Appendix A: First-order theta derivative relations
1.1 A1: Genus 2
Below, a list of relations between vectors of first-order theta derivatives in genus 2 is presented (for brevity, the notation \(\theta ^{\{\kappa _1,\kappa _2\}}\) for \(\theta [\{\kappa _1,\kappa _2\}](0;\tau )\) is used)
$$\begin{aligned} \partial _{v} \theta ^{\emptyset }&= \frac{\theta ^{\{2,3\}}\theta ^{\{2,4\}}\theta ^{\{2,5\}}}{\theta ^{\{3,4\}}\theta ^{\{3,5\}}\theta ^{\{4,5\}}} \partial _{v} \theta ^{\{1\}} - \frac{\theta ^{\{1,3\}}\theta ^{\{1,4\}}\theta ^{\{1,5\}}}{\theta ^{\{3,4\}}\theta ^{\{3,5\}}\theta ^{\{4,5\}}} \partial _{v} \theta ^{\{2\}} \\&= \frac{\theta ^{\{1,3\}}\theta ^{\{3,4\}}\theta ^{\{3,5\}}}{\theta ^{\{1,4\}}\theta ^{\{1,5\}}\theta ^{\{4,5\}}} \partial _{v} \theta ^{\{2\}} - \frac{\theta ^{\{1,2\}}\theta ^{\{2,4\}}\theta ^{\{2,5\}}}{\theta ^{\{1,4\}}\theta ^{\{1,5\}}\theta ^{\{4,5\}}} \partial _{v} \theta ^{\{3\}} \\&= \frac{\theta ^{\{1,4\}}\theta ^{\{2,4\}}\theta ^{\{4,5\}}}{\theta ^{\{1,2\}}\theta ^{\{1,5\}}\theta ^{\{2,5\}}} \partial _{v} \theta ^{\{3\}} - \frac{\theta ^{\{1,3\}}\theta ^{\{2,3\}}\theta ^{\{3,5\}}}{\theta ^{\{1,2\}}\theta ^{\{1,5\}}\theta ^{\{2,5\}}} \partial _{v} \theta ^{\{4\}} \\&= \frac{\theta ^{\{1,5\}}\theta ^{\{2,5\}}\theta ^{\{3,5\}}}{\theta ^{\{1,2\}}\theta ^{\{1,3\}}\theta ^{\{2,3\}}} \partial _{v} \theta ^{\{4\}} - \frac{\theta ^{\{1,4\}}\theta ^{\{2,4\}}\theta ^{\{3,4\}}}{\theta ^{\{1,2\}}\theta ^{\{1,3\}}\theta ^{\{2,3\}}} \partial _{v} \theta ^{\{5\}}, \end{aligned}$$
and the same with the standard representation of characteristics
$$\begin{aligned} \partial _{v} \theta [{}^{11}_{01}]&= \big (\theta [{}^{11}_{00}]\theta [{}^{10}_{00}]\theta [{}^{10}_{01}]\big )^{-1} \Big ( \theta [{}^{00}_{01}]\theta [{}^{00}_{00}]\theta [{}^{01}_{00}]\partial _{v} \theta [{}^{01}_{01}] - \theta [{}^{00}_{11}]\theta [{}^{00}_{10}]\theta [{}^{01}_{10}]\partial _{v} \theta [{}^{01}_{11}] \Big ) \\&= \big (\theta [{}^{00}_{10}]\theta [{}^{01}_{10}]\theta [{}^{10}_{01}]\big )^{-1} \Big ( \theta [{}^{00}_{11}]\theta [{}^{11}_{00}]\theta [{}^{10}_{00}]\partial _{v} \theta [{}^{01}_{11}] - \theta [{}^{11}_{11}]\theta [{}^{00}_{00}]\theta [{}^{01}_{00}]\partial _{v} \theta [{}^{10}_{11}] \Big ) \\&= \big (\theta [{}^{11}_{11}]\theta [{}^{01}_{10}]\theta [{}^{01}_{00}]\big )^{-1} \Big ( \theta [{}^{00}_{10}]\theta [{}^{00}_{00}]\theta [{}^{10}_{01}]\partial _{v} \theta [{}^{10}_{11}] - \theta [{}^{00}_{11}]\theta [{}^{00}_{01}]\theta [{}^{10}_{00}]\partial _{v} \theta [{}^{10}_{10}] \Big ) \\&= \big (\theta [{}^{11}_{11}]\theta [{}^{00}_{11}]\theta [{}^{00}_{01}]\big )^{-1} \Big ( \theta [{}^{01}_{10}]\theta [{}^{01}_{00}]\theta [{}^{10}_{00}]\partial _{v} \theta [{}^{10}_{10}] - \theta [{}^{00}_{10}]\theta [{}^{00}_{00}]\theta [{}^{11}_{00}]\partial _{v} \theta [{}^{11}_{10}] \Big ). \end{aligned}$$
Note that the matrices of characteristics are constructed from the homology basis given in Sect.2.3. Any two vectors of first-order theta derivatives could serve as a basis, and all other vectors are expressed in terms of these two.
1.2 A2: Genus 3
A list of relations between \(\partial _{v} \theta ^{\{1\}}\) and other vectors of first-order theta derivatives in genus 3 is presented below
$$\begin{aligned} \partial _{v} \theta ^{\{1\}}&= \frac{\theta ^{\{1,3,4\}}\theta ^{\{1,3,5\}}\theta ^{\{3,6,7\}}}{\theta ^{\{1,4,5\}}\theta ^{\{5,6,7\}}\theta ^{\{4,6,7\}}} \partial _{v} \theta ^{\{1,2\}} - \frac{\theta ^{\{1,2,4\}}\theta ^{\{1,2,5\}}\theta ^{\{2,6,7\}}}{\theta ^{\{1,4,5\}}\theta ^{\{5,6,7\}}\theta ^{\{4,6,7\}}} \partial _{v} \theta ^{\{1,3\}} \\&= \frac{\theta ^{\{1,4,2\}}\theta ^{\{1,4,5\}}\theta ^{\{4,6,7\}}}{\theta ^{\{1,2,5\}}\theta ^{\{2,6,7\}}\theta ^{\{5,6,7\}}} \partial _{v} \theta ^{\{1,3\}} - \frac{\theta ^{\{1,3,2\}}\theta ^{\{1,3,5\}}\theta ^{\{3,6,7\}}}{\theta ^{\{1,2,5\}}\theta ^{\{2,6,7\}}\theta ^{\{5,6,7\}}} \partial _{v} \theta ^{\{1,4\}} \\&= \frac{\theta ^{\{1,5,2\}}\theta ^{\{1,5,3\}}\theta ^{\{5,6,7\}}}{\theta ^{\{1,2,3\}}\theta ^{\{2,6,7\}}\theta ^{\{3,6,7\}}} \partial _{v} \theta ^{\{1,4\}} - \frac{\theta ^{\{1,4,2\}}\theta ^{\{1,4,3\}}\theta ^{\{4,6,7\}}}{\theta ^{\{1,2,3\}}\theta ^{\{2,6,7\}}\theta ^{\{3,6,7\}}} \partial _{v} \theta ^{\{1,5\}} \\&= \frac{\theta ^{\{1,6,2\}}\theta ^{\{1,6,3\}}\theta ^{\{6,4,7\}}}{\theta ^{\{1,2,3\}}\theta ^{\{2,4,7\}}\theta ^{\{3,4,7\}}} \partial _{v} \theta ^{\{1,5\}} - \frac{\theta ^{\{1,5,2\}}\theta ^{\{1,5,3\}}\theta ^{\{5,4,7\}}}{\theta ^{\{1,2,3\}}\theta ^{\{2,4,7\}}\theta ^{\{3,4,7\}}} \partial _{v} \theta ^{\{1,6\}} \\&= \frac{\theta ^{\{1,7,2\}}\theta ^{\{1,7,3\}}\theta ^{\{7,4,5\}}}{\theta ^{\{1,2,3\}}\theta ^{\{2,4,5\}}\theta ^{\{3,4,5\}}} \partial _{v} \theta ^{\{1,6\}} - \frac{\theta ^{\{1,6,2\}}\theta ^{\{1,6,3\}}\theta ^{\{6,4,5\}}}{\theta ^{\{1,2,3\}}\theta ^{\{2,4,5\}}\theta ^{\{3,4,5\}}} \partial _{v} \theta ^{\{1,7\}}, \end{aligned}$$
and the same with the standard representation of characteristics
$$\begin{aligned}&\partial _{v} \theta [{}^{011}_{101}] \\&\quad = \big (\theta [{}^{000}_{101}]\theta [{}^{111}_{011}]\theta [{}^{100}_{011}]\big )^{-1} \Big (\theta [{}^{011}_{111}]\theta [{}^{000}_{111}]\theta [{}^{100}_{001}] \partial _{v} \theta [{}^{111}_{001}] - \theta [{}^{101}_{111}]\theta [{}^{110}_{111}]\theta [{}^{010}_{001}] \partial _{v} \theta [{}^{001}_{001}] \Big )\\&\quad = \big (\theta [{}^{000}_{110}]\theta [{}^{111}_{000}]\theta [{}^{100}_{011}]\big )^{-1} \Big (\theta [{}^{101}_{111}]\theta [{}^{000}_{101}]\theta [{}^{100}_{011}] \partial _{v} \theta [{}^{001}_{001}] - \theta [{}^{101}_{101}]\theta [{}^{110}_{110}]\theta [{}^{001}_{000}] \partial _{v} \theta [{}^{001}_{011}] \Big )\\&\quad = \big (\theta [{}^{101}_{101}]\theta [{}^{010}_{001}]\theta [{}^{100}_{001}]\big )^{-1} \Big (\theta [{}^{110}_{111}]\theta [{}^{000}_{111}]\theta [{}^{111}_{011}] \partial _{v} \theta [{}^{001}_{011}] - \theta [{}^{101}_{111}]\theta [{}^{011}_{111}]\theta [{}^{100}_{011}] \partial _{v} \theta [{}^{010}_{011}] \Big )\\&\quad = \big (\theta [{}^{101}_{101}]\theta [{}^{001}_{000}]\theta [{}^{111}_{000}]\big )^{-1} \Big (\theta [{}^{110}_{110}]\theta [{}^{000}_{110}]\theta [{}^{100}_{011}] \partial _{v} \theta [{}^{010}_{011}] - \theta [{}^{110}_{111}]\theta [{}^{000}_{111}]\theta [{}^{100}_{010}] \partial _{v} \theta [{}^{010}_{010}] \Big )\\&\quad = \big (\theta [{}^{110}_{111}]\theta [{}^{010}_{001}]\theta [{}^{111}_{011}]\big )^{-1} \Big (\theta [{}^{111}_{110}]\theta [{}^{001}_{110}]\theta [{}^{100}_{010}] \partial _{v} \theta [{}^{010}_{010}] - \theta [{}^{110}_{110}]\theta [{}^{000}_{110}]\theta [{}^{101}_{010}] \partial _{v} \theta [{}^{011}_{010}] \Big ). \end{aligned}$$
Appendix B: Proof of Proposition 2
Proof
Applying (20) to decompositions of \(\partial _v \theta [{\mathcal {I}}]\) into three pairs: 1) \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}]\) and \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}]\), 2) \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}]\) and \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _3\}]\), 3) \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}]\) and \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _3\}]\), one obtains three equalities:
$$\begin{aligned} \partial _v \theta [{\mathcal {I}}]= & {} \big (\theta [{\mathcal {I}}\cup \{j_m,j_n\}]\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}]\big )^{-1} \\&\times \Big ( \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _3)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}] \\&- \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _3)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}] \Big ) \\= & {} \big (\theta [{\mathcal {I}}\cup \{j_m,j_n\}]\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}]\big )^{-1} \\&\times \Big ( \theta [{\mathcal {I}}\cup \{\kappa _3,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _3,j_n\}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _3)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}] \\&- \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _3\}] \Big ) \\= & {} \big (\theta [{\mathcal {I}}\cup \{j_m,j_n\}]\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}]\big )^{-1} \\&\times \Big ( \theta [{\mathcal {I}}\cup \{\kappa _3,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _3,j_n\}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _3)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}] \\&- \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _3\}] \Big ). \end{aligned}$$
They produce two relations between vectors \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}]\), \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}]\), and \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _3\}]\), and these two relations are equivalent, so one gets (25) from Proposition 2. Indeed, extract the two relations and solve them for \(\partial _{v} \theta [{\mathcal {I}}_1\cup \{\kappa _3\}]\)
