Hyperelliptic integrals and generalized arithmetic-geometric mean
Consider \(d=2g+2\) real numbers (there is ultimately no real loss of generality in assuming that \(d\) is even) \(a_1<a_2<\cdots<a_d\) and, for \(k\geq 1\), put \(f(x)=(x-a_1)\cdots(x-a_d)\) and define the integral \[ I_{j,k}(a_1,\ldots,a_d)=\int_{a_j}^{a_{j+1}} {{x^{k-1}dx}\over{\sqrt{|f(x)|}}} \] On the hyperelliptic curve \(C=\{y^2=f(x)\}\) one has a basis of the space of global \(1\)-forms given by \(\psi_k=x^{k-1}{{dx}\over{y}}\) and a standard symplectic basis \(A_1,\ldots,A_g,B_1,\ldots,B_g\) for \(H_1(C,{\mathbb Z})\). Let us define \[ A_{k,j}:=\int_{A_j}\psi_k=(-1)^{j+1}I_{2j-1,k} \] and \[ B_{k,j}:=\int_{B_j}\psi_k=\sum_{l=j}^k (-1)^{l-1}I_{2l,k}, \] so that the matrices \(A=(A_{j,k})\) and \(B=(B_{j,k})\) determine the normalised period matrix of \(C\) as \((1, iA^{-1}B)\). The authors compute \(\det A\) and \(\det B\). First they write down an integral formula, namely (for \(A\)) \[ \det A=(-1)^{\lfloor g/2\rfloor} \int_{x_1=a_1}^{a_2}\cdots\int_{x_g=a_{2g-1}}^{a_{2g}} \prod_{1\leq k<j\leq g} {{(x_j-x_k)}\over{\sqrt{|f(x_1)\cdots f(x_g)|}}}dx_1\cdots dx_g. \] Then they use the Thomae formula to get an expression in terms of a generalised arithmetic-geometric mean (AGM) of \(2^g\) numbers, defined by Dupont [Moyenne de Borchardt sur les complexes, Preprint -- per bibl.]. A function \(v_0: V={\mathbb F}_2^g\to {\mathbb R}_+\) gives a sequence \((v_n)\) of functions defined by \[ v_{n+1}: \beta\to 2^{-g}\sum_{\gamma\in V}\sqrt{v_n(\gamma)v_n(\beta+\gamma)}, \] and this sequence converges to a constant function whose value is the AGM of the numbers \(\{v_0(\beta)\}\). The result is illustrated by a numerical example in which the computation of \(\det A\) by this method is seen to be far faster than evaluating each entry of \(A\), even for \(g=3\).
scientific article; zbMATH DE number 6084536