1 x 1 and P 1 x 2 If ϕ i : P 1 x i P 1 z, i =1, 2 is exceptional, then P 1 x 1 P 1 z P 1 x 2 has a unique absolutely irreducible component, an exceptional cover of P 1 x i, i =1, 2. Suppose ϕ i : X i Z, i =1, 2, are abs. irreducible covers. The minimal (F q ) Galois closure ˆX of both is any F q component of ˆX1 Z ˆX2. Attached group, Ĝ = Ĝ(ϕ 1,ϕ 2 ) = G( ˆX/Z): Fiber product of G( ˆX 1 /Z) and G( ˆX 2 /Z) over maximal H through which they both factor. Typeset by FoilTEX 28

29 D(avenport)Pairs: new pr-except. correspondences Definition 16. (ϕ 1,ϕ 2 ) is a DP (resp. i(sovalent)dp) if ϕ 1 (X 1 (F q t)) = ϕ 2 (X 2 (F q t)) for -ly many t (resp. ranges assumed with same multiplicity; T. Bluer s name). Equivalent to being a DP: pr Xi X 1 Z X 2 X i, is pr-exceptional, and the exceptionality sets E pri (F q ), i =1, 2, have nonempty (so infinite) intersection E pr1 (F q ) E pr2 (F q ) def = E ϕ1,ϕ 2 (F q ). Typeset by FoilTEX 29

30 Part IV: (Chow) motives: Diophantine category of Poincare series over (Z, F q ) Let W D,Fq (u) = t=1 N D(t)u t be a Poincaré series for a diophantine problem D over a finite field F q.we call these Weil vectors. Example: F (x, z) F q [x, z], N D (t) = {z F m z x q t F m x,f(x, z) =0}. q t Weil Relation between W D1,F q (u) and W D2,F q (u): -ly many coefficients of W D1,F q (u) W D2,F q (u) equal 0. Effectiveness result: For any Weil vector, the support set of t Z of 0 coefficients differs by a finite set from a union of full Frobenius progressions. Typeset by FoilTEX 30

31 Motivic formulation Question 17. If Poincare series of X over F q has t-th coefficient equal q t +1 for -ly many t, is there a chain of except. correspondences from X to P 1? Equivalent to characterizing X for which t=1 tr Fr [ 2 q t 0 ( 1)i Hl i(x)]ut has a relation with the series with X = P 1 : Chow motive coefficients. There are p-adic versions: Replace F q t by higher residue fields with the Witt vectors R t with residue class F q t; and use integration instead of counting. Typeset by FoilTEX 31

32 Result of Denef-Loeser [Fr77], [DL01], [Ni04] Consider a number field version, by R p the completion the integers of K with respect to prime p. def Then, W D,Rp (u) = v=1 N D,R (v)uv p with N (v) D,Rp using values in R p /p v that lift to values in R p. To make this useful motivically requires doing this for those D with a map to a fixed space Z/K. Given D, There is a string of relative to Z Chow motives (over K) {[M v ]} v=0, so for almost all p, W (u) = D,Rp t=1 tr Fr [M p t]u t. Typeset by FoilTEX 32

33 Role of idps Given Weil Vector W (D, F q ) over (Z, F q ) and ϕ : X Z can define pullback W ϕ (D, F q ) over (X, F q ). Assume ϕ i : X i Z, i =1, 2, is an idp over F q, X 1 = X 2 and D has a map to Z. Then, (ϕ 1,ϕ 2 ) produces new Weil vectors W ϕ i D,F q, i = 1, 2, and a relation between W ϕ 1 D,F q (u) and W ϕ 2 D,F q (u): -ly many coefficients of W ϕ 1 D,F q (u) W ϕ 2 D,F q (u) equal 0. Typeset by FoilTEX 33

34 PartV:CMandGL 2 exceptional genus 0 covers Test for a cover ϕ : X Z decomposing. Check X Z X\Δ for irreducible components Z of form X Z X. If none, then ϕ is indecomposable. Otherwise, ϕ factors through X Z (Gutierrez, et.al. from [FrM69]). Denote the minimal Galois extension of K over which ϕ decomposes into absolutely indecomposable covers by K ϕ (ind): The indecomposability field of ϕ. Proposition 18. For any cover ϕ : X Z over a field K, K ϕ (ind) ˆK ϕ (2). Typeset by FoilTEX 34