$$\begin{aligned} \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _3\}]= & {} \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _3)}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2)}]} \nonumber \\&\bigg (\frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}] \theta [{\mathcal {I}}\cup \{\kappa _3,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _3,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]} \nonumber \\&- \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}] \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]} \bigg )\nonumber \\&\times \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}] + \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _3)}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2)}]} \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}] \nonumber \\= & {} - \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _3)}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2)}]} \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}] \nonumber \\&+ \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _3)}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2)}]} \nonumber \\&\bigg (\frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}] \theta [{\mathcal {I}}\cup \{\kappa _3,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _3,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}] \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]} \nonumber \\&+ \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}] \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]} \bigg )\nonumber \\&\times \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}]. \end{aligned}$$
(66)
This implies a relation between \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}]\) and \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}]\)
$$\begin{aligned}&\frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _3)}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2)}]} \\&\qquad \bigg (\frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}] \theta [{\mathcal {I}}\cup \{\kappa _3,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _3,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]} \\&\qquad - \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}] \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]} + 1\bigg ) \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}] \\&\quad = \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _3)}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2)}]} \\&\qquad \bigg (\frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}] \theta [{\mathcal {I}}\cup \{\kappa _3,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _3,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}] \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]} \\&\qquad + \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}] \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]} - 1\bigg ) \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}]. \end{aligned}$$
And this relation is an identity, since the expressions in parentheses vanish identically due to FTT Corollary 1, namely
$$\begin{aligned}&\frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]^2\theta [{\mathcal {I}}\cup \{\kappa _3,j_n\}]^2}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]^2\theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]^2} = \frac{\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}]^2\theta [{\mathcal {I}}\cup \{\kappa _3,j_m\}]^2}{\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}]^2\theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}]^2} = \frac{e_{\kappa _2} - e_{\kappa _1}}{e_{\kappa _3} - e_{\kappa _2}},\\&\frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]^2\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]^2}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]^2\theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]^2} = \frac{\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}]^2\theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]^2}{\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}]^2\theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}]^2} = \frac{e_{\kappa _3} - e_{\kappa _1}}{e_{\kappa _3} - e_{\kappa _2}},\\&\frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]^2\theta [{\mathcal {I}}\cup \{\kappa _3,j_n\}]^2}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]^2\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]^2} = \frac{\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}]^2\theta [{\mathcal {I}}\cup \{\kappa _3,j_m\}]^2}{\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}]^2\theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]^2} = \frac{e_{\kappa _2} - e_{\kappa _1}}{e_{\kappa _3} - e_{\kappa _1}}. \end{aligned}$$
Thus,
$$\begin{aligned}&\frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}] \theta [{\mathcal {I}}\cup \{\kappa _3,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _3,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]} \\&\quad - \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}] \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]} + 1 \\&\qquad = \frac{e_{\kappa _2} - e_{\kappa _1}}{e_{\kappa _3} - e_{\kappa _2}} - \frac{e_{\kappa _3} - e_{\kappa _1}}{e_{\kappa _3} - e_{\kappa _2}} + 1 = 0, \\&\frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}] \theta [{\mathcal {I}}\cup \{\kappa _3,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _3,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}] \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]} \\&\quad + \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}] \theta [{\mathcal {I}}\cup \{\kappa _1,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _1,j_n\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}] \theta [{\mathcal {I}}\cup \{\kappa _2,j_m\}]\theta [{\mathcal {I}}\cup \{\kappa _2,j_n\}]} - 1 \\&\qquad = \frac{e_{\kappa _2} - e_{\kappa _1}}{e_{\kappa _3} - e_{\kappa _1}} + \frac{e_{\kappa _3} - e_{\kappa _2}}{e_{\kappa _3} - e_{\kappa _1}} - 1 = 0. \end{aligned}$$
Therefore, (66) contains one relation between vectors \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}]\), \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}]\), and \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _3\}]\), and every two of the vectors are linearly independent. The relation simplifies to the form
$$\begin{aligned}&\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _3)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1\}] \\&\qquad - \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _3)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2\}] \\&\qquad + \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _3\}] = 0. \end{aligned}$$
\(\square \)
As an example, in genus 2, one finds
$$\begin{aligned}&\theta ^{\{1,4\}}\theta ^{\{1,5\}}\theta ^{\{2,3\}} \partial _{v} \theta ^{\{1\}} - \theta ^{\{2,4\}}\theta ^{\{2,5\}}\theta ^{\{1,3\}} \partial _{v} \theta ^{\{2\}} \\&\qquad + \theta ^{\{3,4\}}\theta ^{\{3,5\}}\theta ^{\{1,2\}} \partial _{v} \theta ^{\{3\}} = 0, \end{aligned}$$
and in genus 3
$$\begin{aligned}&\theta ^{\{2,6,7\}}\theta ^{\{5,2,7\}}\theta ^{\{3,4,7\}} \partial _{v} \theta ^{\{1,2\}} - \theta ^{\{3,6,7\}}\theta ^{\{5,3,7\}}\theta ^{\{2,4,7\}} \partial _{v} \theta ^{\{1,3\}} \\&\qquad + \theta ^{\{4,6,7\}}\theta ^{\{5,4,7\}}\theta ^{\{2,3,7\}} \partial _{v} \theta ^{\{1,4\}} = 0. \end{aligned}$$
Recall that \(\theta ^{\{i,j\}}\) stands for \(\theta [\{i,j\}](0;\tau )\), as well as \(\partial _{v} \theta ^{\{i\}}\) stands for \(\mathrm {grad}_v \theta [\{i\}](v;\tau ) |_{v=0}\).
Appendix C: Proof of Proposition 3
Proof
The set \({\mathcal {I}}=\{i_1,\,\dots \), \(i_{g-3}\}\) corresponds to a characteristic of multiplicity 2, then joining two indices from \(\{\kappa _1,\,\kappa _2,\,\kappa _3,\,\kappa _4,\,\kappa _5\}\) results into a partition \({\mathcal {I}}\cup \{\kappa _i,\kappa _j\}\) corresponding to a characteristic of multiplicity 1. Write down equality (27) with the following gradients:
-
(i)
\(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _2\}]\), \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _3\}]\) and \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _5\}]\) with the common set \({\mathcal {I}}\cup \{\kappa _1\}\) denoted by \({\mathcal {I}}\) in (27),
-
(ii)
\(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _2\}]\), \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2,\kappa _3\}]\) and \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2,\kappa _5\}]\) with the common set \({\mathcal {I}}\cup \{\kappa _2\}\),
-
(iii)
\(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _5\}]\), \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2,\kappa _5\}]\) and \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _4,\kappa _5\}]\) with the common set \({\mathcal {I}}\cup \{\kappa _5\}\),
and obtain three relations, namely:
$$\begin{aligned}&\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _3,\kappa _4,\kappa _5)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _2\}] \\&\qquad - \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _4,\kappa _5)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _3\}] \\&\qquad + \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _4,\kappa _5)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _4,\kappa _5)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _3,\kappa _4)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _5\}] = 0, \\&\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _3,\kappa _4,\kappa _5)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _2\}] \\&\qquad - \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _4,\kappa _5)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2,\kappa _3\}] \\&\qquad + \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _4,\kappa _5)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _4,\kappa _5)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _3,\kappa _4)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2,\kappa _5\}] = 0,\\&\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _3,\kappa _4)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _5\}] \\&\qquad - \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _3,\kappa _4)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2,\kappa _5\}] \\&\qquad + \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2,\kappa _3)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _4,\kappa _5\}] = 0. \end{aligned}$$
