CoCalc -- Collaborative Calculation in the Cloud
Sharedwww / job / SteinCV.psOpen in CoCalc
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(wship,)f(Spring)g(1999.)894 1746 y Fh(\017)48 b Fj(Vice)33
b(Chancellor)e(researc)m(h)k(gran)m(t)d(\(computing)f(equipmen)m(t\),)i
(1999.)894 1867 y Fh(\017)48 b Fj(Graduate)32 b(studen)m(t)i(researc)m
(her,)h(F)-8 b(all)30 b(1998.)894 1987 y Fh(\017)48 b
Fj(Outstanding)32 b(mathematics)f(senior,)i(1994.)894
2107 y Fh(\017)48 b Fj(Applied)32 b(math)g(mo)s(deling)e(con)m(test,)k
(meritorious)c(ranking,)i(1994.)30 2271 y Ff(IN)k(PROGRESS)169
b Fh(\017)49 b Fd(L)-5 b(e)g(ctur)g(es)43 b(on)g(Serr)-5
b(e's)42 b(c)-5 b(onje)g(ctur)g(es)p Fj(,)43 b(with)e(K.)16
b(A.)42 b(Rib)s(et,)g(to)f(app)s(ear)g(in)g(the)992 2391
y(IAS/P)m(ark)34 b(Cit)m(y)f(Mathematics)f(Institute)g(Lecture)i
(Series.)894 2511 y Fh(\017)48 b Fd(Mo)-5 b(d)28 b Fj(5)g
Fd(appr)-5 b(o)g(aches)26 b(to)j(mo)-5 b(dularity)28
b(of)f(ic)-5 b(osahe)g(dr)g(al)27 b(Galois)g(r)-5 b(epr)g(esentations)p
Fj(,)992 2632 y(with)32 b(K.)16 b(M.)34 b(Buzzard.)894
2752 y Fh(\017)48 b Fd(Explicit)36 b(appr)-5 b(o)g(aches)34
b(to)j(mo)-5 b(dular)35 b(ab)-5 b(elian)35 b(varieties)p
Fj(,)f(UC)g(Berk)m(eley)h(Ph.D.)992 2873 y(thesis)e(under)h(H.)16
b(W.)33 b(Lenstra.)894 2993 y Fh(\017)48 b Fd(He)-5 b(cke:)45
b(The)34 b(mo)-5 b(dular)34 b(forms)g(c)-5 b(alculator)p
Fj(,)33 b(computer)f(soft)m(w)m(are.)894 3113 y Fh(\017)48
b Fd(The)34 b(mo)-5 b(dular)35 b(forms)f(datab)-5 b(ase:)1317
3234 y Fg(http://shimura.math.berk)q(eley)q(.ed)q(u/~w)q(as/T)q(able)q
(s)894 3354 y Fh(\017)48 b Fd(Comp)-5 b(onent)34 b(gr)-5
b(oups)34 b(of)h(optimal)f(quotients)h(of)g(Jac)-5 b(obians)p
Fj(.)894 3474 y Fh(\017)48 b Fd(Visibility)25 b(of)g(Shafar)-5
b(evich-T)e(ate)22 b(gr)-5 b(oups)25 b(of)g(mo)-5 b(dular)24
b(ab)-5 b(elian)24 b(varieties)p Fj(,)f(with)992 3595
y(A.)33 b(Agash)m(\023)-46 b(e.)894 3715 y Fh(\017)48
b Fd(Computing)26 b(analytic)h(invariants)f(of)h(mo)-5
b(dular)27 b(eigenforms)p Fj(,)d(with)g(H.)g(V)-8 b(errill.)894
3836 y Fh(\017)48 b Fd(L)-5 b(e)g(ctur)g(es)49 b(on)f(mo)-5
b(dular)48 b(forms)g(and)g(Galois)h(r)-5 b(epr)g(esentations)p
Fj(,)50 b(with)d(K.)16 b(A.)992 3956 y(Rib)s(et,)32 b(in)m(tended)h
(for)f(Springer-V)-8 b(erlag's)31 b(Univ)m(ersitext)j(series.)30
4119 y Ff(PUBLICA)-9 b(TIONS)96 b Fh(\017)49 b Fd(Empiric)-5