35 Most of rest of genus 0 except. covers/q [Fr78], [GSM04]: From Weierstrass -functions. P 1 f ±w P 1 {±z} mod {±1} mod L z /L w C w /L w C z /L z. Case CM: deg(f) =r, a prime Case GL 2 : deg(f) =r 2, a prime squared mod {±1} [O67], [Se68], [Se81], [R90], [Se03] case of Serre s O(pen)I(mage)T(heorem). CM case can describe inversion period from Euler s Theorem, essentially equivalent to the theory of complex multiplication. Typeset by FoilTEX 35

36 GL 2 gist [Fr05, ], Serre s GL 2 OIT [Se68, etc] [f] P 1 j by the j-invariant of the 4 branch points; G f =(Z/r) 2 s {±1}; yet for a non-cm j-invariant (say in Q), then for a.a. r, then for f def = f j,r, Ĝ f =(Z/r) 2 s GL 2 (Z/r). Exceptionality versus indecomposability: Given f j,r and the set A of A GL 2 (Z/r)/{±1} for which A acts irreducibly on (Z/r) 2. Consider P fj,r,a those primes p with the Frobenius of f j,r : P 1 w P 1 z mod p in A. For such p f j,r mod p is exceptional; and (equivalently) f j,r mod p is indecomposable, but decomposes over F p. Typeset by FoilTEX 36

37 Two automorphic function questions [Fr05, 6] poses an analog of [Se03] to find an automorphic funct. (should exist according to Langlands) for primes of except. for j Ogg s curve 3 + [Se81, extensive discuss]. Would give an explicit structure to the primes of exceptionality. For any exceptional f j,r mod p, form a Poincaré series with the period of exceptionality its coefficients. Conjecture, this series is rational. This result would then remove from consideration the arbitrary identification of P 1 w with P 1 z. Typeset by FoilTEX 37

38 Bibliography; Parts 0 and I: [DL63] H. Davenport and D.J. Lewis, Notes on Congruences (I), Quart. J. Math. Oxford (2) 14 (1963), [Fr70] M.D. Fried, On a conjecture of Schur, Mich. Math. J. 17 (1970), [Fr74] M. Fried, On a Theorem of MacCluer, Acta. Arith. XXV (1974), [Fr78] M. Fried, Galois groups and Complex Multiplication, T.A.M.S. 235 (1978) [Fr05] M. Fried, The place of exceptional covers among all diophantine relations, J. Finite Fields 11 (2005) [GMS03] R. Guralnick, P. Müller and J. Saxl, The rational function analoque of a question of Schur and exceptionality of permutations representations, Memoirs of AMS (2003),ISBN [LMT93] R. Lidl, G.L. Mullen and G. Turnwald, Dickson Polynomials, Pitman monographs, Surveys in pure and applied math,65, Longman Scientific, [GLTZ07] R. Guralnick, T. Tucker and M. Zieve (behind the scenes Lenstra), Exceptional covers and bijections on Rational Points, to appear IRMN, [Mc67] C. MacCluer, On a conjecture of Davenport and Lewis concerning exceptional polynomials, Acta. Arith. 12 (1967), [Sch23] I. Schur, Über den Zusammenhang zwischen einem Problem der Zahlentheorie and einem Satz über algebraische Functionen, S.-B. Preuss. Akad. Wiss., Phys.-Math. Klasse (1923), Typeset by FoilTEX 38

39 Bibliography; Parts II and V: [DL01] J. Denef and F. Loeser, Definable sets, motives and p-adic integrals, JAMS 14 (2001), [Fr76] M. Fried, Solving diophantine problems over all residue class fields of a number field..., Annals Math. 104 (1976), [FGS93] M.D. Fried, R. Guralnick and J. Saxl, Schur covers and Carlitz s conjecture, Israel J. Math. 82 (1993), [GTZ07] R. Guralnick, T. Tucker and M. Zieve, Exceptional covers and bijections on rational points, to appear in IRMN. [Le95] H.W. Lenstra Jr., Talk at Glasgow conference, Finite Fields III, (1995). [Ni04] J. Nicaise, Relative motives and the theory of pseudo-finite fields, to appear in IMRN. [O67] A.P. Ogg, Abelian curves of small conductor, Crelle s J 226 (1967), [R90] K. Ribet, Review of new edition of [Se68], BAMS 22 (1990), [Se68] J.-P. Serre, Abelian l-adic representations and elliptic curves, 1st ed., McGill University Lecture Notes, Benjamin, New York Amsterdam, 1968, in collaboration with Willem Kuyk and John Labute. [Se81] J.-P. Serre, Quelques Applications du Théorème de Densité de Chebotarev, Publ. Math. IHES 54 (1981), [Se03] J.-P. Serre, On a Theorem of Jordan, BAMS 40 #4 (2003), Typeset by FoilTEX 39