CoCalc -- Collaborative Calculation in the Cloud
Sharedwww / job / SteinProp.psOpen in CoCalc
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393 y Fq(Researc)l(h)46 b(Plan)1732 525 y Fr(William)28
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b Fr(.)53 b(Let)35 b Fp(J)1789 407 y Fm(0)1829 393 y
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1164 y Fp(L)p Fr(\()p Fp(A;)15 b Fr(1\))p 1430 1205 287
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1079 y Fh(Q)2464 1174 y Fi(p)p Fg(j)p Fi(N)2602 1147
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(princi-)148 1561 y(ple;)k(the)f(T)-8 b(amaga)m(w)m(a)31
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b(expressed)h(as)h(the)g(lattice)f(index)f(of)i(t)m(w)m(o)148
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y Fr(in)f Fp(J)2168 5249 y Fm(0)2208 5235 y Fr(\()p Fp(N)10
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4 3 bop 148 94 a Fs(William)28 b(A.)j(Stein)3122 b Fr(4)148
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Fp(p)p Fr(-torsion)148 808 y(are,)g(according)g(to)g(E.)f(Kani's)f
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(understand)d(what)i(happ)s(ens)148 940 y(in)e(all)h(dimensions.)289
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y(the)37 b(Shafarevic)m(h-T)-8 b(ate)37 b(group)e(of)i(an)f(ab)s(elian)
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b(con)m(trast,)k(o)m(v)m(er)e(all)148 1468 y(optimal)28
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1600 y(a)i(Shafarevic)m(h-T)-8 b(ate)31 b(group)f(is)g(7.)148
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(irreducible)d(represen)m(tation)148 2318 y Fp(\032)26
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2582 y(This)d(case)i(of)f(Artin's)f(conjecture)i(is)e(kno)m(wn)h(when)f
(the)h(image)h(of)f Fp(\032)g Fr(in)f(PGL)2944 2596 y
Fm(2)2984 2582 y Fr(\()p Fn(C)p Fr(\))h(is)f(solv)-5
b(able)32 b(\(see)i([27)q(]\),)148 2714 y(and)c(in)f(in\014nitely)e
(man)m(y)k(cases)g(when)f(the)g(image)h(of)f Fp(\032)h
Fr(is)e(not)i(solv)-5 b(able)29 b(\(see)i([7)q(]\).)289
2846 y(In)20 b(1998,)25 b(K.)20 b(Buzzard)g(suggested)i(a)e(w)m(a)m(y)i
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m(wing)f(theorem.)148 3317 y Fn(Theorem)34 b(6.)46 b
Fl(The)33 b(ic)-5 b(osahe)g(dr)g(al)36 b(A)n(rtin)c(r)-5
