Brauer-Siegel theorem for families of elliptic surfaces over finite fields
The main objects of the note under review are an elliptic curve \(E\) defined over \(K=\mathbb F_q(X)\), the field of rational functions of a smooth projective \(\mathbb F_q\)-curve \(X\), and the corresponding elliptic surface \(\mathcal E\to X\). Varying \(X\) in a family of coverings \[ X=X_0\longleftarrow X_1\longleftarrow\dots\longleftarrow X_i\longleftarrow\dots , \] one can consider the corresponding family of elliptic surfaces \(\mathcal E_i\) obtained from \(\mathcal E_0\) by base change. For the elliptic curve \(E_i\), the generic fibre of \(\mathcal E_i\), one can consider the first nonzero coefficient \(c_{E_i}\) in the expansion of \(L_{E_i}(s)\) in the Taylor series at \(s=1\) and look at asymptotic behaviour of the ratio \(\log |c_{E_i}|/g(X_i)\). A conjecture stated by \textit{B.~È.~ Kunyavskiĭ} and \textit{M.~A.~Tsfasman} [Int. Math. Res. Not. 2008, Article ID rnn009, 9 p. (2008; Zbl 1225.11087)] (reformulated in an appropriate way) suggests an explicit expression for the limit of this ratio. Such a statement could be viewed as a higher-dimensional analogue of the Brauer--Siegel theorem. However, this conjecture remains completely open (even for the case of constant families because of a gap in the proof presented in the above cited paper which was noticed by the author). The note under review provides an important step towards the above mentioned conjecture. Its main result (announced in Theorem 2) provides, among other things, an upper bound for the limit defined above in terms of the value at 1 of the limit \(L\)-function of the family \(\{E_i\}\) (under the assumption that the ratio \(r_{E_i}/g(X_i)\) tends to zero, which holds, in particular, for all constant families). The proofs will be published elsewhere.
scientific article; zbMATH DE number 5621809