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Fz(F)593 940 y Fr(2)p Fq(g)593 1012 y Fr(2)702 987 y
Ft(also)j(de\014nes)h(a)f(transv)m(ection)h Fs(t)1894
1002 y Fq(v)1966 987 y Fo(2)30 b Fs(S)6 b(p)p Ft(\(2)p
Fs(g)t(;)17 b Fz(F)2429 1002 y Fr(2)2468 987 y Ft(\),)34
b(whic)m(h)h(is)e(an)h(in)m(v)m(olution)f(if)g Fs(v)h
Fo(6)p Ft(=)c(0,)-118 1103 y(b)m(y)993 1234 y Fs(t)1028
1249 y Fq(v)1113 1234 y Ft(:)44 b Fs(V)66 b Fo(\000)-16
b(!)44 b Fs(V)5 b(;)212 b(t)1848 1249 y Fq(v)1889 1234
y Ft(\()p Fs(w)s Ft(\))27 b(:=)h Fs(w)c Ft(+)e Fs(E)6
b Ft(\()p Fs(w)s(;)17 b(v)t Ft(\))p Fs(v)t(:)-118 1385
y Ft(In)28 b([)26 1385 y
SDict begin H.S end
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224 1317 a 224 1385 a
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<</S/S/W 1>>/Subtype /Link/Dest (cite.CG) cvn H.B /ANN pdfmark end
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y
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 642 1317 a 642 1385 a
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<</S/S/W 1>>/Subtype /Link/Dest (cite.DG) cvn H.B /ANN pdfmark end
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1385 a Ft(])g(section)f(9.3)h(it)e(is)h(sho)m(wn)i(that,)g(giv)m(en)e
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1732 y Fr(\002)996 1717 y Ft(:)44 b Fz(C)1148 1676 y
Fq(N)7 b Fr(+1)1350 1717 y Fo(\000)-16 b(!)44 b Fz(C)1636
1676 y Fq(N)7 b Fr(+1)1793 1717 y Fs(;)212 b(M)2126 1732
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Fs(M)1521 1903 y Fm(0)1511 1964 y Fr(\002)1598 1939 y
Ft(=)e Fs(U)1768 1954 y Fq(w)1824 1939 y Fs(M)1918 1954
y Fr(\002)1978 1939 y Fs(U)2054 1903 y Fm(\000)p Fr(1)2044
1963 y Fq(w)2178 1939 y Ft(for)h(some)g Fs(w)h Fo(2)e
Fz(F)2829 1892 y Fr(2)p Fq(g)2829 1963 y Fr(2)2905 1939
y Ft(.)43 b(Th)m(us)30 b(it)f(is)g(easy)h(to)f(\014nd)-118
2056 y(explicitly)i(the)i(action)e(of)h(\000)937 2071
y Fq(g)977 2056 y Fs(=)p Ft(\000)1087 2071 y Fq(g)1127
2056 y Ft(\(2)p Fs(;)17 b Ft(4\))32 b(on)g Fz(P)1589
2020 y Fr(2)1624 1996 y Fk(g)1660 2020 y Fm(\000)p Fr(1)1755
2056 y Ft(.)-118 2171 y
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2171 a -118 2171 a
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b(of)g(ppa)m(v's.)51 b Ft(F)-8 b(rom)32 b(the)i(de\014nition)f(of)h
(the)g(thetan)m(ulls)g(\002[)p Fs(\033)t Ft(])g(it)f(is)g(ob)m(vious)i
(that)e(if)g(the)-118 2375 y(p)s(erio)s(d)f(matrix)f
Fs(\034)45 b Ft(is)33 b(in)f(blo)s(c)m(k)h(form,)f(with)h(diagonal)d
(blo)s(c)m(ks)k Fs(\034)2271 2390 y Fq(a)2313 2375 y
Fs(;)17 b(\034)2399 2390 y Fq(b)2466 2375 y Ft(\(with)33
b Fs(\034)2769 2390 y Fq(c)2832 2375 y Fo(2)c Fz(H)3015
2390 y Fq(c)3083 2375 y Ft(and)k Fs(a)22 b Ft(+)h Fs(b)29
b Ft(=)f Fs(g)t Ft(\))k(and)-118 2492 y(zero)h(en)m(tries)g(elsewhere,)
h(then)1285 2622 y(\002[)p Fs(\033)t Ft(]\()p Fs(\034)11
b Ft(\))28 b(=)g(\002[)p Fs(\033)1893 2637 y Fq(a)1935
2622 y Ft(]\()p Fs(\034)2042 2637 y Fq(a)2084 2622 y
Ft(\)\002[)p Fs(\033)2280 2637 y Fq(b)2315 2622 y Ft(]\()p
Fs(\034)2422 2637 y Fq(b)2456 2622 y Ft(\))-118 2773
y(where)34 b(if)d Fs(\033)h Ft(=)27 b(\()p Fs(s)527 2788
y Fr(1)567 2773 y Fs(;)17 b(:)g(:)g(:)e(;)i(s)831 2788