b(epr)g(esentations)36 b(of)d(c)-5 b(onductor)34 b Fr(1376)27
b(=)e(2)3115 3284 y Fm(5)3175 3317 y Fk(\001)c Fr(43)33
b Fl(ar)-5 b(e)34 b(mo)-5 b(dular.)289 3524 y Fr(W)d(e)26
b(exp)s(ect)e(our)f(metho)s(d)h(to)g(yield)f(sev)m(eral)h(more)g
(examples.)38 b(These)24 b(ongoing)g(computations)g(are)g(la)m(ying)148
3656 y(a)k(small)e(part)h(of)h(the)g(tec)m(hnical)f(foundations)f
(necessary)i(for)f(a)h(full)d(pro)s(of)i(of)g(the)h(Artin)e(conjecture)
i(for)f(o)s(dd)148 3788 y(t)m(w)m(o)k(dimensional)c Fp(\032)p
Fr(,)j(as)g(w)m(ell)e(as)i(stim)m(ulating)e(the)i(dev)m(elopmen)m(t)g
(of)g(new)f(algorithms)f(for)i(computing)f(with)148 3920
y(mo)s(dular)g(forms)g(using)g(mo)s(dular)g(sym)m(b)s(ols)g(in)g(c)m
(haracteristic)i Fp(`)p Fr(.)148 4163 y Fo(3.2)113 b(Cyclotomic)34
b(p)s(oin)m(ts)j(on)h(mo)s(dular)e(curv)m(es)148 4315
y Fr(If)42 b Fp(E)48 b Fr(is)41 b(an)i(elliptic)d(curv)m(e)i(o)m(v)m
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4461 y Fi(p)3728 4447 y Fr(\).)67 b(F)-8 b(or)148 4579
y(whic)m(h)33 b(primes)g Fp(p)h Fr(do)s(es)g(there)h(exist)f(an)g
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b Fn(Q)p Fr(\()p Fp(\026)525 4725 y Fi(p)564 4711 y Fr(\)?)78
b(When)43 b Fp(p)j Fr(=)f(2)p Fp(;)15 b Fr(3)p Fp(;)g
Fr(5)45 b(the)e(corresp)s(onding)e(mo)s(duli)f(space)k(has)e(gen)m(us)h
(zero)h(and)e(in\014nitely)148 4843 y(man)m(y)35 b(examples)f(exist.)53
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(computations)i(he)f(enlisted)f(me)i(to)148 4975 y(do,)i(suggest)f
(that)g(these)g(are)f(the)h(only)e(primes)g Fp(p)g Fr(for)h(whic)m(h)f
(suc)m(h)h(elliptic)e(curv)m(es)j(exist.)55 b(In)34 b([19)q(],)j(Merel)
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Fp(p)p Fr(,)h(on)g(a)g(case-b)m(y-case)j(basis.)39 b(Theoretical)29
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b(and)g(c)m(haracter)i(groups)e(of)g(tori,)h(giv)m(e)f(the)h(follo)m
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b Fk(\021)g Fr(3)41 b(\(mo)s(d)30 b(4\))g Fl(b)-5 b(e)30
b(a)g(prime)g(satisfying)g Fr(7)c Fk(\024)f Fp(p)g(<)g
Fr(1000)p Fl(.)43 b(Ther)-5 b(e)30 b(ar)-5 b(e)31 b(no)f(el)5
b(liptic)30 b(curves)148 5841 y(over)j Fn(Q)p Fr(\()p
Fp(\026)518 5855 y Fi(p)558 5841 y Fr(\))f Fl(al)5 b(l)34
b(of)e(whose)i Fp(p)p Fl(-torsion)g(is)e(r)-5 b(ational)35
