On p-adic elliptic logarithms and p-adic approximation lattices

Transcription

1 On -adic ellitic logarithms and -adic aroximation lattices Christian Wuthrich Aril 6, 006 Abstract Let E be an ellitic curve over Q and let be a rime number. Based on numerical evidence, we formulate a conjecture on the height of rational oints on E whose coordinates have high owers of in the denominator. On one hand, this conjecture is linked to a -adic ellitic analogue of a conjecture of Lang-Waldschmidt on linear forms of logarithms. On the other hand, we reformulate the conjecture in terms of -adic aroximation lattices; namely the lattice tye of a certain oint on P (Q ) should be maximal. We show that the average lattice tye of oints on P (Q ) is indeed maximal. Introduction Let E be an ellitic curve defined over the field of rational numbers Q given by a fixed Weierstrass equation y + a x y + a 3 y = x 3 + a x + a 4 x + a 6. We suose that the rank r of the Mordell-Weil grou E(Q) is ositive. Let be a rime number. For each k we define the following subgrou of E(Q) : B (k) = {P E(Q) ord (x(p )) k} If the equation is minimal at, we may also characterise B (k) as E(Q) E( b k Z ) where E b is the formal grou associated to E/Q. The index of B (k) in E(Q) divides c #Ẽ ns(f ) k with c being the Tamagawa number and Ẽns(F) the grou of non-singular oints on the reduction of E at. Furthermore, we define ˆm k to be jq ff ˆm (k) = min ĥ(p ) 0 P B(k) Here ĥ(p ) denotes the canonical height of P. Conjecture. There exists constants C > c deending on E and such that for all k, we have r log() k + C log` ˆm (k) r log() k + c where r is the rank of E. We may also announce a weaker form of the conjecture : Conjecture. For any, there exists constants D > d such that for all k D > log( ˆm k ) log() k > d > 0.

2 This aer is about different reformulations of these conjectures and evidence in their favour. Here is a first easy result in this direction. Proosition. The conjecture holds if the rank r is equal to. Proof. Let P be a oint of minimal height in B (). By the quadraticity of the canonical height, we know that P k = k P is a oint of minimal height in B (k). Now we comute q log` ˆm (k) = log ĥ(p k ) = log` k ĥ(p) = log() k + log ĥ(p). Using the convex body theorem of Minkowski we will show in section the uer bound in the stronger conjecture, namely Proosition. There exists a constant C, deending on E and, such that r log() k + C log`m (k). In section 3, we will resent numerical evidence in favour of the conjectures. The values obtained are in good agreement with both conjectures and not surrisingly the error terms C and c in conjecture seem to be related to the regulator of the curve. There are links between these conjectures and linear forms in -adic ellitic logarithms. Unfortunately the results in this field known to the author do not ermit to rove any of the two conjectures. The stronger of the two conjectures is similar to a well-known conjecture of Lang and Waldschmidt. We resent a version of it in section 4. Since the growth of the formula is linear in conjecture, we may create a generating function and the statement of the conjectures becomes a question about the domain of convergence of this analytic function. The most imortant reformulation is concerned with so-called -adic aroximation lattices. Let k be an integer. Given a oint z = (z, : z :... : z r) in the rojective sace P r (Q ) over the -adic numbers, one may consider the lattice L(z, k) = {(x, x,... x r) Z r x z + + x r z r 0 (mod k )}. The conjectures can now be reformulated as a formula for the growth of the length of the minimal vector minl(z, k) = min{ x 0 x L(z, k)} in the lattice L(z, k) when z is formed by the values of the -adic ellitic logarithm evaluated on a certain set of r linearly indeendent oints in E(Q). For the sake of simlicity, the theory is only develoed in the case r =. We say that z P (Q ) is of lattice tye α if the length of the minimal vector satisfies log(minl(z, k)) = α k + O() as k. The notion is indeendent on the choice of the norm on Z. The main result is concerned with the average value of the lattice tye on P (Q ). In theorem 0 and theorem, we rove the following recise statement. If z is an element of P (Z/ k Z) write simly L( z) for L(z, k) where z is any lift of z to P (Z ). Theorem 3. Let µ( k ) be the average of the logarithms of the length of the minimal vectors in all the lattices L( z) where z runs over all elements in P (Z/ k Z). Then log() k O(k k/ ) µ( k ) log() k O(k k/ ).