Next, eliminate vectors \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _5\}]\) and \(\partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2,\kappa _5\}]\) from the relations, and come to
$$\begin{aligned}&\Big ( - \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _3)}] \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2,\kappa _4)}] \nonumber \\&\qquad + \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2,\kappa _3)}] \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _4)}] \Big ) \nonumber \\&\qquad \times \frac{\theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _3,\kappa _4,\kappa _5)}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3,\kappa _4)}]} \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _2\}] \nonumber \\&\qquad + \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _3)}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _4,\kappa _5)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _3\}] \nonumber \\&\qquad - \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2,\kappa _3)}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _4,\kappa _5)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2,\kappa _3\}] \nonumber \\&\qquad + \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _4,\kappa _5)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _4,\kappa _5)}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2,\kappa _3)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _4,\kappa _5\}] = 0.\nonumber \\ \end{aligned}$$
(67)
With the help of FTT Corollary 1 the following holds:
$$\begin{aligned}&- \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _3)}] \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2,\kappa _4)}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3,\kappa _4)}] \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _2)}]} \\&\qquad + \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2,\kappa _3)}] \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _4)}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3,\kappa _4)}] \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _2)}]} + 1 \\&\quad = - \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _3)}] \theta [{\mathcal {I}}\cup \{j_n,\kappa _1,\kappa _5\}] \theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _3)}] \theta [{\mathcal {I}}\cup \{j_m,\kappa _1,\kappa _5\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3,\kappa _4)}] \theta [{\mathcal {I}}\cup \{j_n,\kappa _4,\kappa _5\}] \theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3,\kappa _4)}] \theta [{\mathcal {I}}\cup \{j_m,\kappa _4,\kappa _5\}]} \\&\qquad \times \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2,\kappa _4)}]\theta [{\mathcal {I}}\cup \{j_n,\kappa _4,\kappa _5\}] \theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2,\kappa _4)}]\theta [{\mathcal {I}}\cup \{j_m,\kappa _4,\kappa _5\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _2)}]\theta [{\mathcal {I}}\cup \{j_n,\kappa _1,\kappa _5\}] \theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _2)}]\theta [{\mathcal {I}}\cup \{j_m,\kappa _1,\kappa _5\}]} \\&\qquad + \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2,\kappa _3)}]\theta [{\mathcal {I}}\cup \{j_n,\kappa _2,\kappa _5\}] \theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2,\kappa _3)}]\theta [{\mathcal {I}}\cup \{j_n,\kappa _2,\kappa _5\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {I}}\cup \{j_n,\kappa _4,\kappa _5\}] \theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _3,\kappa _4)}]\theta [{\mathcal {I}}\cup \{j_n,\kappa _4,\kappa _5\}]} \\&\qquad \times \frac{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _4)}]\theta [{\mathcal {I}}\cup \{j_n,\kappa _4,\kappa _5\}] \theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _4)}]\theta [{\mathcal {I}}\cup \{j_n,\kappa _4,\kappa _5\}]}{\theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _2)}]\theta [{\mathcal {I}}\cup \{j_n,\kappa _2,\kappa _5\}] \theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _2)}]\theta [{\mathcal {I}}\cup \{j_n,\kappa _2,\kappa _5\}]} + 1 \\&\qquad \quad = - \frac{(e_{\kappa _4}-e_{\kappa _2})(e_{\kappa _3}-e_{\kappa _1})}{(e_{\kappa _2}-e_{\kappa _1})(e_{\kappa _4}-e_{\kappa _3})} + \frac{(e_{\kappa _4}-e_{\kappa _1})(e_{\kappa _3}-e_{\kappa _2})}{(e_{\kappa _2}-e_{\kappa _1})(e_{\kappa _4}-e_{\kappa _3})} + 1 = 0, \end{aligned}$$
and (67) simplifies to
$$\begin{aligned}&- \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _2)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _2)}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _3,\kappa _4,\kappa _5)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _2\}] \\&\qquad + \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _1,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _1,\kappa _3)}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _2,\kappa _4,\kappa _5)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _3\}] \\&\qquad - \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _2,\kappa _3)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _2,\kappa _3)}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _4,\kappa _5)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _2,\kappa _3\}] \\&\qquad + \theta [{\mathcal {J}}^{(j_n\rightarrow \kappa _4,\kappa _5)}]\theta [{\mathcal {J}}^{(j_m\rightarrow \kappa _4,\kappa _5)}] \theta [{\mathcal {J}}^{(j_m,j_n \rightarrow \kappa _1,\kappa _2,\kappa _3)}] \partial _{v} \theta [{\mathcal {I}}\cup \{\kappa _4,\kappa _5\}] = 0, \end{aligned}$$
which coincides with (28).
Linear independence of any three vectors in (28), say \(\partial _v \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _2\}]\), \(\partial _v \theta [{\mathcal {I}}\cup \{\kappa _1,\kappa _3\}]\) and \(\partial _v \theta [{\mathcal {I}}\cup \{\kappa _2,\kappa _3\}]\), follows from the fact that there is no relation between these theta derivatives due to Proposition 2’, according to which a relation between three theta derivatives exists only if characteristics of these theta derivatives are such that the intersection of the corresponding sets of indices has cardinality \(g-2\). Instead, intersection \({\mathcal {I}}\) of the sets corresponding to characteristics \([{\mathcal {I}}\cup \{\kappa _1,\kappa _2\}]\), \([{\mathcal {I}}\cup \{\kappa _1,\kappa _3\}]\), \([{\mathcal {I}}\cup \{\kappa _2,\kappa _3\}]\) has cardinality \(n-3\). \(\square \)
Appendix D: Second derivative theta relations
1.1 D1: Genus 3
Formula (45) gives 35 representations (excluding different representations of ratios of theta constants following from FTT Corollary 1) for each entry of \(\partial ^2_v \theta ^{\emptyset }\), all the relations are equivalent. In particular, with \({\mathcal {I}}_0=\{1,2,3\}\), \(j_1=6\), \(j_2=5\)
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta ^{\emptyset }= & {} \big (\theta ^{\{1,2,3\}}\theta ^{\{4,5,7\}}\theta ^{\{4,6,7\}}\big )^{-1} \\&\times \Big ({-} \theta ^{\{3,5,6\}}\theta ^{\{3,4,7\}}\theta ^{\{1,2,5\}}\theta ^{\{1,2,6\}} \big (\theta ^{\{1,5,6\}}\theta ^{\{2,5,6\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{2,3\}} \partial _{v_{n_2}} \theta ^{\{1,3\}} + \partial _{v_{n_2}} \theta ^{\{2,3\}} \partial _{v_{n_1}} \theta ^{\{1,3\}}\big )\\&+ \theta ^{\{2,5,6\}}\theta ^{\{2,4,7\}}\theta ^{\{1,3,5\}}\theta ^{\{1,3,6\}} \big (\theta ^{\{1,5,6\}}\theta ^{\{3,5,6\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{2,3\}} \partial _{v_{n_2}} \theta ^{\{1,2\}} + \partial _{v_{n_2}} \theta ^{\{2,3\}} \partial _{v_{n_1}} \theta ^{\{1,2\}}\big )\\&- \theta ^{\{1,5,6\}}\theta ^{\{1,4,7\}}\theta ^{\{2,3,5\}}\theta ^{\{2,3,6\}} \big (\theta ^{\{2,5,6\}}\theta ^{\{3,5,6\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{1,3\}} \partial _{v_{n_2}} \theta ^{\{1,2\}} + \partial _{v_{n_2}} \theta ^{\{1,3\}} \partial _{v_{n_1}} \theta ^{\{1,2\}}\big ) \end{aligned}$$
or with characteristics in the standard form
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta [{}^{111}_{101}]= & {} \big (\theta [{}^{101}_{101}] \theta [{}^{100}_{010}]\theta [{}^{100}_{011}]\big )^{-1} \\&\times \Big ({-}\theta [{}^{101}_{000}]\theta [{}^{111}_{000}]\theta [{}^{110}_{111}]\theta [{}^{110}_{110}] \big (\theta [{}^{011}_{100}]\theta [{}^{011}_{000}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{001}_{101}] \partial _{v_{n_2}} \theta [{}^{001}_{001}] + \partial _{v_{n_2}} \theta [{}^{001}_{101}] \partial _{v_{n_1}} \theta [{}^{001}_{001}]\big )\\&+ \theta [{}^{011}_{000}]\theta [{}^{001}_{000}]\theta [{}^{000}_{111}]\theta [{}^{000}_{110}] \big (\theta [{}^{011}_{100}]\theta [{}^{101}_{000}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{001}_{101}] \partial _{v_{n_2}} \theta [{}^{111}_{001}] + \partial _{v_{n_2}} \theta [{}^{001}_{101}] \partial _{v_{n_1}} \theta [{}^{111}_{001}]\big )\\&- \theta [{}^{011}_{100}]\theta [{}^{001}_{100}]\theta [{}^{000}_{011}]\theta [{}^{000}_{010}] \big (\theta [{}^{011}_{000}]\theta [{}^{101}_{000}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{001}_{001}] \partial _{v_{n_2}} \theta [{}^{111}_{001}] + \partial _{v_{n_2}} \theta [{}^{001}_{001}] \partial _{v_{n_1}} \theta [{}^{111}_{001}]\big ); \end{aligned}$$
