Transcendence of elliptic modular functions in characteristic p

José Felipe Voloch

Classical transcendence theory began with the study of special values of the exponential function and later the study of special values of the logarithm and, more generally elliptic and abelian logarithms, was added (see [B]). Characteristic p analogues have been considered, firstly with the analogues of the exponential function arising from the theory of Drinfeld modules (see [Y]) and also with p-adic exponentiation and its abelian analogue ([MP],[V]). Let us also note the general transcendence criterion obtained in [CKMR].

The purpose of this paper is to study the transcendence properties in positive characteristic of the power series introduced by Tate which give a non-archimedian analogue of elliptic functions. Let p be a prime number, k be the algebraic closure of F_{p} and q a transcendental over k, i.e., a variable. The following series determine elements of k[[q]]:

a_{4} = −5 ∑_{n≥1} n^{3}q^{n}/(1 − q^{n}),

a_{6} = (−1/12) ∑_{n≥1}(7n^{5} + 5n^{3})q^{n}/(1 − q^{n}).

Swinnerton-Dyer has shown that a_{4} and a_{6} are algebraically dependent in positive characteristic, in contrast with characteristic zero [S-D]. In fact this is clear for p < 11, for example a_{4} = a_{6} for p = 2 and a_{4} = 0 for p = 5. Let K = k(a_{4}, a_{6}) and L = k((q)). Note that K is a subfield of L. Our first result is:

Theorem A. q is transcendental over K.

We will prove this theorem below. First we will introduce some more notation and state our second result. Consider the following series:

x = x(q, u) = ∑_{n∈Z} q^{n}u/(1 − q^{n}u)^{2} − 2 ∑_{n≥1} nq^{n}/(1 − q^{n}),

y = y(q, u) = ∑_{n∈Z} q^{2n}u^{2}/(1 − q^{n}u)^{3} + ∑_{n≥1} nq^{n}/(1 − q^{n}).

They converge for any u in L^{∗}, not a power of q, and satisfy y^{2} + xy = x^{3} + a_{4}x + a_{6}, therefore giving an analytic parametrization over L of the elliptic curve E (the Tate curve), defined over K by this equation. For details on the Tate curve, the reader may consult [S], chapter V.

Theorem B. If u ≠ 0, q^{n}, n ∈ Z is algebraic over L and such that (x(q, u), y(q, u)) is a point on E algebraic over K and of infinite order, then u is transcendental over K.

Note that theorem A is the analogue of transcendence of periods of an elliptic curve with algebraic coefficients and theorem B the analogue of the transcendence of the elliptic logarithm of algebraic points on such elliptic curves. See [B], chapter 6.

After seeing this paper, D. Thakur [T] found another proof of theorem A using the transcendence criterion of [CKMR].

