Workshop on Arithmetic Geometry in Kanazawa (Nov, 2009)

On the history of the Sato-Tate conjecture

Tetsushi Ito (Kyoto Univ, Math Dept)
November 24(Tue), 10:50-12:00
Ishikawa prefecture bunkyo hall, Kanazawa, Japan
1

This talk
• No serious mathematics• Just relax, smile, have tea & snacks

Are you ready for this? Go ahead!
2

Welcome to Japan,
andwelcome to Kanazawa!

3

By the way,4

Where is Kanazawa?

5

6Kanazawa• is the biggest city in the Hokuriku region,• has a population of 450,000,•

is a castletown ruled over by the Maeda family forthree centuries after the ﬁrst lord Toshiie Maeda.
• In the Edo Era (1603-1868), Maeda family was thebiggest feudal lord (daimyo) except for the Tokugawa
Shogunate.

“Kaga Hyakuman-goku” (cf. Banquet :Kanazawa Daimyo Jaya)

7

Who is Maeda?

8

Toshiie Maeda (1537-99, ﬁrst lord)

9And then,10Why Math in Kanazawa?

11

Takeshi Saito was born in Kanazawa.

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Necessary conditions for Good Math
• Historical and beautiful city. “small Kyoto”
Kenrokuen is one of Japan’s “three most beautiful
landscape gardens”• Delicious seafoods and local speciality• At least 4 Starbucks in the city13

First of all
Thanks to all the speakers, participants, especially
Teruyoshi and Kai-Wen for helping me a lot.

Five-year program (2009-2012)
Supp by JSPS Grant-in-Aid for Young Scientists (S)“Comprehensive studies on Shimura varieties, arithmetic geometry, Galois representations, and automorphic representations”

Kyoto (Nov 2008), Ehime (Feb 2009), Kyoto (Apr2009), Kesennuma (July 2009), Kanazawa (Nov 2009)...

14

Kyoto (Nov 2008)

15

Kyoto (Nov 2008) continued

16

Ehime (Feb 2009)17

Kyoto (Apr 2009)
Mini-workshop on Iwasawa theory18

Kesennuma (July 2009)19

Bad news...
Unfortunate eﬀect of economic crisis and change ofadministration party•

JSPS called oﬀ application to “Young Scientists(S)” after next year (Oct 16).
•

The Japanese government decided to reduce thescientiﬁc research funds (Kakenhi) (Nov 13).
• Situation of our grant after next year is unclear...

20

Good news!
Finally, I can gather excellent young researchers inarithmetic geometry and related area.
Thank you very much for coming.

I’m very looking

forward to attending the lectures!

21

Plan of this talk1. General introduction — ﬁnished
2. History of “Sato’s conjecture”
Tea & Coﬀee Break (10min)
4. Birth of “Sato’s conjecture”
5. Outline of the proof of Sato-Tate conj
(oversimpliﬁed exposition)22

I’m sorry...
Big overlap between this talk and my talks at• RIMS Workshop (Dec 2006)• Workshop on the Sato-Tate conjecture(Jan 2007, Tokyo Inst Tech)• Ehime workshop (Feb 2009)• Annual meeting of the Math Soc of Japan(Mar 2009) (∃ ashamed video streaming)

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History of the Sato-Tate conjecture
Recall: An elliptic curve E over Q is deﬁned by a cubic
equation of the form:
E : y 2 = x3 + ax + b(a, b ∈ Z, 4a3 + 27b2 6= 0)

We would like to study the arithmetic of E.
→ Key : Reciprocity Law
24

The Problem of Reciprocity Law
Problem: For a scheme X of ﬁnite type over Spec Z,
count #X(Fp) and vary p.
Very diﬃcult problem even if X is ﬁnite over Spec Z.
The answer for
X = Spec Z[T ]/(T 2 − a)is given by the Law of Quadratic Reciprocity. Striking
fact is that the answer depends only on p (mod 4a).

