On the inverse formula (Q1849101)

When an abelian variety has a principal polarization, it is self-dual. However, when the normalization is not principal, the problem of giving an explicit formula for the ``inversion'' (by which the author means taking the opposite with respect to the group law), on the Heisenberg group associated to the polarizing bundle, becomes subtle and important. The author generalizes results, both classical [cf. \textit{C. Birkenhake, H. Lange}, Complex abelian varieties. 2nd augmented ed. Grundlehren der Mathematischen Wissenschaften 302. (Berlin: Springer) (2004; Zbl 1056.14063)] and proved by \textit{D. Mumford} [Invent. Math. 1, 287--354 (1966); ibid. 3, 75--135, 215--244 (1967; Zbl 0219.14024)] and \textit{G. R. Kempf} [Am. J. Math. 111, No.1, 65--94 (1989; Zbl 0673.14023)] in any characteristic (different from 2), and is able to give very precise results, even for (ample) line bundles that satisfy fewer technical conditions than the previously considered ones (he also compares definitions, and can show that certain conditions follow, in certain cases). These formulas should be very useful, potentially in mathematical physics where the Heisenberg group and the sections of various theta divisors, viewed as `duals' of conformal blocks [cf. e.g. \textit{T. Abe}, On \(\text{SL}(2)\)--\(\text{GL}(n)\) strange duality, preprint] are of great interest. To give an oversimplified statement of the main result, given a separable ample line bundle \(L\) over an abelian variety \(X\), with the only requirement of being symmetric with respect to the `minus' involution \(\iota\), \(\iota^\ast L\cong L\), the author considers generalized Heisenberg groups \({\mathcal G}(\delta )\) where \(\delta =(\delta_1,\ldots ,\delta_g)\) is the polarization and a theta structure \({\mathcal G}(\delta ) {\cong\atop{{\rightarrow\atop{\;\;}}}} {\mathcal G}(L)\). He gives the appropriate notion of even/odd line bundle algebraically equivalent to \(L\), counts the number of symmetric sections in \(H(L)\) that are invariant under the natural Heisenberg action, using characters of the Heisenberg group, and shows that for such an even \({\mathcal L}\cong T^\ast_a(L)\), taking the opposite results in adding \(2u_0\) to \(u\) (where \(u\) is the point in \(X\) corresponding to the projection of the Heisenberg group and \(u_0\) corresponds to \(a\) and dividing by the character corresponding to \((u+u_0)\). He compares this with the classical and with Kempf's formula. The methods are clever uses of linear-algebraic properties for the representation of the Heisenberg group (on spaces of sections), in the presence of suitably chosen symplectic forms and attendant Arf invariants.