3 Numerical comutation of a large amount of exact values of µ(n) (here n may be any integer and the definition of µ(n) is similar to µ( k )) suggests that the value µ(n) log(m) converges to a certain value close to This is contained in conjecture 4. The theorem can be interreted loosely by saying that the average lattice tye of a -adic number is log(), which is also the maximal ossible lattice tye. Hence if we believe that the -adic ellitic logarithms of elements in E(Q) are in some sense random numbers, or say not too articular, we must believe in the conjectures to be true. Though a roof of them seems not within the reach of the resented methods. Other norms It is well-known (see [Sil9]) that for all k (exect when k = and =, which we may exclude), the grou B (k) is free of rank r. We choose a basis {P,... P r} of the free art of E(Q) and consider it as a lattice with the bilinear form rovided by the canonical height airing. The subgrous B (k) form then a sequence of sublattices and we investigate the length ˆm (k) of the minimal vector of the B (k). The function q P h = ĥ(p ) induces a norm on E(Q). Let be any other norm on E(Q). Define m (k) to be the length of the minimal vector of B (k) with resect to this norm, i.e. m (k) = min{ P 0 P B (k)}. There exists two constants c and c such that c Hence we deduce the inequalities q ĥ(p ) P c q ĥ(p ). log c + log` ˆm (k) log`m (k) log c + log` ˆm (k) and we may therefore relace in both conjectures ˆm (k) by m (k), if we allow the constants C and c to deend on the chosen norm. This roves the following lemma. Lemma 4. The conjecture is equivalent to the statement that, for any norm on E(Q), there exists constants C and c with for all k. (k) r log() k + C log`m r log() k + c In what follows we will mainly consider the usual norm with resect to a chosen basis {P,... P r} of the free art of E(Q) : q α P + α rp r = α + α r This interretation gives us easily the following Proosition 5. There exists a constant C such that r log() k + C log`m (k). In articular, the first half of the conjecture is true. 3

4 Proof. By the convex body theorem of Minkowski (see for instance [Cas97], Theroem 3.II), there exists a constant γ such that γ det(b (k)) r m (k). Since we know that the index of B (k) is of the form c k for some constant c Q, which is deending on, we have log(γ) + r log(c k ) log`m (k). Note that the constant C is effectively comutable. 3 Numerical results suorting the conjectures Even though the conjectures may aear too daring at first, the numerical evidence in favour of them is overwhelming. First, we stick to a single curve E : y + y = x 3 + x x, labelled 389a in the tables of Cremona. The oints P = (0, 0) and P = (, 0) form a basis of E(Q) of minimal canonical height. If for instance = 3, we may exlicitly comute the the subgrou B 3(): the oints Q = P + P = (, 9 ) and 9 7 Q = P P = ( 0, 8 ) form a Z-basis. 9 7 As we noticed in lemma 4, we may as well work with the minima of the -norm with resect to the given basis {Q, Q }. Moreover, we may ski the comutations of the minimal vectors of B (k). In fact the first vector of an LLL-reduced basis of B (k) will do as the following lemma shows Lemma 6. Let k be the first vector of an LLL-reduced basis of B (k), then m (k) A k with A = (r )/. This is Lemma 3.4 in [dw89]. In articular, the growth of log k as k is the same as for log( ˆm (k)). The figure shows the values of log( ˆm (k)) for the rimes between and 30 and for k u to 00. The lines reresent the redicted sloes log() k. Conjecture states that for every given the dots do not differ from the line by more than a fixed constant. Next, we rovide evidence for the consistency by varying the curve but fixing the rime = 3. We use five curves of rank and five curves of rank 3. Namely they are 389a, 433a, 446d, 563a, 57b and 5077a, 97a, 64a, 3766a. Figure shows the values of log( ˆm 3(k)) for all of these curves. one sees immediately that the values of the curves of rank (corresonding to the darker oints) are close to the line log(3) k, while the curves of rank 3 stay near the line of sloe 3 log(3). In order to refine this statement, we give a list here of the maximum of the difference for the curves listed above and k 00. δ k = log( ˆm 3(k)) log() k r 4