and with \({\mathcal {I}}_0=\{1,2,4\}\), \(j_1=6\), \(j_2=5\)
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta ^{\emptyset }= & {} \big (\theta ^{\{1,2,4\}}\theta ^{\{3,5,7\}}\theta ^{\{3,6,7\}}\big )^{-1} \\&\times \Big ({-} \theta ^{\{4,5,6\}}\theta ^{\{3,4,7\}}\theta ^{\{1,2,5\}}\theta ^{\{1,2,6\}} \big (\theta ^{\{1,5,6\}}\theta ^{\{2,5,6\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{2,4\}} \partial _{v_{n_2}} \theta ^{\{1,4\}} + \partial _{v_{n_2}} \theta ^{\{2,4\}} \partial _{v_{n_1}} \theta ^{\{1,4\}}\big )\\&+ \theta ^{\{2,5,6\}}\theta ^{\{2,3,7\}}\theta ^{\{1,4,5\}}\theta ^{\{1,4,6\}} \big (\theta ^{\{1,5,6\}}\theta ^{\{4,5,6\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{2,4\}} \partial _{v_{n_2}} \theta ^{\{1,2\}} + \partial _{v_{n_2}} \theta ^{\{2,4\}} \partial _{v_{n_1}} \theta ^{\{1,2\}}\big )\\&- \theta ^{\{1,5,6\}}\theta ^{\{1,3,7\}}\theta ^{\{2,4,5\}}\theta ^{\{2,4,6\}} \big (\theta ^{\{2,5,6\}}\theta ^{\{4,5,6\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{1,4\}} \partial _{v_{n_2}} \theta ^{\{1,2\}} + \partial _{v_{n_2}} \theta ^{\{1,4\}} \partial _{v_{n_1}} \theta ^{\{1,2\}}\big ) \end{aligned}$$
or with characteristics in the standard form
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta [{}^{111}_{101}]= & {} \big (\theta [{}^{101}_{111}]\theta [{}^{100}_{000}] \theta [{}^{100}_{001}]\big )^{-1} \\&\times \Big ({-} \theta [{}^{101}_{010}]\theta [{}^{111}_{000}]\theta [{}^{110}_{111}]\theta [{}^{110}_{110}] \big (\theta [{}^{011}_{100}]\theta [{}^{011}_{000}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{001}_{111}] \partial _{v_{n_2}} \theta [{}^{001}_{011}] + \partial _{v_{n_2}} \theta [{}^{001}_{111}] \partial _{v_{n_1}} \theta [{}^{001}_{011}]\big )\\&+ \theta [{}^{011}_{000}]\theta [{}^{001}_{010}]\theta [{}^{000}_{101}]\theta [{}^{000}_{100}] \big (\theta [{}^{011}_{100}]\theta [{}^{101}_{010}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{001}_{111}] \partial _{v_{n_2}} \theta [{}^{111}_{001}] + \partial _{v_{n_2}} \theta [{}^{001}_{111}] \partial _{v_{n_1}} \theta [{}^{111}_{001}]\big )\\&- \theta [{}^{011}_{100}]\theta [{}^{001}_{110}]\theta [{}^{000}_{001}]\theta [{}^{000}_{000}] \big (\theta [{}^{011}_{000}]\theta [{}^{101}_{010}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{001}_{011}] \partial _{v_{n_2}} \theta [{}^{111}_{001}] + \partial _{v_{n_2}} \theta [{}^{001}_{011}] \partial _{v_{n_1}} \theta [{}^{111}_{001}]\big ). \end{aligned}$$
1.2 D2: Genus 4
There exist 9 characteristics of multiplicity 2 of the form \([\{\iota \}]\), and the corresponding second-order theta derivatives are given by the formula (46). Let \({\mathcal {I}}_0=\{1,2,3,4\}\) and \(j_1=5\), \(j_2=6\), then one obtains with \(\iota =1\)
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta ^{\{1\}}= & {} \frac{1}{\theta ^{\{1,2,3,4\}}\theta ^{\{6,7,8,9\}}\theta ^{\{5,7,8,9\}}} \\&\times \Big (- \theta ^{\{1,4,5,6\}}\theta ^{\{4,7,8,9\}}\theta ^{\{1,2,3,5\}}\theta ^{\{1,2,3,6\}} \big (\theta ^{\{1,2,5,6\}}\theta ^{\{1,3,5,6\}}\Big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{1,3,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,4\}} + \partial _{v_{n_2}} \theta ^{\{1,3,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,4\}}\big )\\&+ \theta ^{\{1,3,5,6\}}\theta ^{\{3,7,8,9\}}\theta ^{\{1,2,4,5\}}\theta ^{\{1,2,4,6\}} \big (\theta ^{\{1,2,5,6\}}\theta ^{\{1,4,5,6\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{1,3,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,3\}} + \partial _{v_{n_2}} \theta ^{\{1,3,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,3\}}\big )\\&- \theta ^{\{1,2,5,6\}}\theta ^{\{2,7,8,9\}}\theta ^{\{1,3,4,5\}}\theta ^{\{1,3,4,6\}} \big (\theta ^{\{1,3,5,6\}}\theta ^{\{1,4,5,6\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{1,2,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,3\}} + \partial _{v_{n_2}} \theta ^{\{1,2,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,3\}}\big ) \end{aligned}$$
or in the standard form
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta [{}^{0111}_{0101}]= & {} \big (\theta [{}^{1111}_{1001}]\theta [{}^{1101}_{0101}] \theta [{}^{1101}_{0111}]\big )^{-1} \\&\times \Big ({-} \theta [{}^{0011}_{1011}]\theta [{}^{1011}_{0111}]\theta [{}^{1001}_{1001}]\theta [{}^{1001}_{1011}] \big (\theta [{}^{1111}_{1111}]\theta [{}^{0011}_{1111}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{0111}_{0001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0001}] + \partial _{v_{n_2}} \theta [{}^{0111}_{0001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0001}]\big )\\&+ \theta [{}^{0011}_{1111}]\theta [{}^{1011}_{0011}]\theta [{}^{1001}_{1101}]\theta [{}^{1001}_{1111}] \big (\theta [{}^{1111}_{1111}]\theta [{}^{0011}_{1011}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{0111}_{0001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0101}] + \partial _{v_{n_2}} \theta [{}^{0111}_{0001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0101}]\big )\\&- \theta [{}^{1111}_{1111}]\theta [{}^{0111}_{0011}]\theta [{}^{0101}_{1101}]\theta [{}^{0101}_{1111}] \big (\theta [{}^{0011}_{1111}]\theta [{}^{0011}_{1011}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{1011}_{0001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0101}] + \partial _{v_{n_2}} \theta [{}^{1011}_{0001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0101}]\big ); \end{aligned}$$
and with \(\iota =2\)
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta ^{\{2\}}= & {} \frac{1}{\theta ^{\{1,2,3,4\}}\theta ^{\{6,7,8,9\}}\theta ^{\{5,7,8,9\}}} \\&\times \Big (- \theta ^{\{2,4,5,6\}}\theta ^{\{4,7,8,9\}}\theta ^{\{1,2,3,5\}}\theta ^{\{1,2,3,6\}} \big (\theta ^{\{1,2,5,6\}}\theta ^{\{2,3,5,6\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{2,3,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,4\}} + \partial _{v_{n_2}} \theta ^{\{2,3,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,4\}}\big )\\&+ \theta ^{\{2,3,5,6\}}\theta ^{\{3,7,8,9\}}\theta ^{\{1,2,4,5\}}\theta ^{\{1,2,4,6\}} \big (\theta ^{\{1,2,5,6\}}\theta ^{\{2,4,5,6,\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{2,3,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,3\}} + \partial _{v_{n_2}} \theta ^{\{2,3,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,3\}}\big )\\&- \theta ^{\{1,2,5,6\}}\theta ^{\{1,7,8,9\}}\theta ^{\{2,3,4,5\}}\theta ^{\{2,3,4,6\}} \big (\theta ^{\{2,3,5,6\}}\theta ^{\{2,4,5,6\}}\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{1,2,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,3\}} + \partial _{v_{n_2}} \theta ^{\{1,2,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,3\}}\big ) \end{aligned}$$
or in the standard form
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta [{}^{0111}_{1101}]= & {} \big (\theta [{}^{1111}_{1001}]\theta [{}^{1101}_{0101}] \theta [{}^{1101}_{0111}]\big )^{-1} \\&\times \Big ({-} \theta [{}^{0011}_{1011}]\theta [{}^{1011}_{0111}]\theta [{}^{1001}_{1001}]\theta [{}^{1001}_{1011}] \big (\theta [{}^{1111}_{1111}]\theta [{}^{0011}_{1111}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{0111}_{1001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0001}] + \partial _{v_{n_2}} \theta [{}^{0111}_{1001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0001}]\big )\\&+ \theta [{}^{0011}_{1111}]\theta [{}^{1011}_{0011}]\theta [{}^{1001}_{1101}]\theta [{}^{1001}_{1111}] \big (\theta [{}^{1111}_{1111}]\theta [{}^{0011}_{1011}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{0111}_{1001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0101}] + \partial _{v_{n_2}} \theta [{}^{0111}_{1001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0101}]\big )\\&- \theta [{}^{1111}_{1111}]\theta [{}^{0111}_{0011}]\theta [{}^{0101}_{1101}]\theta [{}^{0101}_{1111}] \big (\theta [{}^{0011}_{1111}]\theta [{}^{0011}_{1011}]\big )^{-1} \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{1011}_{0001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0101}] + \partial _{v_{n_2}} \theta [{}^{1011}_{0001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0101}]\big ). \end{aligned}$$
Let \({\mathcal {I}}_0=\{i_1\), \(i_2\), \(i_3\), \(i_4\}\), \({\mathcal {J}}_0=\{j_1\), \(j_2\), \(j_3\), \(j_4\), \(j_5\}\) and \(j_m=j_1\), \(j_n=j_2\). Second-order theta derivatives with characteristic \(\emptyset \) are given by the following formula