b(al)26 b(evidenc)-5 b(e)26 b(for)h(the)g(Bir)-5 b(ch)27
b(and)f(Swinnerton-Dyer)g(c)-5 b(onje)g(ctur)g(es)27
b(for)992 4240 y(mo)-5 b(dular)31 b(Jac)-5 b(obians)30
b(of)h(genus)g(2)h(curves)p Fj(,)d(with)f(E.)16 b(V.)30
b(Flynn,)f(F.)g(Lepr)m(\023)-46 b(ev)m(ost,)992 4360
y(E.)16 b(F.)33 b(Sc)m(haefer,)h(M.)f(Stoll,)e(J.)16
b(L.)33 b(W)-8 b(etherell,)32 b(submitted.)894 4480 y
Fh(\017)48 b Fd(Parity)39 b(structur)-5 b(es)40 b(and)d(gener)-5
b(ating)38 b(functions)g(fr)-5 b(om)38 b(Bo)-5 b(ole)g(an)37
b(rings)p Fj(,)g(with)992 4601 y(D.)16 b(P)-8 b(.)33
b(Moulton,)f(submitted.)894 4721 y Fh(\017)48 b Fd(F)-7
b(al)5 b(lacies,)56 b(Flaws,)g(and)d(Flim\015am)f(#92:)81
b(A)n(n)53 b(Inductive)g(F)-7 b(al)5 b(lacy)p Fj(,)56
b(with)992 4842 y(A.)33 b(Riskin,)f(College)f(Math.)i(Journal)f(26:5)f
(\(1995\),)h(382.)p eop
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2 1 bop -10 261 a Fk(Wil)10 b(liam)72 b(A.)f(Stein)1703
b Fj(Octob)s(er)32 b(27,)g(1999)209 467 y Fi(\(510\))i(883-9938)88
b Fh(\017)c Fg([email protected])91 b Fh(\017)119
b Fg(http://math.berkeley.edu/)q(~wa)q(s)p 31 506 3886
4 v 30 706 a Ff(TEA)m(CHING)325 b Fe(NSF)38 b(Sp)s(onsored)h(IAS/P)m
(ark)e(Cit)m(y)f(Mathematics)g(Institute)894 826 y Fh(\017)48
b Fd(T)-7 b(e)i(aching)44 b(Assistant)p Fj(,)k(Summer)c(1999.)80
b(Led)45 b(problem)f(sessions)i(and)f(pre-)992 946 y(pared)24
b(notes)g(for)f(Ken)h(Rib)s(et's)f(course)i(on)f(Serre's)g(conjectures)
i(for)d(adv)-5 b(anced)992 1067 y(n)m(um)m(b)s(er)33
b(theory)g(graduate)g(studen)m(ts.)895 1221 y Fe(Univ)m(ersit)m(y)i(of)
j(California)e(at)h(Berk)m(eley)894 1342 y Fh(\017)48
b Fd(Curriculum)43 b(Development)p Fj(,)f(F)-8 b(all)39
b(1997{Summer)g(1998.)69 b(Dev)m(elop)s(ed)42 b(cur-)992
1462 y(riculum)26 b(materials)g(and)j Fc(Ma)-7 b(tlab)28
b Fj(soft)m(w)m(are)h(for)f(w)m(orkshop)i(based)f(calculus)992
1582 y(and)k(linear)e(algebra)g(courses)j(at)f(UC)g(Berk)m(eley)-8
b(.)894 1703 y Fh(\017)48 b Fd(Instructor)p Fj(,)g(Summer)43
b(1997.)79 b(T)-8 b(augh)m(t)45 b(discrete)g(mathematics)e(to)h(a)g
(class)992 1823 y(of)c(20{30)f(undergraduates.)69 b(Duties)40
b(included)g(preparing)g(and)h(deliv)m(ering)992 1944
y(lectures,)33 b(writing)e(and)i(grading)e(exams,)i(and)g(conducting)g
(o\016ce)g(hours.)894 2064 y Fh(\017)48 b Fd(T)-7 b(e)i(aching)40