5 Figure : The values of log( ˆm (k)) for the curve 389a Curve max(δ k ) 389a a d a.67 57b a a a a Linear forms in -adic ellitic logarithms Let L : be(z ) Z be the formal -adic ellitic logarithm on E. Write u i = L (Q i) where {Q,..., Q r} is a basis (of the free art) of B (). Let Λ = α u + + α ru r with α i integers not all equal to zero. We write α = max{ α i i r} for the su-norm on B (). To say that Λ = k is the same as to say that Q = α Q + α rq r belongs to B (k). According to the strong conjecture, there should exist a constant c such that log α log`m (k) r log() k + c where m (k) is with resect to the norm. We may rewrite this as log Λ = log k r log α + rc and conclude that the conjecture imlies the existence of a constant c such that log Λ r log`max{ α i } + c. The constant c would deend on the curve E, the base field (which we fixed anyway to Q), the rime and the chosen u i. The roosition 5 shows that this is the strongest ossible conjecture in this direction. 5

6 Figure : The values of log( ˆm 3 (k)) for some curves of rank and 3 This conjecture is far away from the known bounds for linear forms in -adic ellitic logarithms. In [Ber78], Bertrand roves (under the assumtion that the curve has comlex multilication) that log Λ > d log`max{ α i } 6. In the more recent article [U96], the authors restrict themselves to the case of rank r =. They show that log Λ > d `log max{ α i } `log log max{ α i } 3. () with a exlicit and astronomic constant d < 0. If the last factor were not there we could rove conjecture. The best we can do is Proosition 7. Let E/Q be an ellitic curve of rank. constant d > 0 such that for all k we have log` ˆm (k) log() k > d log(k). 3 For any rime, there exists a Proof. Defining the left hand side of () to be log() k, we obtain the estimate log` ˆm (k) log() k > d log log`m (k) 3. Now we may use the roosition 5 to get the bound log` ˆm (k) log() k > d. log` log() k + C 3 which is of the desired form. We hoe that a forthcoming aer of Noriko Hirata-Kohno will actually roof better bounds that should imly conjecture. 6

7 The conjecture is a actually slight modification of the conjecture of Lang-Waldschmidt (see age in [Lan78]). We refer to the conjecture otimiste of Pilion in [Phi99] which concerns the ordinary -adic logarithm rather than the ellitic -adic logarithm. His conjecture gives a more recise form of the constant c but a slightly weaker growth coefficient. We state here a reformulated and weakened form of this conjecture : Conjecture 3. (Version of Lang-Waldschmidt) Let a,... a r be fixed ositive integers and a fixed rime. Given any ε > 0, there exists a constant c ε with the following roerty: if the integers b,... b r are such that 0 b log (a ) + + b r log (a r) k Z then, writing b for the maximum of b i as i r, we have log( b ) log() k + cε. ( + ε) r Needless to say that this conjecture seems to be out of reach by the current methods of linear forms of logarithms. 5 Siegel s theorem Of course, there is a close link to the following theorem of Siegel (see [Sil9, Theorem IX.3.]) on integral oints in E(Q). Define the logarithmic -adic distance on E(Q ) to be δ (P Q) = log() k if and only if P Q b E( k Z ) \ b E( k+ Z ) Let Q be a oint in E(Q) and let P n be a sequence of oints in E(Q) aroaching Q in the -adic toology. Siegel s theorem asserts that, if the naive height of P n tends to infinity as n then δ (P n Q) lim n h = 0 naive(p n) In fact the weak conjecture imlies that the quotient δ(pn Q) θ (P n) = log(ĥ(pn)) is bounded from above and the strong conjecture would claim that the lim su of these quotients is less or equal to r. It is lausible that a roof of either one of the conjectures would give rise to a better way of comuting S-integral oints in E(Q). 6 Generating function There is an obvious way of encoding the conjectures into an analytic function. Given an ellitic curve E/Q and a rime, we may write ζ (T ) = X k 0 ˆm (k) T k Z[T ] where ˆm (0) is simly the minimum of the canonical height for all oints in E(Q) of infinite order. Obviously ζ (T ) = 0 if the rank of the curve is zero. Proosition 8. Let > be a rime. If the curve E has rank, then ζ (T ) = + b T T where = eg(e/q) is the square root of the regulator of E(Q) and b is the index of B () in E(Q) modulo its torsion subgrou. 7