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta ^{\emptyset }= & {} \big (\theta ^{\{i_1,i_2,i_3,i_4\}} (\theta ^{\{j_2,j_3,j_4,j_5\}} \theta ^{\{j_1,j_3,j_4,j_5\}})^2\big )^{-1} \\&\times \bigg ({-} \frac{\theta ^{\{i_3,i_4,j_1,j_2\}} \theta ^{\{i_1,i_2,j_1,j_2\}}\theta ^{\{i_1,j_3,j_4,j_5\}}\theta ^{\{i_2,j_3,j_4,j_5\}}}{\theta ^{\{i_2,i_4,j_1,j_2\}} \theta ^{\{i_1,i_4,j_1,j_2\}}\theta ^{\{i_2,i_3,j_1,j_2\}}\theta ^{\{i_1,i_3,j_1,j_2\}}} \\&\times \theta ^{\{i_1,i_2,i_4,j_1\}} \theta ^{\{i_1,i_2,i_4,j_2\}} \theta ^{\{i_1,i_2,i_3,j_1\}} \theta ^{\{i_1,i_2,i_3,j_2\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{i_2,i_3,i_4\}} \partial _{v_{n_2}} \theta ^{\{i_1,i_3,i_4\}} + \partial _{v_{n_2}} \theta ^{\{i_2,i_3,i_4\}} \partial _{v_{n_1}} \theta ^{\{i_1,i_3,i_4\}}\big ) \\&+ \frac{\theta ^{\{i_2,i_4,j_1,j_2\}} \theta ^{\{i_1,i_3,j_1,j_2\}}\theta ^{\{i_1,j_3,j_4,j_5\}}\theta ^{\{i_3,j_3,j_4,j_5\}}}{\theta ^{\{i_3,i_4,j_1,j_2\}} \theta ^{\{i_1,i_4,j_1,j_2\}}\theta ^{\{i_2,i_3,j_1,j_2\}}\theta ^{\{i_1,i_2,j_1,j_2\}}} \\&\times \theta ^{\{i_1,i_3,i_4,j_1\}} \theta ^{\{i_1,i_3,i_4,j_2\}} \theta ^{\{i_1,i_2,i_3,j_1\}} \theta ^{\{i_1,i_2,i_3,j_2\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{i_2,i_3,i_4\}} \partial _{v_{n_2}} \theta ^{\{i_1,i_2,i_4\}} + \partial _{v_{n_2}} \theta ^{\{i_2,i_3,i_4\}} \partial _{v_{n_1}} \theta ^{\{i_1,i_2,i_4\}}\big )\\&- \frac{\theta ^{\{i_2,i_3,j_1,j_2\}} \theta ^{\{i_1,i_4,j_1,j_2\}}\theta ^{\{i_1,j_3,j_4,j_5\}}\theta ^{\{i_4,j_3,j_4,j_5\}}}{\theta ^{\{i_3,i_4,j_1,j_2\}} \theta ^{\{i_2,i_4,j_1,j_2\}}\theta ^{\{i_1,i_2,j_1,j_2\}}\theta ^{\{i_1,i_3,j_1,j_2\}}} \\&\times \theta ^{\{i_1,i_2,i_4,j_1\}} \theta ^{\{i_1,i_2,i_4,j_2\}} \theta ^{\{i_1,i_3,i_4,j_1\}} \theta ^{\{i_1,i_3,i_4,j_2\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{i_2,i_3,i_4\}} \partial _{v_{n_2}} \theta ^{\{i_1,i_2,i_3\}} + \partial _{v_{n_2}} \theta ^{\{i_2,i_3,i_4\}} \partial _{v_{n_1}} \theta ^{\{i_1,i_2,i_3\}}\big )\\&- \frac{\theta ^{\{i_1,i_4,j_1,j_2\}} \theta ^{\{i_2,i_3,j_1,j_2\}}\theta ^{\{i_2,j_3,j_4,j_5\}}\theta ^{\{i_3,j_3,j_4,j_5\}}}{\theta ^{\{i_3,i_4,j_1,j_2\}} \theta ^{\{i_2,i_4,j_1,j_2\}}\theta ^{\{i_1,i_2,j_1,j_2\}}\theta ^{\{i_1,i_3,j_1,j_2\}}} \\&\times \theta ^{\{i_2,i_3,i_4,j_1\}} \theta ^{\{i_2,i_3,i_4,j_2\}} \theta ^{\{i_1,i_2,i_3,j_1\}} \theta ^{\{i_1,i_2,i_3,j_2\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{i_1,i_3,i_4\}} \partial _{v_{n_2}} \theta ^{\{i_1,i_2,i_4\}} + \partial _{v_{n_2}} \theta ^{\{i_1,i_3,i_4\}} \partial _{v_{n_1}} \theta ^{\{i_1,i_2,i_4\}}\big )\\&+ \frac{\theta ^{\{i_1,i_3,j_1,j_2\}} \theta ^{\{i_2,i_4,j_1,j_2\}}\theta ^{\{i_2,j_3,j_4,j_5\}}\theta ^{\{i_4,j_3,j_4,j_5\}}}{\theta ^{\{i_3,i_4,j_1,j_2\}} \theta ^{\{i_1,i_4,j_1,j_2\}}\theta ^{\{i_1,i_2,j_1,j_2\}}\theta ^{\{i_2,i_3,j_1,j_2\}}} \\&\times \theta ^{\{i_1,i_2,i_4,j_1\}} \theta ^{\{i_1,i_2,i_4,j_2\}} \theta ^{\{i_2,i_3,i_4,j_1\}} \theta ^{\{i_2,i_3,i_4,j_2\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{i_1,i_3,i_4\}} \partial _{v_{n_2}} \theta ^{\{i_1,i_2,i_3\}} + \partial _{v_{n_2}} \theta ^{\{i_1,i_3,i_4\}} \partial _{v_{n_1}} \theta ^{\{i_1,i_2,i_3\}}\big )\\&- \frac{\theta ^{\{i_1,i_2,j_1,j_2\}} \theta ^{\{i_3,i_4,j_1,j_2\}}\theta ^{\{i_3,j_3,j_4,j_5\}}\theta ^{\{i_4,j_3,j_4,j_5\}}}{\theta ^{\{i_2,i_4,j_1,j_2\}} \theta ^{\{i_1,i_4,j_1,j_2\}}\theta ^{\{i_1,i_3,j_1,j_2\}}\theta ^{\{i_2,i_3,j_1,j_2\}}} \\&\times \theta ^{\{i_2,i_3,i_4,j_1\}} \theta ^{\{i_2,i_3,i_4,j_2\}} \theta ^{\{i_1,i_3,i_4,j_1\}} \theta ^{\{i_1,i_3,i_4,j_2\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{i_1,i_2,i_4\}} \partial _{v_{n_2}} \theta ^{\{i_1,i_2,i_3\}} + \partial _{v_{n_2}} \theta ^{\{i_1,i_2,i_4\}} \partial _{v_{n_1}} \theta ^{\{i_1,i_2,i_3\}}\big )\bigg ). \end{aligned}$$
For example with \({\mathcal {I}}_0=\{1,2,3,4\}\), \(j_1=5\), \(j_2=6\) one gets
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta ^{\emptyset }= & {} \big (\theta ^{\{1,2,3,4\}} (\theta ^{\{6,7,8,9\}} \theta ^{\{5,7,8,9\}})^2\big )^{-1} \\&\times \bigg ({-} \frac{\theta ^{\{3,4,5,6\}} \theta ^{\{1,2,5,6\}}\theta ^{\{1,7,8,9\}}\theta ^{\{2,7,8,9\}}}{\theta ^{\{2,4,5,6\}} \theta ^{\{1,4,5,6\}}\theta ^{\{2,3,5,6\}}\theta ^{\{1,3,5,6\}}} \theta ^{\{1,2,4,5\}} \theta ^{\{1,2,4,6\}} \theta ^{\{1,2,3,5\}} \theta ^{\{1,2,3,6\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{2,3,4\}} \partial _{v_{n_2}} \theta ^{\{1,3,4\}} + \partial _{v_{n_2}} \theta ^{\{2,3,4\}} \partial _{v_{n_1}} \theta ^{\{1,3,4\}}\big ) \\&+ \frac{\theta ^{\{2,4,5,6\}} \theta ^{\{1,3,5,6\}}\theta ^{\{1,7,8,9\}}\theta ^{\{3,7,8,9\}}}{\theta ^{\{3,4,5,6\}} \theta ^{\{1,4,5,6\}}\theta ^{\{2,3,5,6\}}\theta ^{\{1,2,5,6\}}} \theta ^{\{1,3,4,5\}} \theta ^{\{1,3,4,6\}}\theta ^{\{1,2,3,5\}} \theta ^{\{1,2,3,6\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{2,3,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,4\}} + \partial _{v_{n_2}} \theta ^{\{2,3,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,4\}}\big )\\&- \frac{\theta ^{\{2,3,5,6\}} \theta ^{\{1,4,5,6\}}\theta ^{\{1,7,8,9\}}\theta ^{\{4,7,8,9\}}}{\theta ^{\{3,4,5,6\}} \theta ^{\{2,4,5,6\}}\theta ^{\{1,2,5,6\}}\theta ^{\{1,3,5,6\}}} \theta ^{\{1,2,4,5\}} \theta ^{\{1,2,4,6\}} \theta ^{\{1,3,4,5\}} \theta ^{\{1,3,4,6\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{2,3,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,3\}} + \partial _{v_{n_2}} \theta ^{\{2,3,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,3\}}\big )\\&- \frac{\theta ^{\{1,4,5,6\}} \theta ^{\{2,3,5,6\}}\theta ^{\{2,7,8,9\}}\theta ^{\{3,7,8,9\}}}{\theta ^{\{3,4,5,6\}} \theta ^{\{2,4,5,6\}}\theta ^{\{1,2,5,6\}}\theta ^{\{1,3,5,6\}}} \theta ^{\{2,3,4,5\}} \theta ^{\{2,3,4,6\}} \theta ^{\{1,2,3,5\}} \theta ^{\{1,2,3,6\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{1,3,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,4\}} + \partial _{v_{n_2}} \theta ^{\{1,3,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,4\}}\big )\\&+ \frac{\theta ^{\{1,3,5,6\}} \theta ^{\{2,4,5,6\}}\theta ^{\{2,7,8,9\}}\theta ^{\{4,7,8,9\}}}{\theta ^{\{3,4,5,6\}} \theta ^{\{1,4,5,6\}}\theta ^{\{1,2,5,6\}}\theta ^{\{2,3,5,6\}}} \theta ^{\{1,2,4,5\}} \theta ^{\{1,2,4,6\}} \theta ^{\{2,3,4,5\}} \theta ^{\{2,3,4,6\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{1,3,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,3\}} + \partial _{v_{n_2}} \theta ^{\{1,3,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,3\}}\big )\\&- \frac{\theta ^{\{1,2,5,6\}} \theta ^{\{3,4,5,6\}}\theta ^{\{3,7,8,9\}}\theta ^{\{4,7,8,9\}}}{\theta ^{\{2,4,5,6\}} \theta ^{\{1,4,5,6\}}\theta ^{\{1,3,5,6\}}\theta ^{\{2,3,5,6\}}} \theta ^{\{2,3,4,5\}} \theta ^{\{2,3,4,6\}} \theta ^{\{1,3,4,5\}} \theta ^{\{1,3,4,6\}} \\&\times \big ( \partial _{v_{n_1}} \theta ^{\{1,2,4\}} \partial _{v_{n_2}} \theta ^{\{1,2,3\}} + \partial _{v_{n_2}} \theta ^{\{1,2,4\}} \partial _{v_{n_1}} \theta ^{\{1,2,3\}}\big )\bigg ) \end{aligned}$$
or with characteristics in the standard form
$$\begin{aligned} \partial _{v_{n_1},v_{n_2}}^2 \theta [{}^{1111}_{0101}]= & {} \big (\theta [{}^{1111}_{1001}] \big (\theta [{}^{1101}_{0101}] \theta [{}^{1101}_{0111}]\big )^2\big )^{-1} \\&\times \Big (- \frac{\theta [{}^{1111}_{0011}]\theta [{}^{1111}_{1111}]\theta [{}^{0111}_{1011}]\theta [{}^{0111}_{0011}]}{\theta [{}^{0011}_{0011}]\theta [{}^{0011}_{1011}]\theta [{}^{0011}_{0111}]\theta [{}^{0011}_{1111}] } \theta [{}^{1001}_{1101}]\theta [{}^{1001}_{1111}]\theta [{}^{1001}_{1001}]\theta [{}^{1001}_{1011}] \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{0111}_{1001}] \partial _{v_{n_2}} \theta [{}^{0111}_{0001}] + \partial _{v_{n_2}} \theta [{}^{0111}_{1001}] \partial _{v_{n_1}} \theta [{}^{0111}_{0001}]\big )\\&+ \frac{\theta [{}^{0011}_{0011}]\theta [{}^{0011}_{1111}]\theta [{}^{0111}_{1011}]\theta [{}^{1011}_{0011}]}{\theta [{}^{1111}_{0011}]\theta [{}^{0011}_{1011}]\theta [{}^{0011}_{0111}]\theta [{}^{1111}_{1111}] } \theta [{}^{0101}_{1101}]\theta [{}^{0101}_{1111}]\theta [{}^{1001}_{1001}]\theta [{}^{1001}_{1011}] \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{0111}_{1001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0001}] + \partial _{v_{n_2}} \theta [{}^{0111}_{1001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0001}]\big )\\&- \frac{\theta [{}^{0011}_{0111}]\theta [{}^{0011}_{1011}]\theta [{}^{0111}_{1011}]\theta [{}^{1011}_{0111}]}{\theta [{}^{1111}_{0011}]\theta [{}^{0011}_{0011}]\theta [{}^{1111}_{1111}]\theta [{}^{0011}_{1111}] } \theta [{}^{1001}_{1101}]\theta [{}^{1001}_{1111}]\theta [{}^{0101}_{1101}]\theta [{}^{0101}_{1111}] \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{0111}_{1001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0101}] + \partial _{v_{n_2}} \theta [{}^{0111}_{1001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0101}]\big )\\&- \frac{\theta [{}^{0011}_{1011}]\theta [{}^{0011}_{0111}]\theta [{}^{0111}_{0011}]\theta [{}^{1011}_{0011}]}{\theta [{}^{1111}_{0011}]\theta [{}^{0011}_{0011}]\theta [{}^{1111}_{1111}]\theta [{}^{0011}_{1111}] } \theta [{}^{0101}_{0101}]\theta [{}^{0101}_{0111}]\theta [{}^{1001}_{1001}]\theta [{}^{1001}_{1011}] \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{0111}_{0001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0001}] + \partial _{v_{n_2}} \theta [{}^{0111}_{0001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0001}]\big )\\&+ \frac{\theta [{}^{0011}_{1111}]\theta [{}^{0011}_{0011}]\theta [{}^{0111}_{0011}]\theta [{}^{1011}_{0111}]}{\theta [{}^{1111}_{0011}]\theta [{}^{0011}_{1011}]\theta [{}^{1111}_{1111}]\theta [{}^{0011}_{0111}] } \theta [{}^{1001}_{1101}]\theta [{}^{1001}_{1111}]\theta [{}^{0101}_{0101}]\theta [{}^{0101}_{0111}] \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{0111}_{0001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0101}] + \partial _{v_{n_2}} \theta [{}^{0111}_{0001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0101}]\big ) \\&- \frac{\theta [{}^{1111}_{1111}]\theta [{}^{111}_{0011}]\theta [{}^{1011}_{0011}]\theta [{}^{1011}_{0111}]}{\theta [{}^{0011}_{1011}]\theta [{}^{0011}_{0011}]\theta [{}^{0011}_{1111}]\theta [{}^{0011}_{0111}] } \theta [{}^{0101}_{1101}]\theta [{}^{0101}_{1111}]\theta [{}^{0101}_{0101}]\theta [{}^{0101}_{0111}] \\&\times \big ( \partial _{v_{n_1}} \theta [{}^{1011}_{0001}] \partial _{v_{n_2}} \theta [{}^{1011}_{0101}] + \partial _{v_{n_2}} \theta [{}^{1011}_{0001}] \partial _{v_{n_1}} \theta [{}^{1011}_{0101}]\big )\Big ). \end{aligned}$$
Appendix E: Examples of the Schottky relations
Here, we analyse some examples of the Schottky invariants presented in the literature and propose some new examples derived on the base of ideas from Remark 11.