b(Assistant)p Fj(,)j(F)-8 b(all)38 b(1995{Spring)h(1997.)67
b(Led)41 b(discussion)g(sections)992 2184 y(for)23 b(the)h(full)d
(range)i(of)g(undergraduate)h(linear)e(algebra)g(and)h(calculus)g
(courses.)894 2305 y Fh(\017)48 b Fe(Ev)-6 b(aluation)36
b(a)m(v)m(erage:)51 b(5.8)38 b(out)f(of)h(7)f(from)g(300)g(studen)m
(ts.)895 2459 y(Northern)f(Arizona)h(Univ)m(ersit)m(y)894
2579 y Fh(\017)48 b Fd(T)-7 b(e)i(aching)p Fj(,)43 b(1994{1995.)71
b(T)-8 b(augh)m(t)42 b(t)m(w)m(o)h(semesters)h(of)e(college)f(algebra)g
(and)992 2700 y(other)46 b(freshman)g(topics)g(to)g(a)f(class)i(of)e
(30{40)g(undergraduates.)85 b(Duties)992 2820 y(included)38
b(all)e(asp)s(ects)j(of)f(organizing)d(a)j(course,)j(including)36
b(preparing)h(and)992 2941 y(deliv)m(ering)32 b(lectures)h(and)f
(grading)g(examinations.)30 3104 y Ff(LECTURES)329 b
Fh(\017)49 b Fj(Deliv)m(ered)29 b(o)m(v)m(er)h(t)m(w)m(en)m(t)m(y)i
(talks)d(and)g(organized)g(graduate)g(studen)m(t)i(seminars)992
3224 y(on)i(mo)s(dular)d(forms)i(and)h(ab)s(elian)d(v)-5
b(arieties)32 b(at)g(UC)h(Berk)m(eley)-8 b(.)894 3345
y Fh(\017)48 b Fd(Particip)-5 b(ant)35 b(in)g(p)-5 b(anel)34
b(discussion)g(on)h(the)g(use)g(of)g(te)-5 b(chnolo)g(gy)34
b(in)h(the)g(class-)992 3465 y(r)-5 b(o)g(om)p Fj(,)32
b(P)m(ark)i(Cit)m(y)e(Math.)h(Inst.,)h(July)-8 b(,)32
b(1999.)894 3585 y Fh(\017)48 b Fd(Demonstr)-5 b(ations)45
b(of)h(numb)-5 b(er)45 b(the)-5 b(ory)47 b(softwar)-5
b(e)p Fj(,)47 b(P)m(ark)e(Cit)m(y)g(Math.)h(Inst.,)992
3706 y(July)-8 b(,)32 b(1999.)894 3826 y Fh(\017)48 b
Fd(Shafar)-5 b(evich-T)e(ate)24 b(gr)-5 b(oups)25 b(of)h(mo)-5
b(dular)25 b(ab)-5 b(elian)25 b(varieties)p Fj(,)f(P)m(ark)f(Cit)m(y)g
(Math.)992 3947 y(Inst.,)34 b(June,)f(1999.)894 4067
y Fh(\017)48 b Fd(Shafar)-5 b(evich-T)e(ate)49 b(gr)-5
b(oups)50 b(of)h(mo)-5 b(dular)51 b(ab)-5 b(elian)50
b(varieties)p Fj(,)k(Adv)-5 b(ances)51 b(in)992 4187
y(Num)m(b)s(er)33 b(Theory)-8 b(,)33 b(Leiden,)g(Netherlands,)g(April,)
f(1999.)894 4308 y Fh(\017)48 b Fd(Visibility)35 b(of)g(Shafar)-5
b(evich-T)e(ate)32 b(gr)-5 b(oups)p Fj(,)33 b(Arizona)e(Win)m(ter)i(Sc)
m(ho)s(ol,)f(1999.)30 4505 y Ff(COMPUTING)216 b Fj(Extensiv)m(e)35
b(exp)s(erience)f(with)e Fg(C++)p Fj(,)h(UNIX,)g Fc(Ma)n(gma)p
Fj(,)g(and)g(P)-8 b(ARI.)30 4702 y Ff(ADDRESS)379 b Fj(2041)32
b(F)-8 b(rancisco)32 b(Street,)h(Apt.)g(14)895 4823 y(Berk)m(eley)-8