8 Proof. Let P 0 be a generator of the free art of E(Q). We may take P 0 of minimal height. The oints b k P 0 are oints of minimal height in B (k). Hence q ζ (T ) = ĥ(p 0) + X q ĥ(b k P 0) T k k q = ĥ(p 0) + b X ( T ) k = + b T T k In articular, for a curve of rank, the generating function ζ (T ) is a rational function with a single simle ole at T = of residue b/. Theorem 9. Let E/Q be a curve of rank r > 0 and let be a rime. ζ (T ) is an analytic function on the disc centred at T = 0 of radius ρ = /r. Conjecture would imly that the radius of convergence of ζ (T ) is less than. Conjecture is equivalent to the statement that ζ (T ) has a simle ole at T = /r. Proof. The first art is a consequence of Proosition 5. We know that there exists a constant C and conjecture claims that there are constants d and c such that log()k + C log m(k) d log()k + c r Moreover the stronger conjecture is equivalent to d =. We comute r e C k r m(k) e c d k and deduce from it that and the theorem follows. m (0) + e C r T r T ζ(t ) m(0) + d T ec d T The results in [U96] as formulated in 7 only give that the radius of convergence of ζ (T ) is less or equal to, which is obvious anyway for a series with integer coefficients. 7 -adic aroximation lattices For the following considerations, we will stick to the situation when E(Q) has rank. We fix a basis {Q, Q } of B (). Let z = L (Q ) and z = L (Q ). Note that we can describe B (k) by the following formula B (k) = {a Q + a Q a z + a z 0 (mod k )}. This follows from the fact that P = a Q +a Q is in B (k) if and only if L (P ) = a z +a z belongs to k. Write z for the oint (z : z ) in P (Q ). Since {Q, Q } is a basis of B (), we know that z or z is of valuation. So we may reduce the oint ( z : z ) to obtain a oint z k of P (Z/ k Z). The oint z k defines a line in Z/ k Z Z/ k Z assing through (0, 0) and B (k) is the reimage of this line under the ma B () Z Z Z/ k Z Z/ k Z. 8

9 This motivates the following definition. Let n > be an integer. Given a oint z in P (Z/nZ), we define a sublattice L( z) = {(a, a ) Z a z + a z 0 (mod n)} of Z of index n. If n is a ower of, a sublattice of this form will be called a -adic aroximation lattice in Z, following [dw89]. See also [Sma98] for a more recent treatment of the toic. The minimum of this lattice with resect to the usual bilinear form is denoted by n q minl( z) = min a + o a 0 (a, a) L( z) Note that there is are some easy relations, like minl(z : z ) = minl(z : z ) = minl(z : z ) = minl(z : z ). By Minkowski s bound, there is an inequality log`minl( z) log`γ n = log(n) + log(γ) with γ = / 4 3 = The value of this lattice constant can be found in the aendix to [Cas97]. We are interested in the following mean µ(n) = #P (Z/nZ) X z P (Z/nZ) Conjecture 4. There exists a constant ˆγ such that log`minl( z). µ(n) log(n) + ˆγ as n. In order to illustrate the above conjecture we include in figure 3 here a grahic of the first few values of µ(n) log(n) log(γ) +. The darker oints corresond to values of n which are rime. The numerical exerience would suggest that the value of ˆγ is around The Figure 3: The difference between µ(n) and log(n) + log(γ) following theorem shows that ˆγ, if it exists, is smaller than log(γ) =