To prove these theorems we will need to develop some results on the arithmetic of E/K, specially regarding higher p-descents. Let us recall some well-known facts first. The Tate curve E : y^{2} + xy = x^{3} + a_{4}x + a_{6} is indeed an elliptic curve with discriminant Δ = q ∏_{n=1}^{∞}(1 − q^{n})^{24} and j-invariant j = q^{−1} + 744 + · · ·. Moreover E is ordinary with Hasse invariant 1. Indeed, this fact gives the algebraic relation between a_{4} and a_{6} when the Hasse invariant is expressed as polynomial in a_{4} and a_{6}. Let E^{(p^{n})} be the image of E under the n-th power of the Frobenius map F^{n} and V_{n} : E^{(p^{n})} → E the dual isogeny, the n-th order Verschiebung, which is separable since E is ordinary. Let K_{n} be the field of definition of the points of ker V_{n}. These fields were studied by Igusa ([I], see also [KM], theorem 12.6.1), who showed that their degree over K grows with n. He actually showed much more but that is all we will need. Note that x(q^{p^{n}}, u), y(q^{p^{n}}, u) parametrize E^{(p^{n})} and that the image of u = q gives a generator P_{n} of ker V_{n}. Let E[p^{n}] denote, as usual, the group scheme which is the kernel of multiplication by p^{n} on E. We have the following exact sequence 0 → ker F^{n} → E[p^{n}] → ker V_{n} → 0. Over K_{n}, ker V_{n} is isomorphic to Z/p^{n}Z and one can choose the isomorphism so that 1 corresponds to the image of q as above. By Cartier duality, we get an isomorphism between ker F^{n} and μ_{p^{n}}. Thus, the above exact sequence gives a class q_{n}, the Serre-Tate parameter, in Ext_{K_{n}}^{1}(Z/p^{n}Z, μ_{p^{n}}) = H^{1}(K_{n}, μ_{p^{n}}) = K_{n}^{∗}/(K_{n}^{∗})^{p^{n}} where here and elsewhere the cohomology groups are in the flat cohomology of group schemes. The absolute Galois group G of K acts on this group, and we wish to describe its action on q_{n}. Firstly, G acts on ker V_{n} by a p-adic character χ : G → Z_{p}^{∗}, which is independent of n and is of infinite order by Igusa’s theorem. Let K_{s} be the separable closure of K. Taking cohomology of the exact sequence 0 → ker F^{n} → E[p^{n}] → ker V_{n} → 0, we get q_{n} as the image of 1 ∈ ker V_{n} in H^{1}(K_{n}, μ_{p^{n}}). We will, more generally, consider the map β_{n} : E^{(p^{n})}(K_{s})/F^{n}(E(K_{s})) → H^{1}(K_{s}, μ_{p^{n}}) = K_{s}^{∗}/(K_{s}^{∗})^{p^{n}} obtained from the exact sequence 0 → ker F^{n} → E → E^{(p^{n})} → 0 and the identification of ker F^{n} and μ_{p^{n}} given above.

Lemma 1. If P ∈ E^{(p^{n})}(K_{s}) and σ ∈ G then σ(β_{n}(P)) = (β_{n}(σ(P))^{χ(σ)}.

Proof: The map E^{(p^{n})}(K_{s}) → H^{1}(K_{s}, ker F^{n}) commutes with the action of G, and G acts on Hom(ker F^{n}, μ_{p^{n}}) = ker V_{n} by χ, hence the lemma.

Lemma 2. The class of u in L^{∗}/(L^{∗})^{p^{n}} is equal to β_{n}(x(q^{p^{n}}, u), y(q^{p^{n}}, u)).

Proof: The Tate parametrization gives that E(L) is isomorphic to L^{∗}/q^{Z} and that E^{(p^{n})}(L) is isomorphic to L^{∗}/q^{p^{n}Z} and that F^{n} is induced by u ↦ u^{p^{n}} in L^{∗}. The lemma follows.

Proof of Theorem A: If P_{n} ∈ E^{(p^{n})}(K_{n}) corresponds to u = q then β_{n}(P_{n}) = q_{n}. Also, G acts on ker V_{n} via χ. Therefore, applying lemma 1 to P_{n} yields σ(q_{n}) = q_{n}^{χ(σ)^{2}}. From lemma 2, q = q_{n} in L^{∗}/(L^{∗})^{p^{n}}. If q is algebraic, then it follows that q = q_{n} in K_{s}^{∗}/(K_{s}^{∗})^{p^{n}} also. This is impossible since, on one hand, G would act on q by a finite quotient and, on the other hand, G acts on q_{n} by cyclic groups that grow with n, since χ is of infinite order.