b(,)33 b(CA)h(94709)895 4943 y(USA)30 5140 y Ff(PERSONAL)312
b Fj(US)33 b(Citizen,)f(b)s(orn)g(F)-8 b(ebruary)g(,)33
b(1974.)p eop
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3 2 bop -10 261 a Fk(Wil)10 b(liam)72 b(A.)f(Stein)1703
b Fj(Octob)s(er)32 b(27,)g(1999)209 467 y Fi(\(510\))i(883-9938)88
b Fh(\017)c Fg([email protected])91 b Fh(\017)119
b Fg(http://math.berkeley.edu/)q(~wa)q(s)p 31 506 3886
4 v 30 706 a Ff(REFERENCES)198 b Fh(\017)49 b Fe(Kevin)37
b(M.)g(Buzzard)992 826 y Fd(+44)d(207)h(594)f(8523)992
946 y Fj(Departmen)m(t)e(of)g(Mathematics)992 1067 y(Huxley)h(Building)
992 1187 y(Imp)s(erial)d(College)992 1308 y(180)i(Queen's)i(Gate)992
1428 y(London,)f(SW7)g(2BZ)992 1548 y(England)992 1669
y Fg([email protected])894 1832 y Fh(\017)48 b Fe(Professor)37
b(Rob)s(ert)g(Coleman)992 1952 y Fd(\(510\))d(642-5101)992
2073 y Fj(Departmen)m(t)e(of)g(Mathematics)h(#3840)992
2193 y(Univ)m(ersit)m(y)g(of)g(California)992 2314 y(Berk)m(eley)-8
b(,)34 b(CA)f(94720-3840)992 2434 y Fg([email protected])q(u)894
2597 y Fh(\017)48 b Fe(Professor)37 b(Hendrik)g(W.)g(Lenstra)992
2718 y Fd(\(510\))d(643-7857)992 2838 y Fj(Departmen)m(t)e(of)g
(Mathematics)h(#3840)992 2958 y(Univ)m(ersit)m(y)g(of)g(California)992
3079 y(Berk)m(eley)-8 b(,)34 b(CA)f(94720-3840)992 3199
y Fg([email protected])894 3362 y Fh(\017)48 b Fe(Professor)37
b(Barry)h(Mazur)992 3483 y Fd(\(617\))c(495-2171)g(ext.)g(512)992
3603 y Fj(Departmen)m(t)e(of)g(Mathematics)992 3724 y(Harv)-5
b(ard)32 b(Univ)m(ersit)m(y)992 3844 y(One)h(Oxford)g(Street)992
3964 y(Cam)m(bridge,)f(MA)h(02138)992 4085 y Fg
([email protected])q(edu)894 4248 y Fh(\017)48
b Fe(Professor)37 b(Lo)-12 b(\177)-44 b(\020c)37 b(Merel)992
4368 y Fg([email protected])894 4532 y Fh(\017)48
b Fe(Professor)37 b(Kenneth)h(A.)f(Rib)s(et)992 4652
y Fd(\(510\))d(642-0648)992 4772 y Fj(Departmen)m(t)e(of)g(Mathematics)
h(#3840)992 4893 y(Univ)m(ersit)m(y)g(of)g(California)992
5013 y(Berk)m(eley)-8 b(,)34 b(CA)f(94720-3840)992 5134
y Fg([email protected])p eop
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4 3 bop -10 261 a Fk(Wil)10 b(liam)72 b(A.)f(Stein)1703
b Fj(Octob)s(er)32 b(27,)g(1999)209 467 y Fi(\(510\))i(883-9938)88
b Fh(\017)c Fg([email protected])91 b Fh(\017)119
b Fg(http://math.berkeley.edu/)q(~wa)q(s)p 31 506 3886
4 v 30 706 a Ff(ABSTRA)m(CTS)241 b Fh(\017)49 b Fd(L)-5