10 Theorem 0. We have log(n) + log(γ) + O(log(n) n / ) µ(n). We do not claim that the error term in the theorem is otimal. The bounds on the numbers of integral oints inside discs used in the roof are the easy and obvious bounds rather than the best known bounds in [Hux03]. Proof. For any r > 0, we denote by D(r) the closed disc of radius r centred at 0. Let B(r) = Z D(r) denote the set of integral oints (x, y) Z inside D(r). Let = γ n. By Minkowski s convex body theorem, we know that every lattice L( z) has at least one oint in B(), but not every oint in B() figures among the smallest vectors; hence µ(n) is smaller than the mean of log(x + y ) on B(), i.e. µ(n) #B() X (x,y) B() For each (x 0, y 0) B() with x 0, y 0 0, we let Q(x 0, y 0) denote the unit square containing (x 0, y 0) such that log(x + y ) has its minimum on Q(x 0, y 0) exactly at (x 0, y 0). For instance if x 0, y 0 > 0, then Q(x 0, y 0) = [x 0, x 0 +] [y 0, y 0 +]. For the remaining oints (x 0, y 0) in B() situated on the axes, we define Q(x 0, y 0) to be the reunion of all such unit squares, e.g. if x 0 > 0, then Q(x 0, 0) = [x 0, x 0 + ] [, ]. Let [ T () = Q(x, y), (x,y) B() log x + y. which is reresented by the grey surface in the icture. Note that D( + ) T () D(). We have Z µ(n) #B() log x + y dx dy T () Z #B() log x + y dx dy = = #B() π D(+) Z + 0 log(r) r dr #B() π ( + ) ( log( + ) ) 0 0 (x,y ) (0,0) D() T() D(+) On the other hand, it is easy to see that #B() is larger than the area of D( ). Hence, we get µ(n) π ( + ) π ( ) ( log( + ) ) = ` + O( ) log() «+ O( ) = log() «log() + O In the next theorem, we rove a lower bound for µ(n). We only treat the case when n is a rime ower as we are mainly interested in such lattices. If the constant ˆγ exists, then this theorem shows that it is larger than

11 Theorem. Let be a rime. Then µ( k ) log() k + log π «+ O(k k/ ) Proof. The roof is far more comlicated than the uer bound in theorem 0. We will need several lemmas of which the first is the following. Lemma. Let k and (x, y) Z. Suose d is the highest ower of dividing the greatest common divisor of x and y. If k > d, then (x, y) belongs to L( z) for exactly d different elements z in P (Z/ k Z). Proof. Write x = d x and y = d y. By interchanging x and y if necessary, we may suose that y is not divisible by. Any z P (Z/ k Z) can be written as z = (a : b) with a = l for some 0 l k and b if divides a. (x, y) L( z) l+d x + b d y 0 (mod k ) l x + b y 0 (mod k d ) l = 0 and x + b y 0 (mod k d ) The last equation has exactly one solution for b modulo k d solutions in Z/ k Z. and hence we may find d Let be the radius of a large circle. Write l for the largest ower of which is smaller than. Define Σ() to be the number of integral oints inside the disc D(), but where we count every oint (x, y) exactly d times if d is the largest ower of dividing both x and y. The oint (0, 0) is not counted at all. Lemma 3. There is a constant A such that for all > 0. Σ() ( + ) π + A log() Proof. For any given radius r, the number of integral oints B(r) is less than π (r+). Hence, using lemma, we have lx «Σ() = #B() + ( d d ) #B d d= π ( + ) + lx ( d d ) π d= «+ d lx «= π ( + ) + ( ) π d d= d π ( ) + ( ) π! + 4 l + 4 l π + 4 π + 4 π + π + 4 π l + 4 π l 4π «( + ) π + 4 π l π l

