Fay's trisecant identity

In algebraic geometry, Fay's trisecant identity is an identity between theta functions of Riemann surfaces introduced by Fay (1973, chapter 3, page 34, formula 45). Fay's identity holds for theta functions of Jacobians of curves, but not for theta functions of general abelian varieties.

The name "trisecant identity" refers to the geometric interpretation given by Mumford (1984, p.3.219), who used it to show that the Kummer variety of a genus g Riemann surface, given by the image of the map from the Jacobian to projective space of dimension 2^g – 1 induced by theta functions of order 2, has a 4-dimensional space of trisecants.

Statement

Suppose that

C is a compact Riemann surface
g is the genus of C
θ is the Riemann theta function of C, a function from C^g to C
E is a prime form on C×C
u,v,x,y are points of C
z is an element of C^g
ω is a 1-form on C with values in C^g

The Fay's identity states that

${\begin{aligned}&E(x,v)E(u,y)\theta \left(z+\int _{u}^{x}\omega \right)\theta \left(z+\int _{v}^{y}\omega \right)\\-&E(x,u)E(v,y)\theta \left(z+\int _{v}^{x}\omega \right)\theta \left(z+\int _{u}^{y}\omega \right)\\=&E(x,y)E(u,v)\theta (z)\theta \left(z+\int _{u+v}^{x+y}\omega \right)\end{aligned}}$

with

${\begin{aligned}&\int _{u+v}^{x+y}\omega =\int _{u}^{x}\omega +\int _{v}^{y}\omega =\int _{u}^{y}\omega +\int _{v}^{x}\omega \end{aligned}}$

gollark: It does mean that you need self-modifying code to subtract non-constant numbers, but such is the price of such elegance.

gollark: This is how I merged `MOV` (in the sense of "set register to fixed value") and `ADD`.

gollark: See, there are exactly 16 registers, one of which, r0, always contains 0, and one of which, rf, is the program counter, and many of the instructions take a 4-bit value representing which register to pull from.

gollark: <@!330678593904443393> You would pass it 6 register indices.

gollark: 32 registers would probably allow room for more fun stuff, like the program metacounter register.

References

Fay, John D. (1973), Theta functions on Riemann surfaces, Lecture Notes in Mathematics, 352, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0060090, ISBN 978-3-540-06517-3, MR 0335789
Mumford, David (1974), "Prym varieties. I", in Ahlfors, Lars V.; Kra, Irwin; Nirenberg, Louis; et al. (eds.), Contributions to analysis (a collection of papers dedicated to Lipman Bers), Boston, MA: Academic Press, pp. 325–350, ISBN 978-0-12-044850-0, MR 0379510
Mumford, David (1984), Tata lectures on theta. II, Progress in Mathematics, 43, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3110-9, MR 0742776

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