b(e)g(ctur)g(es)35 b(on)f(Serr)-5 b(e's)34 b(c)-5 b(onje)g(ctur)g(es)34
b(\(77)h(p)-5 b(ages\):)42 b Fj(This)32 b(is)g(an)h(exp)s(ository)f
(pa-)992 826 y(p)s(er)27 b(based)g(on)f(Ken)h(Rib)s(et's)e(lectures)i
(at)f(the)h(1999)e(P)m(ark)i(Cit)m(y)g(Mathematics)992
946 y(Institute;)33 b(it)f(will)e(b)s(e)j(published)f(in)g(the)h
(conference)h(pro)s(ceedings.)894 1110 y Fh(\017)48 b
Fd(Mo)-5 b(d)31 b Fj(5)h Fd(appr)-5 b(o)g(aches)30 b(to)h(mo)-5
b(dularity)31 b(of)h(ic)-5 b(osahe)g(dr)g(al)29 b(Galois)i(r)-5
b(epr)g(esentations)992 1230 y(\(16)36 b(p)-5 b(ages\):)46
b Fj(Consider)35 b(a)f(con)m(tin)m(uous)h(o)s(dd)f(irreducible)f
(represen)m(tation)i Fb(\032)c Fj(:)992 1350 y(Gal)o(\()p
1183 1272 85 4 v Fe(Q)p Fb(=)p Fe(Q)p Fj(\))c Fh(!)g
Fj(GL)1730 1365 y Fa(2)1769 1350 y Fj(\()p Fe(C)p Fj(\))p
Fb(:)i Fj(A)f(sp)s(ecial)g(case)h(of)f(a)g(general)g(conjecture)h(of)f
(Artin)992 1471 y(is)h(that)h(the)g Fb(L)p Fj(-function)f
Fb(L)p Fj(\()p Fb(\032;)17 b(s)p Fj(\))30 b(asso)s(ciated)f(to)g
Fb(\032)h Fj(is)f(en)m(tire.)43 b(Buzzard)31 b(and)e(I)992
1591 y(giv)m(e)34 b(new)g(examples)g(of)f(represen)m(tations)i
Fb(\032)e Fj(that)h(satis\014es)g(this)f(conjecture.)992
1712 y(These)42 b(w)m(ere)g(obtained)e(b)m(y)i(applying)d(a)i(recen)m
(t)h(theorem)e(of)g(Buzzard)h(and)992 1832 y(T)-8 b(a)m(ylor)42
b(to)g(a)g(mo)s(d)g(5)g(reduction)g(of)g Fb(\032)p Fj(,)j(com)m(bined)d
(with)g(a)g(computational)992 1952 y(v)m(eri\014cation)32
b(of)g(mo)s(dularit)m(y)f(of)h(a)g(related)g(mo)s(d)g(5)g(represen)m
(tation.)894 2116 y Fh(\017)48 b Fd(Explicit)40 b(appr)-5
b(o)g(aches)40 b(to)h(mo)-5 b(dular)40 b(ab)-5 b(elian)39
b(varieties)i(\(130)f(p)-5 b(ages\):)54 b Fj(I)40 b(giv)m(e)992
2236 y(algorithms)28 b(for)h(computing)g(on)h(mo)s(dular)e(ab)s(elian)g
(v)-5 b(arieties)29 b(of)h(an)m(y)h(dimen-)992 2356 y(sion.)82
b(More)45 b(precisely)-8 b(,)49 b(I)d(describ)s(e)g(ho)m(w)g(to)f
(compute)g(congruences,)51 b(the)992 2477 y(mo)s(dular)20
b(degree,)25 b(the)e(rational)d(part)i(of)f(the)i(sp)s(ecial)e(v)-5
b(alue)22 b(of)g(the)g Fb(L)p Fj(-function)992 2597 y(and)43
b(of)g(its)g(t)m(wists,)k(the)c(comp)s(onen)m(t)h(group)f(at)g(primes)f
(of)h(m)m(ultiplicativ)m(e)992 2718 y(reduction,)j(the)e(p)s(erio)s(d)f
(lattice,)h(and)g(the)g(real)e(and)i(imaginary)c(v)m(olumes.)992
2838 y(There)32 b(are)f(still)e(man)m(y)i(in)m(v)-5 b(arian)m(ts)30