12 We note that the definition of l imlies that l < and l < log()/ log(). We obtain Σ() ( + ) π + 4 π log() + «log() log() Hence for > 0, we may take A = 0. For any given oint (x, y) write r for x + y. We wish to comute the average µ() of log(r) on the non-zero integral oints in D(), but again counting each oint d times just as in the definition of Σ(). More recisely, we define µ() by Σ() µ() = X (x,y) B() log(r) + lx ( d d ) d= X «(x,y) B d log( d r) () where the X means that we are excluding (x, y) = (0, 0) from the sum. Lemma 4. We have µ() ( + ) π Σ() ««log() log() + O Proof. An argument similar to the one used in the comutations of theorem 0 shows that for any radius ρ > and any a > 0, we have X (x,y) B(ρ) log(a r) π Z ρ 0 log(a r) r dr = π (log(a (ρ )) ) (ρ ) Let l be the largest ower such that l <. The value of µ() would only decrease if we relace l by l in the definition (). Thus we have Σ() π µ() (log( ) ) ( ) l + ( ) X log d d «d d d= = (log() + log( ) ) ( ) l + ( ) X d (log() + log( d ) d ) ( ) d= = log() + O log() «+ ( ) (log() ) Xl l + ( ) X d= d log d= ( d ) d 4 + 4d d

13 For the first sum, we find l ( ) X d= 4 + 4d d ( ) l «+ O log().! 4 l + 4 l The second sum can be written as l X d= log( d ) ( d 4 + 4d ) in which the first factor is negative and it takes the smallest value for d =. The second factor is ositive and it likewise it is maximal when d =. Hence we have l X log( d ) ( 4 + 4d ) d d= l log( ) ( ) = O(log()) O( ) ( + O( )) = O log(). Let k be a large integer. Then define to be the real number satisfying ( + ) π + A log() = k ( + ), if k is sufficiently large so that > 0. #P (Z/ k Z). Note that the exression on the right is equal to Lemma 5. We have = /π k/ + O(k). Proof. We write X = k and C =. If we denote by T the exression such that = π C X ( T ) then + A log() π ( + ) = ««C X = = + T + 3 T 3 + T with growing this exression tends to, hence T is tending to 0. A π ( + log(x) + log(c) + log( T ) ) C X = ( T ) ( T + 3 T + 4 T 3 + ) = T + which roves that T = O( log(x) X ). Thus = C k/ + C X T = C k/ + O(log(X)) = C k/ + O(k). We finally start now, the roof of the theorem. Every lattice L( z) has at most two shortest vector different excet for maybe two values of z, corresonding to ±i, if they belong to Z. In this situation there are four shortest vectors. We have chosen in such a way that the number of oints counted with the multilicity at which they may aear at most as a shortest vector of a lattice L( z) is smaller than the total number of such lattices. Hence we have that 3