b(over)e Fn(Q)p Fr(\()p Fp(\026)2321 5855 y Fi(p)2361
5841 y Fr(\))p Fl(.)p eop
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5 4 bop 148 94 a Fs(William)28 b(A.)j(Stein)3122 b Fr(5)289
393 y(The)36 b(case)h(in)e(whic)m(h)g Fp(p)h Fr(is)f(congruen)m(t)i(to)
g(1)f(mo)s(dulo)f(4)h(presen)m(ts)h(additional)d(di\016culties)f(that)k
(in)m(v)m(olv)m(e)148 525 y(sho)m(wing)30 b(that)h Fp(Y)20
b Fr(\()p Fp(p)p Fr(\))30 b(has)h(no)f Fn(Q)p Fr(\()1314
468 y Fk(p)p 1390 468 46 4 v 57 x Fp(p)o Fr(\)-rational)h(p)s(oin)m
(ts.)39 b(W)-8 b(e)32 b(hop)s(e)d(to)j(tac)m(kle)f(these)g(in)e(the)i
(near)f(future.)148 768 y Fo(3.3)113 b(Gen)m(us)38 b(one)g(curv)m(es)
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Fp(C)28 b Fr(o)m(v)m(er)23 b Fn(Q)e Fr(is)g(the)h(order)f(of)h(the)f
(cok)m(ernel)i(of)e(the)h(degree)h(map)e(Div)3665 934
y Fb(Q)3730 920 y Fr(\()p Fp(C)7 b Fr(\))25 b Fk(!)148
1052 y Fn(Z)p Fr(;)31 b(rationalit)m(y)e(of)h(the)h(canonical)e
(divisor)f(implies)g(that)i(the)h(index)d(divides)g(2)p
Fp(g)c Fk(\000)19 b Fr(2,)31 b(where)f Fp(g)k Fr(is)29
b(the)h(gen)m(us)148 1184 y(of)i Fp(C)7 b Fr(.)43 b(When)31
b Fp(g)f Fr(=)d(1)32 b(this)e(is)g(no)h(condition)f(at)i(all;)f(Artin)f
(conjectured,)j(and)d(Lang)i(and)e(T)-8 b(ate)33 b([14)q(])f(pro)m(v)m
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Fp(m)e Fr(there)h(is)f(a)i(gen)m(us)f(one)g(curv)m(e)g(of)g(index)f
Fp(m)h Fr(o)m(v)m(er)h(some)f(n)m(um)m(b)s(er)f(\014eld.)59
b(Their)148 1448 y(construction)27 b(yields)f(gen)m(us)i(one)g(curv)m
(es)g(o)m(v)m(er)g Fn(Q)f Fr(only)g(for)g(a)h(few)g(v)-5
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1580 y(can)k(\014nd)e(gen)m(us)h(one)h(curv)m(es)g(o)m(v)m(er)g
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y Fn(Theorem)34 b(8.)46 b Fl(L)-5 b(et)30 b Fp(K)37 b
Fl(b)-5 b(e)30 b(any)h(numb)-5 b(er)30 b(\014eld.)42
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Fp(K)37 b Fl(of)30 b(every)g(o)-5 b(dd)32 b(index.)289
1993 y Fr(The)37 b(pro)s(of)e(in)m(v)m(olv)m(es)i(sho)m(wing)f(that)h
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b(Euler)g(system)i(of)148 2125 y(Heegner)27 b(p)s(oin)m(ts)d(do)h(not)h
(v)-5 b(anish)24 b(com)m(bined)g(with)g(explicit)g(Heegner)i(p)s(oin)m
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2257 y(that)g(curv)m(es)g(of)f(ev)m(ery)h(index)e(o)s(ccur,)i(and)f(to)