b(that)h(I)g(ha)m(v)m(e)h(not)f(b)s(een)h(able)e(to)g(compute)992
2958 y(in)d(all)f(cases,)31 b(including)26 b(the)j(exact)g(structure)g
(of)f(the)g(torsion)g(subgroup)g(and)992 3079 y(the)39
b(regulator.)61 b(The)40 b(second)g(part)e(of)h(m)m(y)f(thesis)i(con)m
(tains)e(in)m(v)m(estigations)992 3199 y(in)m(to)27 b(sev)m(eral)i(op)s
(en)f(problems,)h(including)d(the)i(Birc)m(h)g(and)g(Swinnerton-Dy)m
(er)992 3319 y(conjecture,)g(Artin's)e(conjecture)h(on)e(complex)h
(Galois)d(represen)m(tations,)28 b(and)992 3440 y(Serre's)34
b(conjecture.)894 3603 y Fh(\017)48 b Fd(He)-5 b(cke:)67
b(The)45 b(mo)-5 b(dular)46 b(forms)f(c)-5 b(alculator:)68
b Fj(Hec)m(k)m(e)47 b(is)d(a)g Fg(C++)i Fj(pac)m(k)-5
b(age)45 b(for)992 3724 y(computing)36 b(with)i(spaces)h(of)e(mo)s
(dular)e(forms)i(and)h(mo)s(dular)d(ab)s(elian)h(v)-5
b(ari-)992 3844 y(eties.)68 b(I)40 b(ha)m(v)m(e)i(b)s(een)g(in)m(vited)
e(to)g(visit)f(the)i Fc(Ma)n(gma)g Fj(group)f(in)g(Sydney)i(in)992
3964 y(order)35 b(to)f(mak)m(e)h(Hec)m(k)m(e)i(a)e(part)f(of)h(their)f
(computer)h(algebra)e(system,)k(and)d(I)992 4085 y(ha)m(v)m(e)g
(already)e(p)s(orted)h(m)m(y)f(co)s(de)h(to)f Fc(Ma)n(gma)p
Fj(.)894 4248 y Fh(\017)48 b Fd(The)39 b(mo)-5 b(dular)40
b(forms)f(datab)-5 b(ase:)54 b Fj(This)38 b(is)g(a)f(collection)g(of)g
(mo)s(dular)f(eigen-)992 4368 y(forms,)30 b(sp)s(ecial)e(v)-5
b(alues)30 b(of)f Fb(L)p Fj(-functions,)h(arithmetic)e(in)m(v)-5
b(arian)m(ts)28 b(of)i(mo)s(dular)992 4489 y(ab)s(elian)25
b(v)-5 b(arieties,)26 b(and)h(other)f(data.)42 b(These)28
b(tables,)f(whic)m(h)g(are)g(freely)f(a)m(v)-5 b(ail-)992
4609 y(able)32 b(on)h(the)g(In)m(ternet,)h(ha)m(v)m(e)f(already)g(b)s
(een)g(used)h(b)m(y)f(man)m(y)g(p)s(eople.)894 4772 y
Fh(\017)48 b Fd(Comp)-5 b(onent)24 b(gr)-5 b(oups)24
b(of)h(optimal)f(quotients)h(of)g(Jac)-5 b(obians)23
b(\(16)i(p)-5 b(ages\):)37 b Fj(Let)22 b Fb(A)992 4893
y Fj(b)s(e)41 b(an)g(optimal)d(quotien)m(t)j(of)g Fb(J)2210
4908 y Fa(0)2249 4893 y Fj(\()p Fb(N)10 b Fj(\).)69 b(The)42
b(main)d(theorem)h(of)h(this)f(pap)s(er)992 5013 y(giv)m(es)d(a)f
(relationship)e(b)s(et)m(w)m(een)k(the)f(mo)s(dular)d(degree)j(of)f
Fb(A)g Fj(and)h(the)f(order)992 5134 y(of)e(the)g(comp)s(onen)m(t)g
(group)g(of)g Fb(A)p Fj(.)48 b(F)-8 b(rom)32 b(this)i(I)g(deduce)i(a)d
(computable)h(for-)p eop