1.1 E1: Examples from papers of Farkas & Rauch
In genus 4 hyperelliptic case, the Schottky relation (61) is given by formula (\(\mathrm {R}_2\)) [6, p. 685], the same in [7, p. 459]. In our notation and choice of homology basis, it gets the form
$$\begin{aligned}&\big (\theta ^{\{2,4,6,8\}} \theta ^{\{1,4,6,8\}} \theta ^{\{2,5,7,9\}} \theta ^{\{1,5,7,9\}} \theta ^{\{2,4,7,9\}}\theta ^{\{1,4,7,9\}}\theta ^{\{2,5,6,8\}}\theta ^{\{1,5,6,8\}}\big )^{1/2}\nonumber \\&\qquad - \big (\theta ^{\{2,4,6,9\}} \theta ^{\{1,4,6,9\}} \theta ^{\{2,5,7,8\}} \theta ^{\{1,5,7,8\}} \theta ^{\{2,4,7,8\}}\theta ^{\{1,4,7,8\}}\theta ^{\{2,5,6,9\}}\theta ^{\{1,5,6,9\}}\big )^{1/2}\nonumber \\&\qquad - \big (\theta ^{\{2,4,6,7\}} \theta ^{\{1,4,6,7\}} \theta ^{\{2,5,8,9\}} \theta ^{\{1,5,8,9\}} \theta ^{\{2,4,8,9\}}\theta ^{\{1,4,8,9\}}\theta ^{\{2,5,6,7\}}\theta ^{\{1,5,6,7\}}\big )^{1/2} = 0.\qquad \end{aligned}$$
(68)
The signs are chosen according to the order of sets of indices. The initial syzygetic group (P) of rank 3 is generated by 3 elements
$$\begin{aligned}&P_1= [\varepsilon _1]+[\varepsilon _2]&\{1,2\}&\\&P_2 = [\varepsilon _1]+[\varepsilon _2]+[\varepsilon _3]&\{1,2,3\}&\\&P_3 = [\varepsilon _1]+[\varepsilon _2]+[\varepsilon _3]+[\varepsilon _4]+[\varepsilon _5]&\{1,2,3,4,5\},&\end{aligned}$$
given with the corresponding sets of indices. One can add sets by the union operation, taking into account that an index occurring twice drops (two equal elements cancel each other out). Recall that characteristic \([{\mathcal {I}}]\) of theta constant corresponds to the set \({\mathcal {I}}+{\mathcal {R}}\), where \({\mathcal {R}}=\{2,4,6,8\}\) is associated with the vector of Riemann constants. For example, the set of indices in characteristic \([\{1,4,6,8\}]\) is obtained as \(\{1,2\} + \{2,4,6,8\}\). In such notation it becomes obvious that all theta constants in (68) have non-singular even characteristics, because all sets corresponding to characteristics contain \(4=g\) indices. Thus, characteristics of theta constants in the first summand of (68) correspond to the following elements of group (P) (in the same order)
$$\begin{aligned} P_0=0,\quad P_1,\quad P_2, \quad P_1 P_2, \quad P_3, \quad P_1 P_3, \quad P_2 P_3, \quad P_1 P_2 P_3. \end{aligned}$$
Here the notation of [1, ch. XVII] is adopted, that is \(P_1P_2\) denotes the sum of characteristics \(P_1\) and \(P_2\). Three cosets, which give rise to three products of theta constants in the Schottky invariant, are produced by the following azygetic triplet of characteristics
$$\begin{aligned}&A_1= 0 = [\{2,4,6,8\}]&\{\} \\&A_2= [\varepsilon _8]+[\varepsilon _9] = [\{2,4,6,9\}]&\{8,9\} \\&A_3= [\varepsilon _7]+[\varepsilon _8] = [\{2,4,6,7\}]&\{7,8\}. \end{aligned}$$
A concise proof of (68) follows from FTT Corollary 1:
$$\begin{aligned}&\bigg (\frac{\theta ^{\{2,4,6,8\}} \theta ^{\{1,3,5,8\}} \theta ^{\{1,3,5,7\}} \theta ^{\{2,4,7,9\}} \theta ^{\{1,4,7,9\}} \theta ^{\{2,3,5,7\}} \theta ^{\{2,3,5,8\}} \theta ^{\{1,4,6,8\}} }{\theta ^{\{2,4,6,7\}} \theta ^{\{1,3,5,7\}} \theta ^{\{1,3,5,8\}} \theta ^{\{2,4,8,9\}} \theta ^{\{1,4,8,9\}} \theta ^{\{2,3,5,8\}} \theta ^{\{2,3,5,7\}} \theta ^{\{1,4,6,7\}} } \\&\qquad \times \frac{\theta ^{\{2,5,6,8\}} \theta ^{\{1,3,4,8\}}\theta ^{\{1,3,4,7\}} \theta ^{\{2,5,7,9\}} \theta ^{\{1,5,7,9\}} \theta ^{\{2,3,4,7\}} \theta ^{\{2,3,4,8\}} \theta ^{\{1,5,6,8\}} }{\theta ^{\{2,5,6,7\}} \theta ^{\{1,3,4,7\}}\theta ^{\{1,3,4,8\}} \theta ^{\{2,5,8,9\}} \theta ^{\{1,5,8,9\}} \theta ^{\{2,3,4,8\}} \theta ^{\{2,3,4,7\}} \theta ^{\{1,5,6,7\}} } \bigg )^{1/2}\\&\qquad - \bigg (\frac{\theta ^{\{2,4,6,9\}} \theta ^{\{1,3,5,9\}} \theta ^{\{1,3,5,7\}} \theta ^{\{2,4,7,8\}} \theta ^{\{1,4,7,8\}} \theta ^{\{2,3,5,7\}} \theta ^{\{2,3,5,9\}} \theta ^{\{1,4,6,9\}} }{\theta ^{\{2,4,6,7\}} \theta ^{\{1,3,5,7\}} \theta ^{\{1,3,5,9\}} \theta ^{\{2,4,8,9\}} \theta ^{\{1,4,8,9\}} \theta ^{\{2,3,5,9\}} \theta ^{\{2,3,5,7\}} \theta ^{\{1,4,6,7\}} } \\&\qquad \times \frac{\theta ^{\{2,5,6,9\}} \theta ^{\{1,3,4,9\}} \theta ^{\{1,3,4,7\}} \theta ^{\{2,5,7,8\}} \theta ^{\{1,5,7,8\}} \theta ^{\{2,3,4,7\}} \theta ^{\{2,3,4,9\}} \theta ^{\{1,5,6,9\}} }{\theta ^{\{2,5,6,7\}}\theta ^{\{1,3,4,7\}} \theta ^{\{1,3,4,9\}} \theta ^{\{2,5,8,9\}} \theta ^{\{1,5,8,9\}} \theta ^{\{2,3,4,9\}}\theta ^{\{2,3,4,7\}} \theta ^{\{1,5,6,7\}} } \bigg )^{1/2} \\&\qquad - 1 = \frac{(e_9-e_7)(e_8-e_6)}{(e_9-e_8)(e_7-e_6)} - \frac{(e_8-e_7)(e_9-e_6)}{(e_9-e_8)(e_7-e_6)} - 1 = 0. \end{aligned}$$
In the context of Remark 11, the group (P) is generated by three even pairwise syzygetic characteristics: \([\{2,4,7,9\}]=[{\mathcal {A}}(\{6,7,8,9\})]\), \([\{1,4,6,8\}]=[{\mathcal {A}}(\{1,2\})]\), \([\{2,5,6,8\}]=[{\mathcal {A}}(\{4,5\})]\). In this case \({\mathcal {I}}_0 = \{6,7,8,9\}\) and three cosets are produced by the even azygetic triplet: \(A_1 = [{\mathcal {I}}_0^{(7,9 \rightarrow 2,4)}]\), \(A_2 = [{\mathcal {I}}_0^{(7,8 \rightarrow 2,4)}]\), \(A_3 = [{\mathcal {I}}_0^{(8,9 \rightarrow 2,4)}]\).