h(determine)e(the)i(consequences)g(of)f(m)m(y)g(non)m(v)-5
b(anishing)23 b(result)h(for)148 2389 y(Selmer)k(groups.)40
b(This)27 b(can)i(b)s(e)f(view)m(ed)h(as)g(a)g(con)m(tribution)e(to)j
(the)f(problem)e(of)i(understanding)e Fp(H)3662 2356
y Fm(1)3701 2389 y Fr(\()p Fn(Q)p Fp(;)15 b(E)5 b Fr(\).)148
2633 y Fo(3.4)113 b(Serre's)37 b(conjecture)h(mo)s(dulo)e
Ff(pq)148 2784 y Fr(Let)c Fp(p)f Fr(and)f Fp(q)k Fr(b)s(e)c(primes,)g
(and)h(consider)f(a)h(con)m(tin)m(uous)g(represen)m(tation)g
Fp(\032)c Fr(:)f Fp(G)2942 2798 y Fb(Q)3034 2784 y Fk(!)g
Fr(GL\(2)p Fp(;)15 b Fn(Z)p Fp(=pq)s Fn(Z)p Fr(\))33
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Fp(q)k Fr(are)e(b)s(oth)f(irreducible.)48 b(Call)32 b
Fp(\032)148 3048 y Fl(mo)-5 b(dular)29 b Fr(if)d(there)h(is)f(a)h(mo)s
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b Fr(is)26 b(the)h(mo)s(d)f Fp(p)148 3180 y Fr(reduction)33
b(of)h Fp(\032)p Fr(,)h(and)e(ditto)h(for)g Fp(q)s Fr(.)51
b(I)33 b(ha)m(v)m(e)j(carried)d(out)h(sp)s(eci\014c)e(computations)i
(suggested)h(b)m(y)e(Mazur)i(in)148 3312 y(hop)s(es)28
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3444 y(computation)32 b(suggests)h(that)f(the)g(righ)m(t)f(conjectures)
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(to)h Fp(\032)g Fr(mo)s(d)f Fp(p)h Fr(and)f(another)h(giving)f(rise)g
(to)148 3708 y Fp(\032)30 b Fr(mo)s(d)f Fp(q)s Fr(;)h(w)m(e)g(hop)s(e)f
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(form)g(giving)f(rise)h(to)h(b)s(oth)f(reductions.)148
3995 y Fq(References)194 4217 y Fr([1])46 b(A.)32 b(Agash)m(\023)-43
b(e,)34 b Fl(On)g(invisible)f(elements)h(of)g(the)g(Tate-Shafar)-5
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i(S)m(\023)-43 b(er.)335 4349 y(I)30 b(Math.)i Fn(328)f
Fr(\(1999\),)i(no.)d(5,)h(369{374.)194 4497 y([2])46
b(A.)23 b(Agash)m(\023)-43 b(e)25 b(and)d(W.)16 b(A.)23
b(Stein,)h Fl(Visibility)h(of)h(Shafar)-5 b(evich-Tate)27
b(gr)-5 b(oups)27 b(of)f(mo)-5 b(dular)28 b(ab)-5 b(elian)27
b(varieties)p Fr(,)335 4629 y(in)i(preparation)h(\(1999\).)194
4777 y([3])46 b(E.)34 b(Artin,)746 4754 y Fl(\177)730
4777 y(Ub)-5 b(er)35 b(eine)g(neue)f(Art)i(von)f(L-Reihen)p
Fr(,)f(Abh.)f(Math.)h(Sem.)f(in)f(Univ.)h(Ham)m(burg)h
Fn(3)g Fr(\(1923\),)335 4909 y(89{108.)194 5057 y([4])46
