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A recursive family of differential polynomials generated by the Sylvester identity, and addition theorems for hyperelliptic Kleinian functions. (English. Russian original) Zbl 1126.33301
Funct. Anal. Appl. 31, No. 4, 240-251 (1997); translation from Funkts. Anal. Prilozh. 31, No. 4, 19-33 (1997).
From the introduction: In the present paper we introduce a recursive family of functions $$\{F_n\}){n=1,2,\dots}$$ generated by the relations $F_{n+1} F_{n-1}= \lambda(F_n\partial_{xy} F_n- \partial_x F_n- \partial_x F_n\partial_y F_n)+\mu f(x, y)F^2_n\tag{1}$ with initial conditions $$F_0= 1$$ and $$F_1= f(x,y)$$. It is shown that, for any $$n$$, the function $$F_n(x, y)$$ is a polynomial with integer coefficients in a function $$f(x, y)$$, its partial derivatives, and the parameters $$\lambda$$ and $$\mu$$. The differential polynomials $$F_n(x, y)= D_n(f(x, y))= D_n(f(x, y);\lambda,\mu)$$ are representable as leading principal minors of a matrix, and the family $$\{D_n(f)\}$$ itself is generated by the Sylvester identity for compound determinants.
We apply this recursion to the classical problem of generalizing the relation ${\sigma(u+ v)\sigma(u- v)\over \sigma(u)^2\sigma(v)^2}= \wp(v)- \wp(u)$ for the elliptic Weierstrass functions which plays a key role in the theory and applications of elliptic functions (here the genus is $$g= 1$$), to the case of hyperelliptic Kleinian $$\sigma$$-functions (with genus $$g> 1$$).
A hyperelliptic $$\sigma$$-function of genus $$g$$ is defined as an element of the ring of Riemann $$\theta$$-functions that is automorphic with respect to the action of the modular group $$\text{Sp}(2g,\mathbb{Z})$$. The logarithmic derivatives of the $$\sigma$$-function, $\wp_{ij}=- {\partial^2\over\partial u_i\partial u_j}\ln \sigma(u),\;\wp_{i,j,k}=-{\partial^2\over\partial u_i\partial u_j\partial u_k}\ln\sigma(u),\;i,j,k= 1,\dots, g,$ and so on, are hyperelliptic Abelian functions, i.e., $$2g$$-periodic meromorphic functions.
If the genus $$g$$ is one, then the field of elliptic functions is generated by Weierstrass elliptic functions $$\wp$$ and $$\wp'$$, which uniformize an elliptic curve. In the case $$g> 1$$, the hyperelliptic functions $$\wp_{ij}$$ and $$\wp_{ijk}$$ uniformize the Jacobian variety $$\text{Jac}(V)$$ of a hyperelliptic curve $$V$$, while the even functions $$\wp_{ij}$$ themselves uniformize the Kummer variety $$\text{Kum}(V)= \text{Jac}(V)/(u\to -u)$$, defined as the quotient of the Jacobian with respect to the hyperelliptic involution.
The following assertion is one of the main results of the paper.
For a hyperelliptic Kleinian $$\sigma$$-function of genus $$g\geq 1$$, the following relation holds: ${\sigma(u+ v)\sigma(u- v)\over \sigma(u)^2\sigma(v)^2}= D_g\Biggl(\wp_{gg}(v)- \wp_{gg}(u);\,{1\over 4},{1\over 2}\Biggr),$ where $$\wp_{gg}(v)- \wp_{gg}(u)$$ is regarded as a function of $$x= u_g+ v_g$$ and $$y= u_g- v_g$$, and $$D_g(\;;\;)$$ is the differential polynomial defined by the recursion (1).

##### MSC:
 33E05 Elliptic functions and integrals 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 14H42 Theta functions and curves; Schottky problem 35Q53 KdV equations (Korteweg-de Vries equations) 14H40 Jacobians, Prym varieties 14K25 Theta functions and abelian varieties 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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