14 µ( k ) µ(), i.e. Σ() #P (Z/ k Z) or Σ() #P (Z/ k Z) + if i belongs to Z. Thus we obtain + µ( k ) µ() ) π ( + ) π + A log() `log() «log() + O ««log() = + O `log() «log() + O r! = log π k «k + O k/ which finishes the roof. 8 The lattice tye Let z P (Z ) = P (Q ). For any k, we obtain a -adic aroximation lattice L(z, k) = L(z mod k ) and we may consider the sequence minl(z, k) = minl(z mod k ). We say that z is of lattice tye α if there is a constant C > 0 such that log(minl(z, k)) k α < C. The revious section suggests that the most frequent lattice tye is log() which is also the largest lattice tye ossible by Minkowski s bound. The next lemma shows that the lattice tye is some sort of a measure of the irrationality of z. Proosition 6. The elements in P (Q ) of lattice tye 0 are exactly P (Q). Proof. If z P (Q) then, we may write z = (a : b) with a and b two integers which are rime to each other. Then every one of the lattices L(z, k) contains the oint ( b, a) and hence minl(z, k) is bounded by a + b, i.e. z is of lattice tye 0. Conversely if z is of lattice tye 0, there is a constant C such that all of the L(z, k) contain an element of norm less than ex(c). Since there are only finitely many integral oints in this disc, there is a oint (x, y) which belongs to L(z, k) for infinitely many k. Write z = ( n : z ) for some n 0 and z Z. Hence n x + z y 0 (mod k ) for infinitely many k. So z must be equal to n x y which lies in Q. The next roosition gives a lot of algebraic numbers of maximal lattice tye. Proosition 7. Let θ Q be an algebraic integer such that Z[θ] is the number ring in a quadratic imaginary field K. Then θ is of maximal lattice tye log(). Proof. The embedding Z[θ] Q defines a rime ideal = Z[θ] Z above. The owers of the ideal k may be written as k = {x + y θ x, y Z with x + y θ k Z }, in other words it is the lattice L(θ, k) in Z[θ] Z. Let h be the class number of K. For i h, let α i be the element of i whose value of N K/Q (α i) N is minimal. Since h is rincial, we may take α h = α to be the generator of h ; moreover the minimal values of the norm for h k+i is taken by α k α i. Hence we obtain for any i h minl(θ, k h + i) = min β k h+i N K/Q (β) = N K/Q (α) k N K/Q (α i) Now, we note that N K/Q ( ) / extends to a norm on the vector sace Z[θ] and as such it is equivalent to the usual norm. So the growth of the minimal vector is given by log minl(l(θ, k h + i)) = log( h k ) + log N K/Q (α i) since the norm of α is equal to h. This roves the roosition. 4

15 Aart from these two roositions there is a long list of questions that one might ask about lattice tyes. Is it ossible to construct -adic integers that do not have a lattice tye? Can transcendental numbers have non-trivial and non-maximal lattice tye? Can one give any inequalities on the lattice tye of sums or roducts of numbers?... The most imortant question with resect to our initial question is of course whether one can find a criterion for deciding if a number has maximal lattice tye. Most of all -adic numbers have maximal lattice tye. The original conjecture for curves of rank can be reformulated as follows. Proosition 8. Let E be an ellitic curve of rank over Q and be a rime number. Let Q and Q be two linearly indeendent oints in B (). Then the strong conjecture is equivalent to the statement that z = (L (Q ) : L (Q )) P (Q ) has maximal lattice tye The notion of lattice tye of dimension can easily be generalised to higher dimensions and the above roosition can be extended accordingly. The main results of section 7 should also be valid in a suitable generalised form. The conjecture of Lang and Waldschmidt as formulated in conjecture 3 imlies that if (log (a ) : log (a )) P (Q ) has a lattice tye then it is of maximal lattice tye and we can ut ε = 0. eferences [Ber78] [Cas97] [dw89] Daniel Bertrand, Aroximations diohantiennes -adiques sur les courbes ellitiques admettant une multilication comlexe, Comositio Math. 37 (978), no., 50. J. W. S. Cassels, An introduction to the geometry of numbers, Classics in Mathematics, Sringer-Verlag, Berlin, 997, Corrected rerint of the 97 edition. B. M. M. de Weger, Algorithms for Diohantine equations, CWI Tract, vol. 65, Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 989. [Hux03] M. N. Huxley, Exonential sums and lattice oints. III, Proc. London Math. Soc. (3) 87 (003), no. 3, [Lan78] Serge Lang, Ellitic curves: Diohantine analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Princiles of Mathematical Sciences], vol. 3, Sringer-Verlag, Berlin, 978. [Phi99] [U96] [Sil9] Patrice Philion, Quelques remarques sur des questions d aroximation diohantienne, Bull. Austral. Math. Soc. 59 (999), no., Gaël émond and Florent Urfels, Aroximation diohantienne de logarithmes ellitiques -adiques, J. Number Theory 57 (996), no., Joseh H. Silverman, The arithmetic of ellitic curves, Sringer-Verlag, New York, 99, Corrected rerint of the 986 original. [Sma98] Nigel P. Smart, The algorithmic resolution of Diohantine equations, London Mathematical Society Student Texts, vol. 4, Cambridge University Press, Cambridge,