b(B.)16 b(J.)30 b(Birc)m(h,)g Fl(El)5 b(liptic)32 b(curves)f(over)i
Fn(Q)p Fl(:)41 b(A)32 b(pr)-5 b(o)g(gr)g(ess)34 b(r)-5
b(ep)g(ort)p Fr(,)33 b(1969)e(Num)m(b)s(er)e(Theory)h(Institute)f
(\(Pro)s(c.)335 5189 y(Symp)s(os.)40 b(Pure)g(Math.,)k(V)-8
b(ol.)41 b(XX,)g(State)h(Univ.)e(New)h(Y)-8 b(ork,)44
b(Ston)m(y)c(Bro)s(ok,)k(N.Y.,)h(1969\),)g(Amer.)335
5321 y(Math.)31 b(So)s(c.,)g(Pro)m(vidence,)g(R.I.,)f(1971,)j(pp.)c
(396{400.)194 5469 y([5])46 b(S.)36 b(Blo)s(c)m(h)g(and)f(K.)h(Kato,)j
Fp(L)p Fl(-functions)e(and)i(Tamagawa)h(numb)-5 b(ers)38
b(of)g(motives)p Fr(,)h(The)c(Grothendiec)m(k)335 5601
y(F)-8 b(estsc)m(hrift,)31 b(V)-8 b(ol.)31 b(I,)g(Birkh\177)-45
b(auser)28 b(Boston,)k(Boston,)g(MA,)f(1990,)h(pp.)e(333{400.)194
5749 y([6])46 b(C.)22 b(Breuil,)g(B.)g(Conrad,)g(F.)g(Diamond,)h(and)e
(R.)h(T)-8 b(a)m(ylor,)24 b Fl(On)g(the)h(mo)-5 b(dularity)27
b(of)e(el)5 b(liptic)24 b(curves)h(over)g Fn(Q)p Fr(,)335
5881 y(in)k(preparation.)p eop
%%Page: 6 6
6 5 bop 148 94 a Fs(William)28 b(A.)j(Stein)3122 b Fr(6)194
393 y([7])46 b(K.)31 b(Buzzard,)g(M.)g(Dic)m(kinson,)f(N.)h
(Shepherd-Barron,)e(and)h(R.)h(T)-8 b(a)m(ylor,)31 b
Fl(On)i(ic)-5 b(osahe)g(dr)g(al)36 b(Artin)d(r)-5 b(epr)g(e-)335
525 y(sentations)p Fr(,)32 b(a)m(v)-5 b(ailable)30 b(at)h
Fa(http://www.math.harvard.e)o(du/~)o(rta)o(ylor)o(/)p
Fr(.)194 658 y([8])46 b(K.)27 b(Buzzard)g(and)f(R.)h(T)-8
b(a)m(ylor,)28 b Fl(Comp)-5 b(anion)32 b(forms)e(and)g(weight)g(one)g
(forms)p Fr(,)f(Annals)c(of)i(Math.)h(\(1999\).)194 805
y([9])46 b(J.)15 b(E.)44 b(Cremona,)i Fl(A)n(lgorithms)f(for)g(mo)-5
b(dular)47 b(el)5 b(liptic)45 b(curves)p Fr(,)h(second)d(ed.,)k(Cam)m
(bridge)42 b(Univ)m(ersit)m(y)335 937 y(Press,)30 b(Cam)m(bridge,)g
(1997.)148 1070 y([10])47 b(J.)15 b(E.)21 b(Cremona)g(and)g(B.)g
(Mazur,)j Fl(Visualizing)g(elements)g(in)g(the)h(Shafar)-5
b(evich-T)e(ate)26 b(gr)-5 b(oup)p Fr(,)24 b(Pro)s(ceedings)335
1202 y(of)31 b(the)f(Arizona)g(Win)m(ter)h(Sc)m(ho)s(ol)f(\(1998\).)148
1335 y([11])47 b(N.)16 b(D.)32 b(Elkies,)f Fl(El)5 b(liptic)34
b(and)h(mo)-5 b(dular)37 b(curves)d(over)g(\014nite)g(\014elds)h(and)g
(r)-5 b(elate)g(d)36 b(c)-5 b(omputational)37 b(issues)p
Fr(,)335 1467 y(Computational)26 b(p)s(ersp)s(ectiv)m(es)f(on)i(n)m(um)
m(b)s(er)e(theory)i(\(Chicago,)h(IL,)f(1995\),)j(Amer.)c(Math.)i(So)s