25

Class Field Theory
Class Field Theory (Takagi, Artin, 1920’s) tells us
the answer depends only on p (mod N ) (for some N )if and only if X ⊗Z Q is a “abelian extension” of Q.
“Sato-Tate Conj”= “Non-abelian Reciprocity Law for elliptic curves”

26Class Field Theory (2)A surprising discovery of Takagi is Class Field Theory
exists for all number ﬁelds K. It was expected only for√
Q or Q( −D) (Kronecker’s Jugendtraum) or unramiﬁed
extensions of K (Hilbert Class Field). The fact
“All abel ext of all number ﬁelds are class ﬁelds”
was not even expected before.

27

Birth of “Langlands’ Functoriality”(cf. R. P. Langlands (1936–))

The proof of CFT was complete by Takagi after considering CFT of all number ﬁelds simultaneously.

“Langlands’ Functoriality” = relation between CFT’s
for diﬀerent groups/number ﬁelds
Inevitable tool to understand many thms & phenomena in number theory & rep theory(Non-abel CFT(= Gal v.s. Autom), BC/AI, Endoscopy,
Sato-Tate, Formal deg conj (cf. Ichino’s talk))28

Prove thms for all K simultaneously
The proof of CFT (for K) doesn’t seem to work onlywith abel ext of K.
• We need to consider CFT for all L ([L : K] < ∞),• We also need non-abel ext L/K at some point.
(unless we have “Kronecker-Weber thm”)

Proof of CFT was buried in the forest of many Functorialities. (eg. Proof of LLC for GL(n) (Harris-Taylor,
Henniart), “Potential Automorphy”)
29Role of “Ramiﬁcation”— seems ubiquitous in number theory
The proof of unramiﬁed CFT doesn’t seem to work
only with unramiﬁed extensions.
• Iwasawa Theory, Exp Rec Law, (ϕ, Γ)-modules— Very ramiﬁed extensions simplify the problem• Euler systems, Taylor-Wiles machinery— role of “slightly ramiﬁed auxiliary ext”
q1, . . . , qr ≡ 1 (mod `N )• Khare-Wintenberger’s proof of Serre’s conj
“Killing ramiﬁcation”
30Even now, CFT tells us many things[1] Taylor, “Reciprocity laws and density theorems”,
Shaw Prize Lecture (2007).
[2] Yoshida, “Enigma of Teiji Takagi and Class Field
Theory (高木貞治と類体論の謎)”, Gendai Shiso (現代思想,
Modern Thought), Dec 2009.
(to appear in bookstores on Nov 27(Fri)).

31

Let’s go back to the Sato-Tate conj.

32

Sato-Tate Conj (Mikio Sato, Mar 1963)
E : y 2 = x3 + ax + b (a, b ∈ Z, 4a3 + 27b2 6= 0)
Elliptic Curve without Complex Multiplication
Deﬁne θp (0 5 θp 5 π) by√
p + 1 − #E(Fp) = 2 p cos θp.
Then, for 0 5 α < β 5 π,¯o¯
# p ¯ p:prime, p 5 N, α 5 θp 5 β2Z β
2 θ dθ
n ¯
o
lim
sin
=
N →∞
π α
# p ¯¯ p:prime, p 5 Nn

The distribution of {θp}p2πis the graph of y = π2 sin2 θ
0

π
2

π
33

Brief History
1920’s Class Field Theory (Takagi, Artin)¯¯√¯1933 Hasse’s thm ¯p + 1 − #E(Fp)¯¯ ≤ 2 p
1955 “Taniyama’s conj” L(s, E) = L(s, f )1957 Comp syst of `-adic Gal rep (Taniyama)
1963 “Sato’s conj”, “Tate’s conj”
1968 Serre’s book “Abel `-adic Rep and Ell Curves”,
Symmetric power L-functions L(s, E, Symn)
1970– Automorphic Rep/L-Fct,
Langlands’ principle of functoriality,
Langlands conj (Non-abelian CFT)
Weil conj/Ramanujan conj
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Brief History (continued)
1980– Development of Shimura var (LNM 900)
Iwasawa Main Conj
Deform of Galois rep, p-adic Hodge theory
1990– Shimura-Taniyama conj (semistable case),(⇒ Fermat’s Last Thm)
LLC for GL(n)2001– Full proof of Shimura-Taniyama conj
p-adic LLC for GL(2, Qp) & (ϕ, Γ)-mod
Framed deformation (→ number theory is reduced
to geometry of Gal deform sp (nilp orbits).)
and...
35

April 2006
Clozel, Harris, Shepherd-Barron, Taylor announced
a proof of the Sato-Tate conj for elliptic curves
E : y 2 = x3 + ax + b(a, b ∈ Z, 4a3 + 27b2 6= 0)

31728·4awhen j(E) = 4a3+27b2 is not an integer.

