Modular elements over function fields
This paper deals with the theory of ``modular elements'' for elliptic curves over global fields of characteristic \(p\). The modular elements under consideration are function field versions of the objects considered by B. Mazur and J. Tate in their study of ``refined'' versions of the \(p\)-adic Birch-Swinnerton-Dyer conjecture [see \textit{B. Mazur} and \textit{J. Tate}, Duke Math. J. 54, 771-750 (1987; Zbl 0636.14004)]. The modular elements of this paper, like those of Mazur and Tate, are associated to modular forms and are related to special values of the associated \(L\)- functions. It is possible to express a function field version of Mazur and Tate's refined conjectures using these elements. This paper recalls Mazur's construction of the modular elements in the function field case and proves some basic integrality results for them; it derives a general functional equation satisfied by the modular elements; and it presents some examples where the function field version of the refined conjecture holds true. The paper does not present the details of these examples, but instead cites the author's thesis [``Refined conjecture of the Birch and Swinnerton-Dyer type'', Thesis, Dep. of Math., Harvard Univ. (June 1990)] and another paper with the same title which has not yet appeared. Let \(k\) be a global field of characteristic \(p\) over the field \(\mathbb{F}_ q\), \({\mathbf A}_ k^ \times\) the ideles of \(k\), \(U_ D\) the compact open subgroup of \({\mathbf A}_ k^ \times\) determined by a divisor \(D\) of \(k\), and \[ W_ D= k^* \setminus {\mathbf A}_ k^ \times /U_ D \] the Weil group of conductor \(D\). If \(f\) is a cusp form on \(\text{GL}_ 2 ({\mathbf A}_ k)\), then the modular element \(\Theta_{D,f}\) is the element of the group ring \({\mathbb{Q}} [W_ D]\) obtained as the sum \[ \Theta_{D,f}:= {1\over {I_ D}} \sum_{w\in W_ D} \Biggl(f \begin{pmatrix} dw &w\\ 0 &1 \end{pmatrix}\Biggr)w \] where \(I_ D=1\) if \(D\) is non-trivial and \(I_ D= q-1\) otherwise. This is in fact a finite sum. The author recalls a result of B. Mazur (from a 1987 letter to J. Tate) which shows that, if \(\chi\) is a primitive quasi-character of \(W_ D\), then \[ \chi(\Theta_{D,f})- \tau_ \chi L(\chi,1) \] where \(\tau_ \chi\) is a Gauss sum and \(L(\chi,1)\) is the special value of the \(L\)- function associated to \(f\). The functional equation derived in the paper relates \(\Theta_{D,f}\) to \(\Theta_{D,w \cdot f}\) where \(w\cdot f\) is the transform of \(f\) by an Atkin-Lehner involution. The integrality results of the paper show that if \(f\) is a cuspidal eigenform corresponding to an elliptic curve over the field \(k= \mathbb{F}_ q (T)\), corresponding to an elliptic curve over \(k\) with square free conductor \(N\), or if \(E\) is a constant elliptic curve, then the denominators of \(\Theta_{D,f}\) are powers of \(p\).
scientific article; zbMATH DE number 483990