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5 4 bop -10 261 a Fk(Wil)10 b(liam)72 b(A.)f(Stein)1703
b Fj(Octob)s(er)32 b(27,)g(1999)209 467 y Fi(\(510\))i(883-9938)88
b Fh(\017)c Fg([email protected])91 b Fh(\017)119
b Fg(http://math.berkeley.edu/)q(~wa)q(s)p 31 506 3886
4 v 992 706 a Fj(m)m(ula)30 b(for)h(the)g(comp)s(onen)m(t)h(group)f(of)
f(an)m(y)i(optimal)d(quotien)m(t)i(of)g Fb(J)3515 721
y Fa(0)3554 706 y Fj(\()p Fb(N)10 b Fj(\))32 b(at)f(a)992
826 y(prime)f(of)g(m)m(ultiplicativ)m(e)e(reduction.)43
b(I)31 b(then)h(compute)f(o)m(v)m(er)h(one)f(thousand)992
946 y(examples)40 b(leading)d(me)i(to)h(conjecture)g(that)f(the)h
(torsion)f(and)g(comp)s(onen)m(t)992 1067 y(groups)33
b(of)f(quotien)m(ts)h(of)f Fb(J)2006 1082 y Fa(0)2046
1067 y Fj(\()p Fb(p)p Fj(\))g(are)h(as)f(simple)g(as)g(p)s(ossible.)894
1230 y Fh(\017)48 b Fd(Visibility)34 b(of)f(Shafar)-5
b(evich-T)e(ate)32 b(gr)-5 b(oups)33 b(of)g(mo)-5 b(dular)33
b(ab)-5 b(elian)33 b(varieties)g(\(20)992 1350 y(p)-5
b(ages\):)48 b Fj(W)-8 b(e)36 b(study)g(Mazur's)h(notion)d(of)h
(visibilit)m(y)d(of)j(Shafarevic)m(h-T)-8 b(ate)36 b(of)992
1471 y(mo)s(dular)27 b(ab)s(elian)h(v)-5 b(arieties,)29
b(and)h(use)g(it)e(to)h(v)m(erify)h(the)g(conjecture)h(of)e(Birc)m(h)
992 1591 y(and)k(Swinnerton-Dy)m(er)g(for)f(sev)m(eral)h(sp)s(eci\014c)
g(ab)s(elian)e(v)-5 b(arieties.)894 1755 y Fh(\017)48
b Fd(L)-5 b(e)g(ctur)g(es)43 b(on)g(mo)-5 b(dular)42
b(forms)h(and)f(Galois)g(r)-5 b(epr)g(esentations)42
b(\(170)h(p)-5 b(ages\):)992 1875 y Fj(In)29 b(1996,)f(Ken)g(Rib)s(et)f
(taugh)m(t)i(an)f(adv)-5 b(anced)29 b(course)g(on)f(mo)s(dular)e(forms)
h(and)992 1995 y(Galois)g(represen)m(tations.)44 b(In)30
b(collab)s(oration)c(with)j(Rib)s(et,)g(I)h(am)f(turning)f(m)m(y)992
2116 y(course)k(notes)f(in)m(to)f(a)h(b)s(o)s(ok)f(that)h(is)f(in)m
(tended)i(for)e(publication)e(in)i(Springer-)992 2236
y(V)-8 b(erlag's)32 b(Univ)m(ersitext)h(series.)894 2399
y Fh(\017)48 b Fd(Empiric)-5 b(al)32 b(evidenc)-5 b(e)31
b(for)h(the)g(Bir)-5 b(ch)32 b(and)g(Swinnerton-Dyer)f(c)-5
b(onje)g(ctur)g(e)32 b(for)992 2520 y(mo)-5 b(dular)47
b(Jac)-5 b(obians)46 b(of)h(genus)g(2)g(curves)g(\(22)g(p)-5
b(ages\):)70 b Fj(W)-8 b(e)46 b(pro)m(vide)h(sys-)992
2640 y(tematic)f(n)m(umerical)g(evidence)j(for)e(the)h(BSD)f
(conjecture)i(in)e(the)h(case)g(of)992 2760 y(dimension)32