They necessarily proved similar results for elliptic curves
over all totally real ﬁelds simultaneously(whose j-invariants are not algebraic integers).

and...
36

July 2009
Barnet-Lamb, Geraghty, Harris, Taylor announced
a proof of the Sato-Tate conj for• all non-CM elliptic curves over tot real ﬁelds, and• all non-CM holom elliptic modular forms (/Q).
Key to the proof, among others• Fund Lemma (Laumon, Ngô, Waldspurger)• Geom of Shimura var and constr of Gal rep(Mantovan, Shin, “Book Project” (Harris))• Modularity Lifting Thm over Hida families (G)• Cohomology of Calabi-Yau families (Katz, B-L)
and...
37

Very very Recently (October 2009),
Barnet-Lamb, Gee, Geraghty obtained a proof of theSato-Tate conj for all non-CM Hilbert modular forms
on GL(2) over totally real ﬁelds.

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Tea & Coﬀee Break
10min

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Birth of “Sato’s conjecture”
1963 (March) Sato made the conj based oncomputer experiments. (Sato received Ph.D from
Univ of Tokyo in 1963, then moved to Osaka.)
1963-68 Tate and Serre gave theoretical explanation
using L-functions (“Tate’s conj ⇒ Sato’s conj”)
Computer experiments by Kanji Namba (logician).
Ref. Namba, “Dedekind η functions and Sato’s sin2conjecture” (in Japanese), 16th Symposium on thehistory of mathematics, Tsuda Univ, 2006.
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Fig 1.

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Fig 2.

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Fig 3. (Letter from Sato to Namba, May 13, 1963)

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Fig 4. (Calculation for X0(11), 1900 primes except 11)0◦ ≤ θ < 10◦1
10◦ ≤ θ < 20◦ 12
20◦ ≤ θ < 30◦ 40
30◦ ≤ θ < 40◦ 68
40◦ ≤ θ < 50◦ 110
50◦ ≤ θ < 60◦ 142

v.s. sin2 θ

60◦ ≤ θ < 70◦70◦ ≤ θ < 80◦80◦ ≤ θ < 90◦90◦ ≤ θ < 100◦
100◦ ≤ θ < 110◦110◦ ≤ θ < 120◦

177
194
198
212
206
174

120◦ ≤ θ < 130◦ 146
130◦ ≤ θ < 140◦ 103
140◦ ≤ θ < 150◦ 72
150◦ ≤ θ < 160◦ 34
160◦ ≤ θ < 170◦9170◦ ≤ θ ≤ 180◦2

v.s. sin θ
44

Fig 5. (Kato’s comment)

2
π

0

π
2

π

looks like

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Ref.
• Tate, “Algebraic Cohomology Classes”, Woodshole(July 1964). (Published as “Algebraic cycles andpoles of zeta functions”)• Serre, “Abelian `-adic Representations and Elliptic
Curves”, Benjamin 1968.
He gave theoretical explanation in terms of Gal rep.
“Sato-Tate” seemed to appear for the ﬁrst time.

46

Tate wrote

Conj 2 = Tate conj (on poles of Hasse-Weil zeta)
47

Unfortunate misunderstanding
Sato’s contribution was sometimes overlooked in number theory.

e.g. Wrong explanation seemed to appear in
Birch, “How the number of points of an elliptic curve
over a ﬁxed prime ﬁeld varies”, JLMS 43 (1968)

Possible reason : Serre was not aware of “which Sato”
made such a striking conjecture.

