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L-functions of universal elliptic curves over Igusa curves. (English) Zbl 0731.14013

Let \(K_ n\) be the function field of the Igusa curve of level \(p^ n\) over \({\mathbb{F}}_ q\), where p is a prime, \(n\in {\mathbb{N}}\) (n\(\geq 2\) if \(p=2)\) and \({\mathbb{F}}_ q\) is a finite extension of \({\mathbb{F}}_ p\). Let \(L(\rho,s)\) be the L-function attached to the representation \(\rho=\rho_{i,d}\) obtained from the natural action of Gal\((\bar K_ n/K_ n)\) on the étale cohomology group \(H^ i(E^ d\otimes \bar K_ n,{\mathbb{Q}}_{\ell})\) (\(\ell \neq p)\), where \(E^ d\) is the d-fold product of \(E/K_ n\) and E is the universal elliptic curve over \(K_ n.\)
The main result gives an expression of \(L(\rho,s)\) as a rational function in \(q^{-s}\) involving the eigenvalues of the Hecke operator \(T_ q\) acting on the space of cusp forms of integral weight k on \(\Gamma_ 1(p^ m)\) (0\(\leq k\leq i\), \(0\leq m\leq n)\). The proof makes essential use of a good reduction theorem by N. M. Katz and B. Mazur [“Arithmetic moduli of elliptic curves”, Ann. Math. Stud. 108 (1985; Zbl 0576.14026)] which allows one to identify the subspace of \(H^ 1\) of the Igusa curve with coefficients in the sheaf attached to \(Sym^ k\rho_{1,1}\) on which \(({\mathbb{Z}}/p^ n{\mathbb{Z}})^*\) acts, with a similar subspace of \(H^ 1\) of the modular curve \(X_ 1(p^ n)\) in characteristic zero. On the other hand, by means of the Eichler-Shimura isomorphism one can relate the latter group to modular forms.
The author points out that his theorem may provide interesting test cases for various conjectures relating L-functions to cycles. For example, if i is odd, then according to the conjecture of Birch and Swinnerton-Dyer the order of vanishing of \(L(\rho_{2i-1},s)\) at \(s=i\) should be equal to the rank of the group \(A^ i(E^ d)\) of cycles on \(E^ d\) of codimension i homologically equivalent to zero modulo rational equivalence, and the existence of modular forms with complex multiplication [G. Shimura, Nagoya Math. J. 43, 199-208 (1971; Zbl 0225.14015)] in conjunction with the author’s result, in fact, provides zeroes of \(L(\rho_{2i-1},s)\) at \(s=i\).

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H52 Elliptic curves
14C25 Algebraic cycles
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