At the same time, ratios from [6, Theorem 1, p. 680] connect theta constants in genera \(g-1\) and g. In the context of the above example, one gets the following equalities of ratios in genera 3 and 4
$$\begin{aligned} \frac{\big (\theta ^{\{2,4,6\}}\big )^2}{\theta ^{\{2,4,6,8\}}\theta ^{\{1,4,6,8\}}}= & {} \frac{\big (\theta ^{\{3,5,7\}}\big )^2}{\theta ^{\{2,5,7,9\}}\theta ^{\{1,5,7,9\}}} = \frac{\big (\theta ^{\{2,5,7\}}\big )^2}{\theta ^{\{2,4,7,9\}}\theta ^{\{1,4,7,9\}}} \nonumber \\= & {} \frac{\big (\theta ^{\{3,4,6\}}\big )^2}{\theta ^{\{2,5,6,8\}}\theta ^{\{1,5,6,8\}}} = \frac{\big (\theta ^{\{2,4,7\}}\big )^2}{\theta ^{\{2,4,6,9\}}\theta ^{\{1,4,6,9\}}}\nonumber \\= & {} \frac{\big (\theta ^{\{3,5,6\}}\big )^2}{\theta ^{\{2,5,7,8\}}\theta ^{\{1,5,7,8\}}} = \ldots \end{aligned}$$
(69)
And similar relations between theta constants in genera 4 and 5
$$\begin{aligned} \frac{\big (\theta ^{\{2,4,6,8\}}\big )^2}{\theta ^{\{2,4,6,8,10\}}\theta ^{\{1,4,6,8,10\}}}= & {} \frac{\big (\theta ^{\{1,4,6,8\}}\big )^2}{\theta ^{\{2,3,6,8,10\}}\theta ^{\{1,3,6,8,10\}}} \nonumber \\= & {} \frac{\big (\theta ^{\{2,5,7,9\}}\big )^2}{\theta ^{\{2,4,7,9,11\}}\theta ^{\{1,4,7,9,11\}}} = \frac{\big (\theta ^{\{1,5,7,9\}}\big )^2}{\theta ^{\{2,3,7,9,11\}}\theta ^{\{1,3,7,9,11\}}} \nonumber \\= & {} \frac{\big (\theta ^{\{2,4,7,9\}}\big )^2}{\theta ^{\{2,4,6,9,11\}}\theta ^{\{1,4,6,9,11\}}}\nonumber \\= & {} \frac{\big (\theta ^{\{1,4,7,9\}}\big )^2}{\theta ^{\{2,3,6,9,11\}}\theta ^{\{1,3,6,9,11\}}} = \ldots \end{aligned}$$
(70)
As stated in [6, p. 685], a genus 3 curve is obtained from a genus 4 curve with 9 branch points by dropping the first two branch points, that is the genus 3 curve has branch points at \(e_3\), \(e_4\), \(e_5\), \(e_6\), \(e_7\), \(e_8\) and \(e_9\) which now are labelled by indices from 1 to 7. Then the following relation
$$\begin{aligned}&\theta ^{\{2,4,6\}} \theta ^{\{3,5,7\}} \theta ^{\{2,5,7\}} \theta ^{\{3,4,6\}}\nonumber \\&\qquad - \theta ^{\{2,4,7\}} \theta ^{\{3,5,6\}} \theta ^{\{2,5,6\}} \theta ^{\{3,4,7\}} - \theta ^{\{2,4,5\}} \theta ^{\{3,6,7\}} \theta ^{\{2,6,7\}} \theta ^{\{3,4,5\}} = 0, \end{aligned}$$
(71)
is derived from (68) with the help of (69). Alternatively, relation (71) can be proven using FTT Corollary 1.
Also, the following extension of the Schottky relation (68) to genus 5 is proposed in [6, 7]
$$\begin{aligned}&\big (\theta ^{\{2,4,6,8,10\}} \theta ^{\{1,4,6,8,10\}} \theta ^{\{2,5,7,9,11\}}\theta ^{\{1,5,7,9,11\}} \nonumber \\&\qquad \times \theta ^{\{2,4,7,9,11\}}\theta ^{\{1,4,7,9,11\}} \theta ^{\{2,5,6,8,10\}} \theta ^{\{1,5,6,8,10\}}\big )^{1/2}\nonumber \\&\qquad - \big (\theta ^{\{2,4,6,8,11\}} \theta ^{\{1,4,6,8,11\}} \theta ^{\{2,5,7,9,10\}}\theta ^{\{1,5,7,9,10\}} \nonumber \\&\qquad \times \theta ^{\{2,4,7,9,10\}}\theta ^{\{1,4,7,9,10\}} \theta ^{\{2,5,6,8,11\}} \theta ^{\{1,5,6,8,11\}}\big )^{1/2}\nonumber \\&\qquad - \big (\theta ^{\{2,4,6,8,9\}} \theta ^{\{1,4,6,8,9\}} \theta ^{\{2,5,7,10,11\}}\theta ^{\{1,5,7,10,11\}} \nonumber \\&\qquad \times \theta ^{\{2,4,7,10,11\}}\theta ^{\{1,4,7,10,11\}} \theta ^{\{2,5,6,8,9\}} \theta ^{\{1,5,6,8,9\}}\big )^{1/2}\nonumber \\&\qquad - \big (\theta ^{\{2,4,6,7,8\}} \theta ^{\{1,4,6,7,8\}} \theta ^{\{2,5,9,10,11\}}\theta ^{\{1,5,7,9,10,11\}} \nonumber \\&\qquad \times \theta ^{\{2,4,7,9,10,11\}}\theta ^{\{1,4,7,9,10,11\}} \theta ^{\{2,5,6,7,8\}} \theta ^{\{1,5,6,7,8\}}\big )^{1/2}= 0, \end{aligned}$$
(72)
which has the form
$$\begin{aligned} \sqrt{r_1} - \sqrt{r_2} - \sqrt{r_3} - \sqrt{r_4} = 0. \end{aligned}$$
Here, the group (P) of rank 3 is generated by the same 3 elements
$$\begin{aligned}&P_1= [\varepsilon _1]+[\varepsilon _2]&\{1,2\}&\\&P_2 = [\varepsilon _1]+[\varepsilon _2]+[\varepsilon _3]&\{1,2,3\}&\\&P_3 = [\varepsilon _1]+[\varepsilon _2]+[\varepsilon _3]+[\varepsilon _4]+[\varepsilon _5]&\{1,2,3,4,5\},&\end{aligned}$$
and four products \(\{r_1\), \(r_2\), \(r_3\), \(r_4\}\) are constructed with the help of four characteristics azygetic in triplets, namely:
$$\begin{aligned}&A_1= 0 = [\{2,4,6,8,10\}]&\{\} \\&A_2= [\varepsilon _{10}]+[\varepsilon _{11}] = [\{2,4,6,8,11\}]&\{10,11\} \\&A_3= [\varepsilon _9]+[\varepsilon _{10}] = [\{2,4,6,8,9\}]&\{9,10\} \\&A_4= [\varepsilon _7]+[\varepsilon _{10}] = [\{2,4,6,7,8\}]&\{7,10\}. \end{aligned}$$
The way of producing (72) is explained in [7, p. 457–458].
1.2 E2: Schottky relations in genus 5
With the help of results proposed in the present paper, the standard Schottky relation can be obtained in an arbitrary genus. In particular, in genus 5 let \({\mathcal {I}}_0 = \{6,8,9,10,11\}\), \({\mathcal {K}}=\{6,9,10,11\}\) and \(A_1\), \(A_2\), \(A_3\) be as above, that is
$$\begin{aligned}&A_1 = [{\mathcal {I}}_0^{(9,11\rightarrow 2,4)}] = [\{2,4,6,8,10\}]&\{\}&\\&A_2 = [{\mathcal {I}}_0^{(9,10\rightarrow 2,4)}] = [\{2,4,6,8,11\}]&\{10,11\}&\\&A_3 = [{\mathcal {I}}_0^{(10,11\rightarrow 2,4)}] = [\{2,4,6,8,9\}]&\{9,10\}.&\end{aligned}$$
Then relation (57) follows from the Schottky relation
$$\begin{aligned} \sqrt{r_1} - \sqrt{r_2} - \sqrt{r_3} = 0, \end{aligned}$$
(73)
which holds with the following products, derived from the group (P) of rank 1 generated by element \({\mathcal {K}}\),
$$\begin{aligned}&r_1 = \big (\theta [{\mathcal {I}}_0^{(9,11\rightarrow 2,4)}]\theta [{\mathcal {I}}_0^{(6,10\rightarrow 2,4)}]\big )^4 = \big (\theta ^{\{2,4,6,8,10\}}\theta ^{\{2,4,8,9,11\}}\big )^4,\\&r_2 = \big (\theta [{\mathcal {I}}_0^{(9,10\rightarrow 2,4)}]\theta [{\mathcal {I}}_0^{(6,11\rightarrow 2,4)}]\big )^4 = \big (\theta ^{\{2,4,6,8,11\}}\theta ^{\{2,4,8,9,10\}}\big )^4,\\&r_3 = \big (\theta [{\mathcal {I}}_0^{(10,11\rightarrow 2,4)}]\theta [{\mathcal {I}}_0^{(6,9\rightarrow 2,4)}]\big )^4 = \big (\theta ^{\{2,4,6,8,9\}}\theta ^{\{2,4,8,10,11\}}\big )^4. \end{aligned}$$
On the other hand, one can construct a group (P) of rank 3 generated by elements \({\mathcal {K}}\), \(\{1,2\}\), \(\{4,5\}\) that results into
$$\begin{aligned} r_1&= \theta ^{\{2,4,6,8,10\}}\theta ^{\{1,4,6,8,10\}}\theta ^{\{2,5,6,8,10\}}\theta ^{\{1,5,6,8,10\}} \\&\quad \times \theta ^{\{2,4,8,9,11\}}\theta ^{\{1,4,8,9,11\}}\theta ^{\{2,4,8,9,11\}}\theta ^{\{1,5,8,9,11\}},\\ r_2&= \theta ^{\{2,4,6,8,11\}}\theta ^{\{1,4,6,8,11\}} \theta ^{\{2,5,6,8,11\}} \theta ^{\{1,5,6,8,11\}} \\&\quad \times \theta ^{\{2,4,8,9,10\}}\theta ^{\{1,4,8,9,10\}}\theta ^{\{2,5,8,9,10\}}\theta ^{\{1,5,8,9,10\}},\\ r_3&= \theta ^{\{2,4,6,8,9\}}\theta ^{\{1,4,6,8,9\}}\theta ^{\{2,5,6,8,9\}}\theta ^{\{1,5,6,8,9\}} \\&\quad \times \theta ^{\{2,4,8,10,11\}}\theta ^{\{1,4,8,10,11\}}\theta ^{\{2,5,8,10,11\}}\theta ^{\{1,5,8,10,11\}}, \end{aligned}$$
or with characteristics in the standard form
$$\begin{aligned} r_1&= \theta [{}^{00000}_{00000}]\theta [{}^{00000}_{10000}]\theta [{}^{01100}_{00000}]\theta [{}^{01100}_{10000}] \theta [{}^{00100}_{00010}]\theta [{}^{00100}_{10010}]\theta [{}^{01000}_{00010}]\theta [{}^{01000}_{10010}],\\ r_2&= \theta [{}^{00001}_{00000}]\theta [{}^{00001}_{10000}]\theta [{}^{01101}_{00000}]\theta [{}^{01101}_{10000}] \theta [{}^{00101}_{00010}]\theta [{}^{00101}_{10010}]\theta [{}^{01001}_{00010}]\theta [{}^{01001}_{10010}],\\ r_3&= \theta [{}^{00000}_{00001}]\theta [{}^{00000}_{10001}]\theta [{}^{01100}_{00001}]\theta [{}^{01100}_{10001}] \theta [{}^{00100}_{00011}]\theta [{}^{00100}_{10011}]\theta [{}^{01000}_{00011}]\theta [{}^{01000}_{10011}]. \end{aligned}$$
The latter products \(\{r_1,r_2,r_3\}\) also satisfy (73).