48

Q. How many Sato’s are there?

49

A. So many50

Sato families
In fact, Sato is the most famous family name in Japan.
1. Sato : 4.7 million families
2. Suzuki : 4.2 million
3. Takahashi : 3.5 million
4. Tanaka : 3.2 million
5. Watanabe : 2.68 million
6. Ito : 2.65 million
・・・

17. Saito : 1.47 million
・・・

1947. Hida : ∃ 1953 Hida families
51

Sato and Tate
Mikio Sato and John Tate received the Wolf prize.
Sato and Tate Receive 20022003 Wolf Prize

The 2002–2003 Wolf Prize in Mathematics has
been awarded to MIKIO SATO, of the Research Institute for Mathematical Sciences, Kyoto University,
Kyoto, Japan; and to JOHN T. TATE, Department ofMathematics, University of Texas, Austin. Sato washonored “for his creation of ‘algebraic analysis’,including hyperfunction and microfunction
theory, holonomic quantum field theory, and aunified theory of soliton equations.” Tate washonored “for his creation of fundamental concepts
in algebraic number theory.” The two share the$100,000 prize.

John T. Tate
For over a quarter of a century, John Tate’s ideas
have dominated the development of arithmetic
algebraic geometry. Tate has introduced pathbreaking techniques and concepts that initiatedmany theories that are very much alive today. These
include Fourier analysis on local fields and adelerings, Galois cohomology, the theory of rigid
analytic varieties, and p -divisible groups andp-adic Hodge decompositions, to name but a few.
Tate has been an inspiration to all those working
in number theory. Numerous notions bear his name:Tate cohomology of a finite group, Tate module ofan abelian variety, Tate-Shafarevich group, LubinTate groups, Neron-Tate heights, Tate motives, theSato-Tate conjecture, Tate twist, Tate elliptic curve,
and others. John Tate is a revered name in algebraic
number theory.
John Tate was born in 1925 in Minneapolis. He
received his A.B. from Harvard College (1946) and
his Ph.D. from Princeton University (1950). He wasa research assistant and instructor at Princeton(1950–53) and a visiting professor at Columbia University (1953–54) before moving to Harvard University. He was a professor at Harvard until 1990, when
he accepted his present position as professor and
Sid W. Richardson Chair in Mathematics at the
University of Texas at Austin. Tate received the AMS
Cole Prize (1956), a Sloan Fellowship (1959–61), anda Guggenheim Fellowship (1965–66). He was electedto the U.S. National Academy of Sciences (1969) andwas named a foreign member of the French Academy of Sciences (1992) and an honorary member ofthe London Mathematical Society (1999).

About the Wolf Prize
Mikio Sato
Mikio Sato’s vision of “algebraic analysis” andmathematical physics initiated several fundamental branches of mathematics. He created thetheory of hyperfunctions and invented microlocal
analysis, which allowed for a description of thestructure of singularities of (hyper)functions on
cotangent bundles. Hyperfunctions, together withintegral Fourier operators, have become a major
tool in linear partial differential equations. Alongwith his students, Sato developed holonomic quantum field theory, providing a far-reaching extensionof the mathematical formalism underlying thetwo-dimensional Ising model, and introduced along
the way the famous tau functions. Sato provided
a unified geometric description of soliton equationsin the context of tau functions and infinitedimensional Grassmann manifolds. This was
extended by his followers to other classes of equations, including self-dual Yang-Mills and Einsteinequations. Sato has generously shared his ideas

Mikio SatoJohn T. Tatewith young mathematicians and has created a flourishing school of algebraic analysis in Japan.
Mikio Sato was born in 1928 in Tokyo. He received his B.Sc. (1952) and his Ph.D. (1963) from the
University of Tokyo. He was a professor at Osaka
University and at the University of Tokyo before
moving to the Research Institute for Mathematical
Sciences at Kyoto University in 1970. He served as
director of that institute from 1987 to 1991. He is
now a professor emeritus at Kyoto University. He
received the Asahi Prize of Science (1969), the Japan
Academy Prize (1976), the Person of Cultural Merits
award of the Japanese Education Ministry (1984),
the Fujiwara Prize (1987), and the Schock Prize of
the Royal Swedish Academy of Sciences (1997). In
1993 he was elected to foreign membership in the
U.S. National Academy of Sciences.

