TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SOURCE-SELECTED-THETA-SPIN.txt
Original file SHA-256: cf4478191ccd75d62a2f049a54193541009bed24877df9cbe0ed9f942d07d717
Inclusive original lines: 28-145
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000028 | 1. THE q|p PARTITION DEFINES ONE THETA CHARACTERISTIC
000029 | 
000030 | On
000031 | 
000032 |     C: eta^2=1-w^6
000033 | 
000034 | write
000035 | 
000036 |     p_a=omega^a,       q_a=-omega^a,       a in Z/3.
000037 | 
000038 | Every branch point r satisfies 2r linearly equivalent to K_C.  Therefore the
000039 | degree one line bundle (1) satisfies
000040 | 
000041 |     2S_0
000042 |       =O_C(2q_0+2q_1+2q_2-2K_C)
000043 |       =O_C(3K_C-2K_C)
000044 |       =K_C.                                                     (2)
000045 | 
000046 | It is thus a theta characteristic.  The meromorphic function
000047 | 
000048 |     (w^3+1)/eta
000049 | 
000050 | has divisor
000051 | 
000052 |     q_0+q_1+q_2-p_0-p_1-p_2.                                   (3)
000053 | 
000054 | Consequently
000055 | 
000056 |     O_C(q_0+q_1+q_2-K_C)
000057 |       isomorphic O_C(p_0+p_1+p_2-K_C).                          (4)
000058 | 
000059 | Thus interchanging the two sides of the partition does not change S_0.
000060 | 
000061 | The theta characteristic is even.  Indeed, an effective degree one theta
000062 | characteristic is O_C(r) for a Weierstrass point r and is one of the six odd
000063 | classes.  Formula (1), or equivalently (4), is not any of these classes: after
000064 | subtracting a point on the chosen side and using K_C equivalent to twice one
000065 | of the other two points, equality would make the difference of two distinct
000066 | points principal.  That would give a degree one map C->P1, impossible for a
000067 | genus two curve.  Hence h^0(S_0)=0 and S_0 is even.
000068 | 
000069 | 
000070 | 2. THE ORIGINAL CUBIC SYMMETRY MAKES THE CHOICE UNIQUE
000071 | 
000072 | The source automorphism
000073 | 
000074 |     c:(w,eta) |-> (omega w,eta)
000075 | 
000076 | cycles each triple, so c^*S_0 isomorphic S_0.  Theta characteristics form an
000077 | affine torsor over Jac(C)[2].  Relative to S_0, every other theta
000078 | characteristic is S_0 tensor L for a unique L in Jac(C)[2].
000079 | 
000080 | The accepted six line calculation gives
000081 | 
000082 |     Jac(C)[2]=V_q direct_sum V_p.                                (5)
000083 | 
000084 | On each two dimensional F2 plane, c is an element of order three with no
000085 | nonzero fixed vector.  Hence
000086 | 
000087 |     Jac(C)[2]^c=0.                                               (6)
000088 | 
000089 | If S_0 tensor L were also c invariant, (6) would force L=0.  Therefore
000090 | 
000091 |     S_0 is the unique c invariant theta characteristic.          (7)
000092 | 
000093 | This uniqueness uses the actual c action and the already proved q|p packet.
000094 | It is not an added theta marking.
000095 | 
000096 | 
000097 | 3. THE MARKED Q ORIENTATION SELECTS THE LIFT OF h=Q^2
000098 | 
000099 | The original quarter map is
000100 | 
000101 |     Q:(z,w,eta) |-> (i z,1/w,-i eta/w^3).                        (8)
000102 | 
000103 | It preserves the q triple and therefore Q^*S_0 isomorphic S_0.  Fix the
000104 | square root isomorphism S_0^2 isomorphic K_C and choose a compatible
000105 | projective spin lift
000106 | 
000107 |     alpha:Q^*S_0 -> S_0.                                        (9)
000108 | 
000109 | There are exactly two normalized choices, alpha and -alpha.
000110 | 
000111 | The Weierstrass point q_0=-1 is fixed by Q.  The coordinate eta is a local
000112 | parameter there.  Formula (8) gives, at q_0 where w^3=-1,
000113 | 
000114 |     eta |-> i eta,
000115 |     d eta |-> i d eta.                                          (10)
000116 | 
000117 | If alpha acts on the spin fiber at q_0 by lambda, compatibility with
000118 | S_0^2=K_C gives
000119 | 
000120 |     lambda^2=i.                                                  (11)
000121 | 
000122 | The fourth iterate of alpha is a global scalar automorphism of S_0, and at
000123 | q_0 it equals
000124 | 
000125 |     lambda^4=-1.
000126 | 
000127 | Therefore
000128 | 
000129 |     alpha^4=-I,       alpha^8=I.                                (12)
000130 | 
000131 | Thus alpha is an order eight spin lift of the order four map Q.  Define
000132 | 
000133 |     beta=alpha composed with Q^*alpha,                            (13)
000134 | 
000135 | which is the induced lift of h=Q^2.  Equations (12) and (13) give
000136 | 
000137 |     beta^2=-I.                                                   (14)
000138 | 
000139 | Replacing alpha by -alpha leaves beta unchanged because the two signs cancel
000140 | in (13).  The actual orientation eta |-> i eta therefore selects the +i local
000141 | phase of beta at q_0 and removes the sign ambiguity that remained when h was
000142 | considered alone.
000143 | 
000144 | Beta is a projective spin lift of order four.  It is not an honest C2
000145 | linearization, whose cocycle square would be +I.