By Shigeki Akiyama (auth.), Kenji Nagasaka, Etienne Fouvry (eds.)

**Contents: S. Akiyama:** On a definite sum of lines of Hecke operators.- **J.-P. Allouche, P. Flajolet, M. Mendès France:** Algebraically self sufficient formal energy sequence: A language idea interpretation.- **J.-P. Allouche, J. Shallit:** Sums of digits and the Hurwitz zeta function.- **D. Bertrand:** Transcendental equipment in mathematics geometry.- **G. Christol:****Globally bounded ideas of differential equations.- **E. **Fouvry:** Nombres presque premiers dans les petits intervalles.- **E. Fouvry, G. Tenenbaum:** Diviseurs de Titchmarsh des entiers sans grand facteur premier.- **A. ****Fujii:** Uniform distributions of the zeros of the Riemann zeta functionality and the suggest price theorems of Dirichlet L-functions (II).- **K. Goto, T. Kano:** a few stipulations on uniform distribution of monotone sequences.- **T. Harase:** Algebraic dependence of formal strength series.- **G. Henniart:** Une conséquence de l. a. théorie du changement de base pour GL(n).- **K. Horie, M. Horie:** at the exponents of excellent category teams of CM-fields.- **M. Ishibashi, S. Kanemitsu:** a few asymptotic formulation of Ramanujan.- **N. Kurokawa:** Analyticity of Dirichlet sequence over best powers.- **K. Matsumoto:** Value-distribution of zeta-functions.- **S. Mizumoto:** Integrality of severe values of triple product L-functions.- **T. Oda:** a number of Hecke sequence for class-1 Whittaker features on GL(n) over p-adic fields.- Programme of the Symposium.

**Read Online or Download Analytic Number Theory: Proceedings of the Japanese-French Symposium held in Tokyo, Japan, October 10–13, 1988 PDF**

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**Additional resources for Analytic Number Theory: Proceedings of the Japanese-French Symposium held in Tokyo, Japan, October 10–13, 1988**

**Sample text**

Elkies To do this we must be able to compute numerically the rational function t : ∼ H/Γ ∗(1)→P1 . Equivalently, we need to associate to each t ∈ P1 a representative of its corresponding Γ ∗(1)-orbit in H. We noted already that this is done, up to a fractional linear transformation over C, by the quotient of two hypergeometric functions in t. To fix the transformation we need images of three points, and we naturally choose the elliptic points t = 0, 1, ∞. These go to fixed points of s2 , s4 , s6 ∈ Γ ∗(1), and to find those fixed points we need an explicit action of Γ ∗(1) on H.

This determines the cover up to K-isomorphism the curve X0 ((τ )) has genus 0, and we can choose coordinates x on that curve and t on X (1) such that t(P3 ) = ∞ and t = x3 − 3cx for some c = 0 — but not the location of the unramified point P2 relative to the other three elliptic points. To determine that we once again use the involution, this time w(τ) , of X0 ((τ )): this involution fixes the point above P2 corresponding to its self-isogeny, and pairs the other two preimages of P2 with the simple preimages of P2 , P2 .

Even in the simplest case Σ = {2, 3} where Γ ∗(1) is a triangle group and all the covers X0∗(l)/X ∗ (1) are in principle determined by their ramifications, finding those covers seems at present a difficult problem once l gets much larger than the few primes we have dealt with here. This is the case even when l is still small enough that X0∗ (l) has genus small enough, say g ≤ 5, that the curve should have a simple model in projective space. For instance, according to 35 the curve X0∗ (73) has genus 1.