On the base of Remark 11, one can produce a group (P) containing odd and even characteristics which are syzygetic in pairs. Let characteristics \(A_1\), \(A_2\), \(A_3\) be constructed by (64), then all characteristics in \((A_1 P)\), \((A_2 P)\), \((A_3 P)\) are even non-singular. Then the Schottky relation (73) holds.
For example, with the same \({\mathcal {I}}_0 = \{6,8,9,10,11\}\) and \({\mathcal {K}}=\{6,9,10,11\}\) as above, let \(j_m=3\), \(j_m=5\) and group (P) be generated by elements
$$\begin{aligned}&P_1= [\varepsilon _1]+[\varepsilon _3]&\{1,3\}&\\&P_2 = [\varepsilon _4]+[\varepsilon _5]&\{4,5\}&\\&P_3 = [\varepsilon _6]+[\varepsilon _9]+[\varepsilon _{10}]+[\varepsilon _{11}]&\{6,9,10,11\}={\mathcal {K}}&\end{aligned}$$
With an azygetic triplet of characteristics
$$\begin{aligned}&A_1 = [{\mathcal {I}}_0^{(9,11\rightarrow 3,5)}] = [\{3,5,6,8,10\}]&\{2,3,4,5\}&\\&A_2 = [{\mathcal {I}}_0^{(9,10\rightarrow 3,5)}] = [\{3,5,6,8,11\}]&\{2,3,4,5,10,11\}&\\&A_3 = [{\mathcal {I}}_0^{(10,11\rightarrow 3,5)}] = [\{3,5,6,8,9\}]&\{2,3,4,5,9,10\}&\end{aligned}$$
one obtains the following products of theta constants
$$\begin{aligned} r_1&= \theta ^{\{3,5,6,8,10\}}\theta ^{\{1,5,6,8,10\}}\theta ^{\{3,4,6,8,10\}}\theta ^{\{1,4,6,8,10\}} \\&\quad \times \theta ^{\{3,5,8,9,11\}}\theta ^{\{1,5,8,9,11\}}\theta ^{\{3,4,8,9,11\}}\theta ^{\{1,4,8,9,11\}},\\ r_2&= \theta ^{\{3,5,6,8,11\}}\theta ^{\{1,5,6,8,11\}} \theta ^{\{3,4,6,8,11\}} \theta ^{\{1,4,6,8,11\}} \\&\quad \times \theta ^{\{3,5,8,9,10\}}\theta ^{\{1,5,8,9,10\}}\theta ^{\{3,4,8,9,10\}}\theta ^{\{1,4,8,9,10\}},\\ r_3&= \theta ^{\{3,5,6,8,9\}}\theta ^{\{1,5,6,8,9\}}\theta ^{\{3,4,6,8,9\}}\theta ^{\{1,4,6,8,9\}} \\&\quad \times \theta ^{\{3,5,8,10,11\}}\theta ^{\{1,5,8,10,11\}}\theta ^{\{3,4,8,10,11\}}\theta ^{\{1,4,8,10,11\}}, \end{aligned}$$
or with characteristics in the standard form
$$\begin{aligned} r_1&= \theta [{}^{10100}_{00000}]\theta [{}^{01100}_{10000}]\theta [{}^{11000}_{00000}]\theta [{}^{00000}_{10000}] \theta [{}^{10000}_{00010}]\theta [{}^{01000}_{10010}]\theta [{}^{11100}_{00010}]\theta [{}^{00100}_{10010}],\\ r_2&= \theta [{}^{10101}_{00000}]\theta [{}^{01101}_{10000}]\theta [{}^{11001}_{00000}]\theta [{}^{00001}_{10000}] \theta [{}^{10001}_{00010}]\theta [{}^{01001}_{10010}]\theta [{}^{11101}_{00010}]\theta [{}^{00101}_{10010}],\\ r_3&= \theta [{}^{10100}_{00001}]\theta [{}^{01100}_{10001}]\theta [{}^{11000}_{00001}]\theta [{}^{00000}_{10001}] \theta [{}^{10000}_{00011}]\theta [{}^{01000}_{10011}]\theta [{}^{11100}_{00011}]\theta [{}^{00100}_{10011}]. \end{aligned}$$
This collection of products \(\{r_1,r_2,r_3\}\) satisfies (73).
1.3 E3: Schottky relations from group of rank 4
From (68) with the help of (70), one can derive a new version of the Schottky relation on the base of a group (P) of rank 4, namely:
$$\begin{aligned} r_1&= \big (\theta ^{\{2,4,6,8,10\}}\theta ^{\{1,4,6,8,10\}}\theta ^{\{2,3,6,8,10\}}\theta ^{\{1,3,6,8,10\}} \\&\quad \times \theta ^{\{2,4,7,9,11\}}\theta ^{\{1,4,7,9,11\}}\theta ^{\{2,3,7,9,11\}}\theta ^{\{1,3,7,9,11\}} \\&\quad \times \theta ^{\{2,4,6,9,11\}}\theta ^{\{1,4,6,9,11\}}\theta ^{\{2,3,6,9,11\}}\theta ^{\{1,3,6,9,11\}} \\&\quad \times \theta ^{\{2,4,7,8,10\}}\theta ^{\{1,4,7,8,10\}}\theta ^{\{2,3,7,8,10\}}\theta ^{\{1,3,7,8,10\}}\big )^{1/2},\\ r_2&= \big (\theta ^{\{2,4,6,8,11\}}\theta ^{\{1,4,6,8,11\}}\theta ^{\{2,3,6,8,11\}}\theta ^{\{1,3,6,8,11\}} \\&\quad \times \theta ^{\{2,4,7,9,10\}}\theta ^{\{1,4,7,9,10\}}\theta ^{\{2,3,7,9,10\}}\theta ^{\{1,3,7,9,10\}} \\&\quad \times \theta ^{\{2,4,6,9,10\}}\theta ^{\{1,4,6,9,10\}}\theta ^{\{2,3,6,9,10\}}\theta ^{\{1,3,6,9,10\}} \\&\quad \times \theta ^{\{2,4,7,8,11\}}\theta ^{\{1,4,7,8,11\}}\theta ^{\{2,3,7,8,11\}}\theta ^{\{1,3,7,8,11\}}\big )^{1/2},\\ r_3&= \big (\theta ^{\{2,4,6,8,9\}}\theta ^{\{1,4,6,8,9\}}\theta ^{\{2,3,6,8,9\}}\theta ^{\{1,3,6,8,9\}} \\&\quad \times \theta ^{\{2,4,7,10,11\}}\theta ^{\{1,4,7,10,11\}}\theta ^{\{2,3,7,10,11\}}\theta ^{\{1,3,7,10,11\}} \\&\quad \times \theta ^{\{2,4,6,10,11\}}\theta ^{\{1,4,6,10,11\}}\theta ^{\{2,3,6,10,11\}}\theta ^{\{1,3,6,10,11\}} \\&\quad \times \theta ^{\{2,4,7,8,9\}}\theta ^{\{1,4,7,8,9\}}\theta ^{\{2,3,7,8,9\}}\theta ^{\{1,3,7,8,9\}}\big )^{1/2}, \end{aligned}$$
or in the standard form
$$\begin{aligned} r_1&= \big (\theta [{}^{00000}_{00000}]\theta [{}^{00000}_{10000}]\theta [{}^{00000}_{01000}]\theta [{}^{00000}_{11000}] \theta [{}^{00110}_{00000}]\theta [{}^{00110}_{10000}]\theta [{}^{00110}_{01000}]\theta [{}^{00110}_{11000}] \\&\quad \times \theta [{}^{00010}_{00000}]\theta [{}^{00010}_{10000}]\theta [{}^{00010}_{01000}]\theta [{}^{00010}_{11000}] \theta [{}^{00100}_{00000}]\theta [{}^{00100}_{10000}]\theta [{}^{00100}_{01000}]\theta [{}^{00100}_{11000}]\big )^{1/2},\\ r_2&= \big (\theta [{}^{00001}_{00000}]\theta [{}^{00001}_{10000}]\theta [{}^{00001}_{01000}]\theta [{}^{00001}_{11000}] \theta [{}^{00111}_{00000}]\theta [{}^{00111}_{10000}]\theta [{}^{00111}_{01000}]\theta [{}^{00111}_{11000}] \\&\quad \times \theta [{}^{00011}_{00000}]\theta [{}^{00011}_{10000}]\theta [{}^{00011}_{01000}]\theta [{}^{00011}_{11000}] \theta [{}^{00101}_{00000}]\theta [{}^{00101}_{10000}]\theta [{}^{00101}_{01000}]\theta [{}^{00101}_{11000}]\big )^{1/2},\\ r_3&= \big (\theta [{}^{00000}_{00001}]\theta [{}^{00000}_{10001}]\theta [{}^{00000}_{01001}]\theta [{}^{00000}_{11001}] \theta [{}^{00110}_{00001}]\theta [{}^{00110}_{10001}]\theta [{}^{00110}_{01001}]\theta [{}^{00110}_{11001}] \\&\quad \times \theta [{}^{00010}_{00001}]\theta [{}^{00010}_{10001}]\theta [{}^{00010}_{01001}]\theta [{}^{00010}_{11001}] \theta [{}^{00100}_{00001}]\theta [{}^{00100}_{10001}]\theta [{}^{00100}_{01001}]\theta [{}^{00100}_{11001}]\big )^{1/2}. \end{aligned}$$
The new collection of products \(\{r_1,r_2,r_3\}\) again satisfies (73). In a similar way, one can increase the rank of group (P) in the Schottky relation.