Proof of Theorem B: Replacing u by u^{p^{m}} for suitable m, we may assume u is separable over L. Let L_{s} denote the separable closure of L. The point in E^{(p^{n})} with parameter u is a point Q_{n} satisfying V_{n}(Q_{n}) = Q_{0}, so is an algebraic point. By lemma 2, β_{n}(Q_{n}) = u in L_{s}^{∗}/(L_{s}^{∗})^{p^{n}}. If σ ∈ G acts trivially on Q_{0}, then σ(Q_{n}) − Q_{n} ∈ ker V_{n}. Therefore there exists an additive p-adic character ψ such that β_{n}(σ(Q_{n})) = β_{n}(Q_{n})q_{n}^{ψ(σ)}. Assume now that u is algebraic over K and assume also that σ fixes u. Then, K_{s}^{∗}/(K_{s}^{∗})^{p^{n}}, u = σ(u) = σ(β_{n}(Q_{n})) = β_{n}(σ(Q_{n}))^{χ(σ)}, by lemma 1. Hence u = u^{χ(σ)}q_{n}^{χ(σ)ψ(σ)} in K_{s}^{∗}/(K_{s}^{∗})^{p^{n}}. Finally choose σ satisfying the above conditions and such that χ(σ) ≠ 1, which exists by Igusa’s theorem. The above equation then gives that there are p-adic integers r ≠ 0, m such that u^{r} = q^{m} in L_{s}^{∗}/(L_{s}^{∗})^{p^{n}}, for all n. Furthermore, we can assume without loss of generality that m is an ordinary integer. If v is the valuation on L_{s}, extending the natural one on L, this gives rv(u) ≡ m( mod p^{n}), for all n sufficiently large, so rv(u) = m. If v(u) ≠ 0, this shows that r in a rational number and raising everything to a suitable power, we can assume that r is an integer. Since the only elements of L_{s} which are p^{n}-th powers for all n are the elements of k, we obtain an equation u^{r} = αq^{m} in L where α ∈ k^{∗}, which gives that Q_{0} is torsion, proving the theorem in this case. If v(u) = 0 then Q_{0} is in the formal group of E which is a Z_{p}-module (see [V]). The equation u^{r} = q^{m} in L_{s}^{∗}/(L_{s}^{∗})^{p^{n}} for all n lead to rQ_{0} = 0 and the same holds for any multiple of r in Z_{p} instead of r. In particular it holds for some non-zero integer, which completes the proof of the theorem.

If u gives a torsion point on E then u = αq^{r}, where α is in k and r is a rational number. Therefore, u is transcendental over K if and only if r ≠ 0.

Finally, one could ask for analogues of the classical statements dealing with linear forms in logarithms. However, the only sense that apparently can be made of that is through p-adic exponentiation, which makes sense for elements of 1 + qk[[q]]. It then follows from the results of [V] that if u_{1}, …, u_{r} ∈ 1 + qk[[q]] give Z-linearly independent points on E, algebraic over K, then u_{1}, …, u_{r} are Z_{p}-multiplicatively independent.

Acknowledgements: The author would like to thank D. Thakur and J. Tate for helpful discussions and the NSF (grant DMS-9301157) and the Alfred P. Sloan Foundation for financial support.

References.

[B] A. Baker, Transcendental number theory, Cambridge Univ. Press, 1975.

[CKMR] G. Christol et al., Suites algèbriques, automates et substituitions, Bull. Soc.

Math. France, 108 (1980) 401-419.

[I] J.-I. Igusa, On the algebraic theory of elliptic modular functions, J. Math. Soc. Japan, 20 (1968) 96-106.

[K] N. M. Katz, Serre-Tate local moduli, Springer LNM 868 (1981) 138-202.

[KM] N. M. Katz, B. Mazur, Arithmetic moduli of elliptic curves, Princeton Univ. Press, 1985.

[MP] M. Mendès France, A. J. van der Poorten, Automata and the arithmetic of formal power series, Acta Arithmetica, XLVI (1986) 211-214.

[S] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, GTM 151, Springer, New York, 1994. [S-D] H. P. F. Swinnerton-Dyer, On l-adic representations and congruences of modular forms, Springer LMN 350 (1973), 1-55.

[T] D. S. Thakur, Automata-style proof of Voloch’s result on transcendence , preprint.

[V] J. F. Voloch, Diophantine Approximation on Abelian varieties in in characteristic p, Amer. J. Math., to appear.

[Y] J. Yu, Transcendence in finite characteristic, in The arithmetic of function fields, D. Goss et al.,eds., de Gruyter, Berlin 1992, pp.253-264. Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA e-mail: voloch@math.utexas.edu