b(t)m(w)m(o.)46 b(This)34 b(conjecture)g(relates)f(six)h(quan)m(tities)
f(asso)s(ciated)g(to)g(a)992 2881 y(Jacobian)40 b(o)m(v)m(er)h(the)g
(rational)d(n)m(um)m(b)s(ers.)68 b(One)41 b(of)f(these)i(quan)m(tities)
e(is)g(the)992 3001 y(size)34 b Fb(S)39 b Fj(of)33 b(the)h(Shafarevic)m
(h-T)-8 b(ate)34 b(group.)47 b(Unable)33 b(to)g(compute)h
Fb(S)39 b Fj(directly)-8 b(,)992 3122 y(w)m(e)46 b(compute)e(the)h
(\014v)m(e)h(other)e(quan)m(tities)h(and)f(solv)m(e)h(for)f(the)h
(conjectural)992 3242 y(v)-5 b(alue)27 b Fb(S)1299 3257
y Fa(?)1363 3242 y Fj(of)g Fb(S)6 b Fj(.)42 b(F)-8 b(or)26
b(all)f(32)i(curv)m(es)i(considered,)g(the)f(real)e(n)m(um)m(b)s(er)i
Fb(S)3578 3257 y Fa(?)3642 3242 y Fj(is)f(v)m(ery)992
3362 y(close)35 b(to)f(either)g(1,)g(2,)h(or)f(4,)g(and)h(agrees)g
(with)f(the)h(size)f(of)g(the)h(2-torsion)e(of)992 3483
y(the)g(Shafarevic)m(h-T)-8 b(ate)33 b(group,)f(whic)m(h)i(w)m(e)f
(could)f(compute.)894 3646 y Fh(\017)48 b Fd(Parity)k(structur)-5
b(es)53 b(and)e(gener)-5 b(ating)51 b(functions)g(fr)-5
b(om)51 b(Bo)-5 b(ole)g(an)51 b(rings)71 b(\(8)992 3766
y(p)-5 b(ages\):)68 b Fj(Let)46 b Fb(S)52 b Fj(b)s(e)46
b(a)f(\014nite)g(set)i(and)e Fb(T)60 b Fj(b)s(e)45 b(a)h(subset)h(of)e
(the)h(p)s(o)m(w)m(er)h(set)992 3887 y(of)33 b Fb(S)6
b Fj(.)45 b(Call)32 b Fb(T)46 b Fj(a)33 b Fd(p)-5 b(arity)36
b(structur)-5 b(e)34 b Fj(for)f Fb(S)39 b Fj(if,)32 b(for)h(eac)m(h)h
(subset)h Fb(b)e Fj(of)g Fb(S)39 b Fj(of)33 b(o)s(dd)992
4007 y(size,)46 b(the)d(n)m(um)m(b)s(er)g(of)g(subsets)i(of)d
Fb(b)h Fj(that)g(lie)e(in)h Fb(T)57 b Fj(is)42 b(ev)m(en.)76
b(W)-8 b(e)43 b(classify)992 4128 y(parit)m(y)29 b(structures)i(using)e
(generating)g(functions)g(from)g(a)g(free)h(b)s(o)s(olean)d(ring.)992
4248 y(W)-8 b(e)36 b(also)f(sho)m(w)i(that)e(if)g Fb(T)49
b Fj(is)35 b(a)h(parit)m(y)f(structure,)j(then,)f(for)e(eac)m(h)i
(subset)g Fb(b)992 4368 y Fj(of)27 b Fb(S)32 b Fj(of)27
b(ev)m(en)i(size,)f(the)f(n)m(um)m(b)s(er)h(of)e(subsets)j(of)e
Fb(b)g Fj(of)g(o)s(dd)g(size)g(that)g(lie)e(in)h Fb(T)41
b Fj(is)992 4489 y(ev)m(en.)52 b(W)-8 b(e)36 b(then)f(giv)m(e)g(sev)m
(eral)h(other)f(prop)s(erties)g(of)f(parit)m(y)h(structures)h(and)992
4609 y(discuss)e(a)e(generalization.)p eop
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end
userdict /end-hook known{end-hook}if
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