y Fq(g)871 2773 y Ft(\))28 b Fo(2)g(f)p Ft(0)p Fs(;)17
b Ft(1)p Fo(g)1273 2737 y Fq(g)1344 2773 y Ft(then)34
b Fs(\033)1622 2788 y Fq(a)1691 2773 y Ft(=)28 b(\()p
Fs(s)1879 2788 y Fr(1)1918 2773 y Fs(;)17 b(:)g(:)g(:)f(;)h(s)2183
2788 y Fq(a)2224 2773 y Ft(\),)33 b Fs(\033)2377 2788
y Fq(b)2439 2773 y Ft(=)28 b(\()p Fs(s)2627 2788 y Fq(a)p
Fr(+1)2759 2773 y Fs(;)17 b(:)g(:)g(:)e(;)i(s)3023 2788
y Fq(g)3063 2773 y Ft(\).)-118 2888 y
SDict begin H.S end
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2888 a
SDict begin 14 H.A end
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SDict begin [/View [/XYZ H.V]/Dest (subsection.1.5) cvn /DEST pdfmark
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b Fz(V)-9 b(anishing)42 b(of)g(thetan)m(ulls.)48 b Ft(There)38
b(are)e(2)1800 2940 y Fq(g)r Fm(\000)p Fr(1)1930 2977
y Ft(\(2)2017 2940 y Fq(g)2082 2977 y Ft(+)25 b(1\))36
b(ev)m(en)i(theta)f(c)m(haracteristics)g([)3460 2940
y Fq(\017)3460 3003 y(\017)3489 2984 y Fi(0)3515 2977
y Ft(])g(\(that)f(is)-118 3093 y(with)g Fs(\017;)17 b(\017)230
3057 y Fm(0)287 3093 y Fo(2)35 b(f)p Ft(0)p Fs(;)17 b
Ft(1)p Fo(g)630 3057 y Fq(g)705 3093 y Ft(and)36 b Fs(\017)25
b Fo(\001)1015 3057 y Fq(t)1044 3093 y Fs(\017)1083 3057
y Fm(0)1141 3093 y Ft(=)34 b(0)i(mo)s(d)f(2\).)54 b(F)-8
b(or)36 b(an)g(ev)m(en)i(theta)e(c)m(haracteristic)g(the)h(corresp)s
(onding)-118 3209 y(thetan)m(ull)g Fs(\022)s Ft([)377
3173 y Fq(\017)377 3235 y(\017)406 3217 y Fi(0)432 3209
y Ft(]\()p Fs(\034)11 b Ft(\),)40 b(a)d(holomorphic)f(function)h(on)h
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3325 y(and)33 b(it)e(is)h(a)h(mo)s(dular)d(form)h(of)h(w)m(eigh)m(t)h
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(the)g(thetan)m(ulls)g(ha)m(v)m(e)h(b)s(een)g(studied)g(extensiv)m(ely)
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SDict begin H.S end
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Ft(vG)3255 3606 y
SDict begin H.R end
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<</S/S/W 1>>/Subtype /Link/Dest (cite.pmv) cvn H.B /ANN pdfmark end
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b(let)g Fs(A)d Ft(=)f Fs(A)719 3805 y Fq(\034)797 3790
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Fs(a)i Ft(and)f Fs(b)p Ft(,)-118 3915 y(with)26 b(p)s(erio)s(d)g
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b(\034)951 3930 y Fq(b)985 3915 y Ft(.)42 b(Then)28 b
Fs(\022)s Ft([)1378 3879 y Fq(\017)1378 3941 y(\017)1407
3922 y Fi(0)1433 3915 y Ft(]\()p Fs(\034)11 b Ft(\))28
b(=)g Fs(\022)s Ft([)1796 3879 y Fq(\017)1825 3855 y
Fh(\()p Fk(a)p Fh(\))1796 3952 y Fq(\017)1825 3932 y
Fi(0)p Fh(\()p Fk(a)p Fh(\))1932 3915 y Ft(]\()p Fs(\034)2039
3930 y Fq(a)2081 3915 y Ft(\))p Fs(\022)s Ft([)2194 3879
y Fq(\017)2223 3855 y Fh(\()p Fk(b)p Fh(\))2194 3952
y Fq(\017)2223 3932 y Fi(0)p Fh(\()p Fk(b)p Fh(\))2325
3915 y Ft(]\()p Fs(\034)2432 3930 y Fq(b)2467 3915 y
Ft(\))e(and)h(the)g(2)2926 3879 y Fq(a)p Fm(\000)p Fr(1)3058
3915 y Ft(\(2)3145 3879 y Fq(a)3196 3915 y Fo(\000)10
b Ft(1\))g Fo(\001)g Ft(2)3467 3879 y Fq(b)p Fm(\000)p
Fr(1)3592 3915 y Ft(\(2)3679 3879 y Fq(b)3723 3915 y