The Israel-based Wolf Foundation was established by the late German-born inventor, diplomat,
and philanthropist Ricardo Wolf. A resident of
Cuba for many years, Wolf became Fidel Castro's
ambassador to Israel, where Wolf lived until his
death in 1981. The Wolf Prizes have been awarded
since 1978 to outstanding scientists and artists“for achievements in the interest of mankind andfriendly relations among peoples, irrespective of nationality, race, color, religion, sex, or political view.”
The prizes of $100,000 are given each year in fourout of five scientific fields, in rotation: agriculture,chemistry, mathematics, medicine, and physics.
In the arts the prize rotates among architecture,music, painting, and sculpture. The 2002–2003
prizes will be conferred by the president of Israel
at a ceremony at the Knesset (the Israeli parliament)
in Jerusalem on May 11, 2003.
—Allyn Jackson

It seemed the ﬁrst time for them to meet each other.
But, “Sato-Tate” was mentioned only for Tate’s work.
52

Yoshihiko Yamamoto (1941-2004)
a student of Sato, wrote Master’s thesis (Feb 1966,
Osaka Univ) on the “Sato conjecture” (in Japanese),but it was not published.

He wrote‘Sato conjecture’: “If E is not of CM type,the distribution of θp is proportional to sin2 θ.”

53

Yamamoto’s Master’s Thesis (Feb 1966, Osaka Univ)

54

Interview with Sato (Notices AMS 54 (2007))In the interview, elliptic curves/modular forms were
never mentioned.

55

More crucially,
Sato didn’t publish papers on “Sato’s conjecture”. Hiscolleagues (e.g. Kuga) wrote few expository articles.

Sato seemed to dislike number theory although hemade signiﬁcant contributions to it (e.g. Kuga-Sato,
Sato-Tate, etc).

There is a big “Sato school” at RIMS on algebraic
analysis, microlocalization, D-modules, hyperfunctions,soliton equations,...

etc.

Nobody was interested in

number theory as far as I know.
56

My Conclusion• According to the scratch papers, Sato seemed toconsider elliptic modular forms (η-products) also.
• Usually, “Sato-Tate” is a conj for ell curves.
It’s not precise. “Sato-Tate for modular forms”
makes perfect sense (theoretically and historically).
• It is not clear (to me) whether Sato conjectured“Sato-Tate Conj is true for some ell curves” or“Sato-Tate Conj is true for all ell curves”.
(We need Gal rep to believe it for all ell curves.)• Anyway, “Sato-Tate” seems an appropriate name.
Both contributions are big (as well as Serre’s).
57

Outline of Proof of Sato-Tate conj
(Oversimpliﬁed Exposition)

Idea• Control the arithmetic of an elliptic curve E via
analytic properties of zeta functions and L-functions
associated to E.
• Analytic properties of L-functions are established
via the Reciprocity Law (= automorphy of Gal rep).

58

Ex. (Hadamard, de la Vallée-Poisson, 1896)
The Riemann zeta function ζ(s) is holomorphic andnon-vanishing on Re(s) ≥ 1 (except s = 1)
⇒ Prime Number
Theorem !
Ãn ¯o log N¯
lim # p ¯ prime ·
=1
N →∞
N
Tate and Serre (1963-68)
deﬁned symmetric power L-functions L(s, E, Symn)
and proved if L(s, E, Symn) is holom and non-van
on Re(s) ≥ 1 + n
2 (for n ≥ 1),then “Sato-Tate conj” is true.

59Symmetric power L-functions
E : y 2 = x3 + ax + b (a, b ∈ Z, 4a3 + 27b2 6= 0)
non-CM elliptic curve√√ −iθp√ iθp
p + 1 − #E(Fp) = 2 p cos θp, αp = pe , βp = pe
Def.
Y

1
L(s, E) :=p (1 − αp p−s)(1 − βp p−s)
n
Y Y
1
n
L(s, E, Sym ) :=n−k −sp k=0 1 − αkβppp
(We ignore bad factors, although I like them.
I’m a bad person.)60Non-abelian CFT ⇒ Sato-Tate(1) L(s, E, Symn) is the L-function ass toSymnρE,` : Gal(Q/Q) −→ GL2(Q`) −→ GLn+1(Q`).
(2) SymnρE,` is irreducible (Serre, 1972).
(3) Non-abelian CFT predicts SymnρE,` is automorphic.
Namely,n , E, Symn) = L(s, π)
nρ
,
Sym
)
=
L(s
+
L(s + nE,`22

for a cusp autom rep π of GLn+1(AQ).
(4) L(s, π) is holom and non-van on Re(s) ≥ 1(Jacquet-Shalika, 1976/77).