(c.,)g(Pro)m(v-)335 1599 y(idence,)i(RI,)h(1998,)h(pp.)d(21{76.)148
1732 y([12])47 b(V.)16 b(A.)28 b(Kolyv)-5 b(agin,)28
b Fl(On)i(the)h(structur)-5 b(e)31 b(of)g(Shafar)-5 b(evich-Tate)32
b(gr)-5 b(oups)p Fr(,)31 b(Algebraic)c(geometry)j(\(Chicago,)335
1864 y(IL,)g(1989\),)j(Springer,)c(Berlin,)f(1991,)33
b(pp.)c(94{121.)148 1997 y([13])47 b(V.)16 b(A.)34 b(Kolyv)-5
b(agin)33 b(and)g(D.)16 b(Y.)34 b(Logac)m(hev,)j Fl(Finiteness)f(of)g
Fj(X)g Fl(over)g(total)5 b(ly)38 b(r)-5 b(e)g(al)37 b(\014elds)p
Fr(,)f(Math.)e(USSR)335 2129 y(Izv)m(estiy)m(a)e Fn(39)f
Fr(\(1992\),)i(no.)d(1,)h(829{853.)148 2262 y([14])47
b(S.)36 b(Lang)h(and)e(J.)i(T)-8 b(ate,)39 b Fl(Princip)-5
b(al)39 b(homo)-5 b(gene)g(ous)40 b(sp)-5 b(ac)g(es)40
b(over)e(ab)-5 b(elian)39 b(varieties)p Fr(,)g(Amer.)d(J.)g(Math.)335
2394 y Fn(80)31 b Fr(\(1958\),)i(659{684.)148 2527 y([15])47
b(B.)30 b(Mazur,)f Fl(R)-5 b(ational)34 b(p)-5 b(oints)33
b(of)e(ab)-5 b(elian)32 b(varieties)g(with)g(values)f(in)g(towers)i(of)
e(numb)-5 b(er)32 b(\014elds)p Fr(,)e(In)m(v)m(en)m(t.)335
2659 y(Math.)h Fn(18)g Fr(\(1972\),)i(183{266.)148 2792
y([16])p 335 2792 273 4 v 335 w(,)23 b Fl(Mo)-5 b(dular)25
b(curves)e(and)h(the)g(Eisenstein)g(ide)-5 b(al)p Fr(,)23
b(Inst.)e(Hautes)2768 2769 y(\023)2760 2792 y(Etudes)f(Sci.)f(Publ.)g
(Math.)j(\(1977\),)335 2924 y(no.)31 b(47,)g(33{186)i(\(1978\).)148
3056 y([17])p 335 3056 V 335 w(,)c Fl(R)-5 b(ational)32
b(iso)-5 b(genies)31 b(of)g(prime)g(de)-5 b(gr)g(e)g(e)31
b(\(with)g(an)g(app)-5 b(endix)33 b(by)d(D.)g(Goldfeld\))p
Fr(,)g(In)m(v)m(en)m(t.)f(Math.)335 3188 y Fn(44)i Fr(\(1978\),)i(no.)e
(2,)g(129{162.)148 3321 y([18])p 335 3321 V 335 w(,)g
Fl(Visualizing)h(elements)h(of)g(or)-5 b(der)34 b(thr)-5
b(e)g(e)34 b(in)f(the)g(Shafar)-5 b(evich-Tate)34 b(gr)-5
b(oup)p Fr(,)32 b(preprin)m(t)c(\(1999\).)148 3454 y([19])47
b(L.)g(Merel,)k Fl(Sur)c(la)h(natur)-5 b(e)49 b(non-cyclotomique)f(des)
g(p)-5 b(oints)49 b(d'or)-5 b(dr)g(e)50 b(\014ni)e(des)g(c)-5
b(ourb)g(es)48 b(el)5 b(liptiques)p Fr(,)335 3586 y(preprin)m(t)29
b(\(1999\).)148 3719 y([20])47 b(J.-F.)39 b(Mestre)f(and)f(J.)h
(Oesterl)m(\023)-43 b(e,)41 b Fl(Courb)-5 b(es)40 b(de)f(Weil)g
(semi-stables)h(de)f(discriminant)i(une)e(puissanc)-5
b(e)335 3851 y Fp(m)p Fl(-i)n(\022)-44 b(eme)p Fr(,)30
b(J.)g(Reine)g(Angew.)h(Math.)g Fn(400)g Fr(\(1989\),)i(173{184.)148