Fo(\000)g Ft(1\))-118 4046 y(thetan)m(ulls)32 b(with)g
Fs(\017)596 4010 y Fr(\()p Fq(a)p Fr(\))715 4046 y Fo(\001)765
4010 y Fq(t)795 4046 y Fs(\017)834 4010 y Fm(0)p Fr(\()p
Fq(a)p Fr(\))977 4046 y Ft(=)c(1)k(or)g Fs(\017)1320
4010 y Fr(\()p Fq(b)p Fr(\))1432 4046 y Fo(\001)1482
4010 y Fq(t)1512 4046 y Fs(\017)1551 4010 y Fm(0)p Fr(\()p
Fq(b)p Fr(\))1687 4046 y Ft(=)c(1)k(v)-5 b(anish)32 b(in)g
Fs(\034)39 b Fo(2)28 b Fz(H)2550 4061 y Fq(g)2590 4046
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m(o)g(2-)-118 4278 y(dimensional)d(ppa)m(v's)k Fs(B)815
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Fr(2)1102 4278 y Ft(.)70 b(In)42 b(case)g Fs(B)1619 4293
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m(w)m(o)h(elliptic)d(curv)m(es)-118 4395 y(\(with)34
b(pro)s(duct)g(p)s(olarisation\),)d(one)k(thetan)m(ull)e(of)h
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b(and)f(this)g(leads)g(to)g(the)h(v)-5 b(anishing)-118
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b(thetan)m(ulls)g(for)g(the)h(ppa)m(v)h Fs(B)1773 4526
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b(form)-118 635 y Fs(y)-66 594 y Fr(2)55 635 y Ft(=)83
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635 y Fo(\000)22 b Ft(24)p Fs(x)707 594 y Fr(6)747 635
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y Fr(4)3414 635 y Fo(\000)g Ft(8)p Fs(x)3618 594 y Fr(4)3658
635 y Fs(t)3693 594 y Fr(3)297 781 y Fo(\000)p Ft(16)p
Fs(x)527 740 y Fr(4)567 781 y Fs(t)602 740 y Fr(2)664
781 y Fo(\000)g Ft(8)p Fs(x)868 740 y Fr(3)907 781 y
Fs(t)942 740 y Fr(4)1004 781 y Ft(+)f(20)p Fs(x)1255
740 y Fr(3)1295 781 y Fs(t)1330 740 y Fr(3)1391 781 y
Ft(+)g(8)p Fs(x)1593 740 y Fr(3)1633 781 y Fs(t)1668
740 y Fr(2)1730 781 y Fo(\000)g Ft(16)p Fs(x)1982 740
y Fr(2)2022 781 y Fs(t)2057 740 y Fr(4)2119 781 y Ft(+)g(8)p
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781 y Fo(\000)h Ft(9)p Fs(x)2661 740 y Fr(2)2700 781
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740 y Fr(2)3088 781 y Fs(t)g Ft(+)g(32)p Fs(xt)3431 740
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b(jection)30 b(on)m(to)f(the)g(pro)5 b(jectiv)m(e)30
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b(form)d(for)h Fs(J)9 b(ac)p Ft(\()p Fs(X)f Ft(\))33
b(with)f(2-torsion)f(section)i(\(0)p Fs(;)17 b Ft(0\):)385
1568 y Fs(J)9 b(ac)p Ft(\()p Fs(X)f Ft(\))28 b(:)83 b
Fs(y)896 1527 y Fr(2)962 1568 y Ft(=)28 b Fs(x)p Ft(\()p
Fs(x)1214 1527 y Fr(2)1276 1568 y Ft(+)22 b(9\(9)p Fs(u)1566
1527 y Fr(2)1627 1568 y Fo(\000)h Ft(58)p Fs(u)e Ft(+)h(9\)\()p
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1568 y Fs(u)3290 1527 y Fr(2)3329 1568 y Ft(\))p Fs(:)-118
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b(has)g(quotien)m(t)g(a)g(rational)d(elliptic)g(surface)k(with)e
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1977 y Fr(6)3542 1962 y Fs(;)17 b(I)8 b(I)g(I)g Ft(.)42
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b(eil)20 b(rank)j(is)g(t)m(w)m(o.)41 b(By)24 b(the)f(Shio)s(da-T)-8
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2194 y Ft(19)k(as)h(required.)-118 2280 y
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