61

Breakthrough after 2006
Clozel, Harris, Shepherd-Barron, Taylor.... established“Potential Automorphy” of SymnρE,`.
(Automorphy was not yet established.)

Thm (CHSBT, BLGHT,...)For n ≥ 1, there exists a tot real Gal ext F/Qs.t. (SymnρE,`)|F is automorphic.

62

Proof of Potential Automorphy• Taylor-Wiles-Kisin method (Rred = T ),Modularity (Automorphy) Lifting Thm“ρ (mod `) : Automorphic ⇒ ρ : Automorphic”• Cohom of Calabi-Yau var (Dwork families)
Find suﬀ many comp syst of `-adic Gal rep,and connect “chain of automorphy”.
Potential Automorphy ⇒ Sato-Tate — “classical”• Cyclic base change (Arthur-Clozel)• Brauer’s induction thm (1946) — He establishedmeromorphy of Artin L-functions.
63

Question
What can we say and can’t we say about
the Sato-Tate conj over K (K is not nec tot real)?

What we can’t say...
Descent of Automorphy (?)
L/K : cyclic ext (not nec tot real)ρ : `-adic Gal rep of K s.t. ρ|L is abs irred???

ρ|L : Automorphic ⇒ ρ : Automorphic
cf. [CHT], Lemma 4.2.2 uses [AC], Thm 4.2 (descentof autom rep) and existence of Gal rep in a crucial way.
64

What we can’t say... (continued)
K : number ﬁeld (not nec tot real)E : non-CM ell curve/K
Potential automorphy of SymnρE,` does notseem to imply the Sato-Tate conj.

According to “Brauer ind argument”, we need “Potential autom & Descent of autom”:
For n ≥ 1, there exists a tot real Gal ext F/Qs.t. (SymnρE,`)|L is autom for all F/L/Q(F/L : solvable)
then, the Sato-Tate conj is true for E.
65

What we can say...
Thm K : number ﬁeld, E : non-CM ell curve/K s.t.
(1) K/Q is cyclic or [K : Q] = 3,
(2) j(E) ∈ Q (but E might not be deﬁned over Q).
Then, the Sato-Tate conj is true for E/K.

Proof
∃ E 0 over Q s.t. ρE,` is a quad twist of ρE 0,`|K
L(s, SymnρE,`) = L(s, (SymnρE 0,`)|K ⊗ χ)
= L(s, (SymnρE 0,`) ⊗ IndK/Qχ)
Use Arthur-Clozel AI (K/Q:cyclic) or JPSS ([K : Q] = 3)
+ [BLGHT] (for E 0/Q) + Rankin-Selberg
66

Problem Give an example of ell curve E over Q(21/5)
s.t. the Sato-Tate conj is known to hold for E/Q(21/5).
Rem.
Galois closure of Q(21/5)/Q is solvable. (can use BC)

Q(21/5)/Q is not Galois. (BC is not available)
Q(21/5) is neither tot real nor CM. (Constr of Gal rep
is not available)

67

Outline of the Kanazawa workshop• Sato-Tate conj and beyond (Barnet-Lamb, Gee)
• p-adic modular forms (‘Hida families’)
and arith applications (Geraghty, Loeﬄer, Sasaki)• Autom Rep and applications (Ichino, Abe, Shin)• Cohom, Arith Geom, Shim var (Lan, Harashita, Mieda)• Iwasawa theory and applications (Pottharst, Ohshita)• p-adic Langlands (Imai), Bhargava’s work (Taniguchi)(→ New density thm, Counting wt 1 mod forms)
Top modular forms (Zakharevich)• Hyperspecial talk (Yoshida – TBA)
68

Finally,69

Thank you very much for your attention!

Enjoy Kanazawa and the Kanazawa
workshop!
70