3984 y([21])47 b(K.)15 b(A.)34 b(Rib)s(et,)e Fl(On)j(mo)-5
b(dular)37 b(r)-5 b(epr)g(esentations)38 b(of)d Fr(Gal\()p
2280 3911 79 4 v Fn(Q)p Fp(=)p Fn(Q)p Fr(\))g Fl(arising)h(fr)-5
b(om)36 b(mo)-5 b(dular)37 b(forms)p Fr(,)e(In)m(v)m(en)m(t.)335
4116 y(Math.)c Fn(100)h Fr(\(1990\),)h(no.)d(2,)h(431{476.)148
4249 y([22])p 335 4249 273 4 v 335 w(,)39 b Fl(R)-5 b(aising)39
b(the)h(levels)e(of)h(mo)-5 b(dular)41 b(r)-5 b(epr)g(esentations)p
Fr(,)42 b(S)m(\023)-43 b(eminaire)36 b(de)h(Th)m(\023)-43
b(eorie)37 b(des)g(Nom)m(bres,)335 4381 y(P)m(aris)30
b(1987{88,)k(Birkh\177)-45 b(auser)29 b(Boston,)i(Boston,)h(MA,)f
(1990,)h(pp.)e(259{271.)148 4514 y([23])47 b(A.)16 b(J.)31
b(Sc)m(holl,)g Fl(A)n(n)h(intr)-5 b(o)g(duction)36 b(to)e(Kato's)h
(Euler)e(systems)p Fr(,)g(Galois)e(Represen)m(tations)g(in)f
(Arithmetic)335 4646 y(Algebraic)g(Geometry)-8 b(,)32
b(Cam)m(bridge)e(Univ)m(ersit)m(y)f(Press,)h(1998,)j(pp.)c(379{460.)148
4779 y([24])47 b(J-P)-8 b(.)36 b(Serre,)h Fl(Sur)g(les)h(r)-5
b(epr)n(\023)-44 b(esentations)40 b(mo)-5 b(dulair)g(es)40
b(de)e(de)-5 b(gr)n(\023)-44 b(e)38 b Fr(2)f Fl(de)h
Fr(Gal\()p 2959 4705 79 4 v Fn(Q)p Fp(=)p Fn(Q)p Fr(\),)f(Duk)m(e)g
(Math.)f(J.)g Fn(54)335 4911 y Fr(\(1987\),)d(no.)e(1,)g(179{230.)148
5044 y([25])47 b(W.)16 b(A.)31 b(Stein,)e Fl(Comp)-5
b(onent)35 b(gr)-5 b(oups)34 b(of)f(optimal)i(quotients)e(of)f(Jac)-5
b(obians)p Fr(,)32 b(preprin)m(t)d(\(1999\).)148 5176
y([26])47 b(R.)28 b(T)-8 b(a)m(ylor)28 b(and)f(A.)15
b(J.)28 b(Wiles,)g Fl(R)n(ing-the)-5 b(or)g(etic)31 b(pr)-5
b(op)g(erties)32 b(of)f(c)-5 b(ertain)31 b(He)-5 b(cke)29
b(algebr)-5 b(as)p Fr(,)30 b(Ann.)d(of)g(Math.)335 5308
y(\(2\))32 b Fn(141)f Fr(\(1995\),)i(no.)d(3,)h(553{572.)148
5441 y([27])47 b(J.)32 b(T)-8 b(unnell,)29 b Fl(A)n(rtin)-9
b('s)33 b(c)-5 b(onje)g(ctur)g(e)35 b(for)f(r)-5 b(epr)g(esentations)36
b(of)e(o)-5 b(ctahe)g(dr)g(al)37 b(typ)-5 b(e)p Fr(,)33
b(Bull.)d(Amer.)i(Math.)g(So)s(c.)335 5573 y(\(N.S.\))f
Fn(5)g Fr(\(1981\),)i(no.)e(2,)g(173{175.)148 5706 y([28])47
b(A.)16 b(J.)24 b(Wiles,)g Fl(Mo)-5 b(dular)28 b(el)5
b(liptic)27 b(curves)g(and)g(Fermat's)h(last)g(the)-5
b(or)g(em)p Fr(,)28 b(Ann.)23 b(of)h(Math.)h(\(2\))g
Fn(141)g Fr(\(1995\),)335 5838 y(no.)31 b(3,)g(443{551.)p
eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF