Arithmetic of L-Functions

Home , Geometry of numbers, Selmer group

ARITHMETIC OF L-FUNCTIONS

CHAO LI

NOTES TAKEN BY PAK-HIN LEE

Abstract. These are notes I took for Chao Li’s course on the arithmetic of L-functions oﬀered at Columbia University in Fall 2018 (MATH GR8674: Topics in Number Theory). WARNING: I am unable to commit to editing these notes outside of lecture time, so they are likely riddled with mistakes and poorly formatted.

Contents 1. Lecture 1 (September 10, 2018) 4 1.1. Motivation: arithmetic of elliptic curves 4 1.2. Understanding ralg(E): BSD conjecture 4 1.3. Bridge: Selmer groups 6 1.4. Bloch–Kato conjecture for Rankin–Selberg motives 7 2. Lecture 2 (September 12, 2018) 7 2.1. L-functions of elliptic curves 7 2.2. L-functions of modular forms 9 2.3. Proofs of analytic continuation and functional equation 10 3. Lecture 3 (September 17, 2018) 11 3.1. Rank part of BSD 11 3.2. Computing the leading term L(r)(E, 1) 12 3.3. Rank zero: What is L(E, 1)? 14 4. Lecture 4 (September 19, 2018) 15 L(E,1) 4.1. Rationality of Ω(E) 15 L(E,1) 4.2. What is Ω(E) ? 17 4.3. Full BSD conjecture 18 5. Lecture 5 (September 24, 2018) 19 5.1. Full BSD conjecture 19 5.2. Tate–Shafarevich group 20 5.3. Tamagawa numbers 21 6. Lecture 6 (September 26, 2018) 22 6.1. Bloch’s reformulation of BSD formula 22 6.2. p-Selmer groups 23 7. Lecture 7 (October 8, 2018) 25 7.1. p∞-Selmer group 25 7.2. Tate’s local duality 26

Last updated: December 14, 2018. Please send corrections and comments to [email protected]. 1 7.3. Local conditions at good primes 27 7.4. Kolyvagin’s method 28 8. Lecture 8 (October 10, 2018) 28 8.1. Kolyvagin’s method 28 8.2. Heegner points on X0(N) 29 8.3. Heegner points on elliptic curves 30 9. Lecture 9 (October 15, 2018) 31 9.1. Gross–Zagier formula 31 9.2. Back to E/Q 32 9.3. Gross–Zagier formula for Shimura curves 33 10. Lecture 10 (October 17, 2018) 34 10.1. Gross–Zagier formula for Shimura curves 34 10.2. Waldspurger’s formula 35 11. Lecture 11 (October 22, 2018) 37 11.1. Examples of Waldspurger’s formula 37 11.2. Back to Kolyvagin 38 12. Lecture 12 (October 24, 2018) 39 12.1. Geometric interpretation of Ribet’s theorem: Ihara’s lemma 40 12.2. Geometry of modular curves in characteristic p 41 13. Lecture 13 (October 29, 2018) 42 13.1. The supersingular locus 42 13.2. Geometry of Shimura curves 43 14. Lecture 14 (October 31, 2018) 45 14.1. Inertia action on cohomology 45 14.2. Nearby cycle sheaves and monodromy ﬁltration 46 14.3. Rapoport–Zink weight spectral sequence 47 15. Lecture 15 (November 7, 2018) 48 16. Lecture 16 (November 12, 2018) 48 17. Lecture 17 (November 14, 2018) 48 18. Lecture 18 (November 19, 2018) 48 18.0. Rankin–Selberg L-functions for GLn × GLn−1 48 18.1. Global zeta integrals 49 18.2. Relation between Z(f, ϕ, s) and L(s, π × π0) 52 18.3. Global L-functions 52 19. Lecture 19 (November 26, 2018) 53 19.1. General Waldspurger formula 53 19.2. Gan–Gross–Prasad conjectures 55 20. Lecture 20 (November 28, 2018) 55 20.1. Bloch–Kato Selmer groups 55 20.2. Bloch–Kato conjecture 58 20.3. Rankin–Selberg motives 58 21. Lecture 21 (December 3, 2018) 58 21.1. BBK for Rankin–Selberg motives 58 21.2. Rank one case 60 22. Lecture 22 (December 5, 2018) 62

2 22.1. Proof strategies 62 22.2. Geometry of unitary Shimura varieties at good reduction 63 22.3. Tate’s conjecture 64 22.4. Second reciprocity 64 23. Lecture 23 (December 10, 2018) 65 23.1. Tate conjecture via geometric Satake 65 23.2. Geometric Satake 66 23.3. Spectral operators 66 23.4. Computing intersection numbers 68

3 1. Lecture 1 (September 10, 2018) 1.1. Motivation: arithmetic of elliptic curves. Let us begin with some motivation by studying the arithmetic of elliptic curves. Consider an elliptic curve E : y2 = x3 + Ax + B, where A, B ∈ Q. Example 1.1. The elliptic curve y2 = x3 − x intersects with the x-axis at (−1, 0), (0, 0) and (1, 0). Given two points P , Q on E, the straight line through P and Q intersects E at a third point R. Declaring that P + Q + R = 0 defines an abelian group law on E and on the set of rational points E(Q). There are some degenerate cases; for example, the tangent line through P 0 ∈ E defines the relation 2P 0 + R0 = 0. A fundamental result in the arithmetic of elliptic curves is that the group of rational points cannot be too huge. Theorem 1.2 (Mordell). E(Q) is finitely generated.

Deﬁnition 1.3. The algebraic rank of E is deﬁned as ralg(E) := rank E(Q) ∈ Z≥0.

In particular, ralg(E) = 0 if and only if E(Q) is ﬁnite. Example 1.4. For E : y2 = x3 − x, E(Q) = Z/2 × Z/2 is generated by the two 2-torsion points (0, 0) and (1, 0). Thus ralg(E) = 0. Example 1.5. For E : y2 = x3 − 25x, one can check that P = (−4, 6) ∈ E(Q), and 1681 62279 2P = ( 144 , − 1728 ) ∈ E(Q). In fact, E(Q) ' Z × Z/2 × Z/2, with generators (−4, 6), (0, 0), (5, 0) respectively, and thus ralg(E) = 1. The greatest mystery in the study of elliptic curves is the following question. Given E/Q, how do we determine ralg(E)? No algorithm exists yet! In fact, we have very little knowledge; we don’t even know which integers can show up as ralg(E). To show how limited our knowledge is:

(n) 2 3 2 (n) Example 1.6. Consider E : y = x − n x. The smallest n for each value of ralg(E ) is (n) ralg(E ) 0 1 2 3 4 5 6 n 1 5 34 1254 29274 48272239 6611714866 (n) but we don’t know of any n which gives ralg(E ) = 7.

1.2. Understanding ralg(E): BSD conjecture. In order to attack the problem of understanding ralg(E), the deep connections with the arithmetic L-function come into play. Analogous to the Riemann zeta function X 1 Y ζ(s) = = (1 − p−s)−1, ns n≥1 p we can deﬁne the L-function associated to E/Q:

X an Y L(E, s) = = L (E, s), ns p n≥1 p 4 where the Euler factor is deﬁned by

( −s 1−2s −1 (1 − app + p ) if p is good, Lp(E, s) = −s −1 (1 − app ) if p is bad, and  p + 1 − |E(F )| if p is good,  p ap = ±1 if p is multiplicative, 0 if p is additive. Later we will give some motivation for deﬁning the L-function this way. The whole point is that these L-functions are much easier to study; for example, ap is a more tractable number, depending on the number of points of E mod p. 3 L(E, s) converges when Re s > 2 . Analogous to the Riemann zeta function, we expect L(E, s) to admit analytic continuation and satisfy a functional equation. In fact, there is the miraculous theorem: Theorem 1.7 (Modularity theorem: Wiles, Taylor, BCDT). There exists a cuspidal eigenform fE ∈ S2(NE) such that X n fE(s) = anq n≥1 where the an’s are as before. 2 3 Example 1.8. Consider E : y = x − x, which has conductor NE = 32. Solving this equation mod p, we see that p 2 3 5 7 11 13 17 19 23 29 31 ap 0 0 -2 0 0 6 2 0 0 -10 0

Notice that ap 6= 0 if and only if p ≡ 1 (mod 4). Under the modularity theorem, the corresponding modular form is Y 4n 2 8n 2 fE = q (1 − q ) (1 − q ) n≥1 = q − 2q5 − 3q9 + 6q13 + 2q17 − q25 − 10q29 + ··· . The modularity theorem is truly miraculous; why does the number of solutions mod p have anything to do with modular forms? For us, we have the

Corollary 1.9. L(E, s) = L(fE, s). In particular, L(E, s) has analytic continuation to C and satisﬁes a functional equation under s ↔ 2 − s.

Deﬁnition 1.10. The analytic rank of E is deﬁned as ran(E) := ords=1 L(E, s) ∈ Z≥0.

Conjecture 1.11 (BSD). ralg(E) = ran(E).

Remark. BSD leads to an algorithm for computing ralg and ran. Theorem 1.12 (Gross–Zagier, Kolyvagin, 1980’s).

(1) If ran(E) = 0, then ralg(E) = 0. (2) If ran(E) = 1, then ralg(E) = 1. But little is known in the higher rank case! 5 1.3. Bridge: Selmer groups. Let us outline how this theorem is proved.

very far E(Q) o / L(E, s)

(∗) (∗∗)

Selp∞ (E) where the connection (∗) is made by p∞-descent, and (∗∗) is by explicit reciprocity laws. Recall the pn-descent sequence

n n 0 → E(Q)/p E(Q) → Selpn (E) → X(E)[p ] → 0 where Selpn (E) is a ﬁnite group and X(E) is the Tate–Shafarevich group.

Conjecture 1.13. X(E) is ﬁnite.

Taking inverse limit over n and tensoring with Qp, we get an injection

E(Q) ⊗ Qp ,→ Selp∞ (E) which is an isomorphism if X(E) is ﬁnite. So

∞ ﬁniteness of X(E) =⇒ ralg(E) = dimQp Selp (E).

∞ dimQp Selp (E) is called the Selmer rank of E. In summary, the algebraic object Selp∞ (E) gives an upper bound for ralg(E), which is sharp if X(E) is ﬁnite. Moreover, Selp∞ (E) is easier to understand due to its local nature. Strategy for proving Theorem 1.12:

(1) ran = 0 =⇒ Selp∞ (E) = 0. (2) ran = 1 =⇒ dim Selp∞ (E) ≤ 1. Then show ralg(E) ≥ 1 by ﬁnding a point of inﬁnite order.

The connection between ran and Selmer groups is obtained by bounding Selmer groups from above using Euler systems (due to Kolyvagin), and the explicit point construction is by considering Heegner points (due to Gross–Zagier). The key in bounding Selmer groups is to construct Galois cohomology classes with controlled local ramification. There are currently three ways to do this: • Kolyvagin: Heegner points in ring class fields • Bertolini–Darmon: Heegner points on different Shimura curves • Kato: Kato classes in cyclotomic fields (in case (1) only) We will focus on the second approach by Bertolini–Darmon, and generalize to more general motives. Let me briefly indicate where the ramification comes from in each case: • Kolyvagin: tamely ramified extensions • Bertolini–Darmon: ramified groups • Kato: wildly ramified extensions 6 1.4. Bloch–Kato conjecture for Rankin–Selberg motives. The second part of the course will be on some recent results on Bloch–Kato conjecture for certain Rankin–Selberg motives. For elliptic curves, the Bloch–Kato conjecture takes the following form:

Conjecture 1.14 (Bloch–Kato). dim Selp∞ (E) = ran(E). This is exactly as predicted by BSD and the ﬁniteness of X. 1 Bloch–Kato gives an alternative deﬁnition of Selmer groups Hf (Q, ρE) where ρE : GQ → GL2(Qp) is the Galois representation on the rational Tate module Tp(E) ⊗ Qp. The Bloch– 1 Kato Selmer groups Hf are Qp-vector spaces associated to arbitrary p-adic Galois representations coming from geometry, which coincide with the usual Selmer groups in the case of elliptic curves. Now we can generalize the earlier picture

fE ∈ S2(N)

u & 1 ∼ Hf (Q, ρE) = Selp∞ (E) L(fE, s) Let Π be a conjugate self-dual cohomological cuspidal automorphic representation on GLn(AK ) where K is a CM quadratic extension of a totally real ﬁeld F . By the global Langlands correspondence (due to many people, e.g. Chenevier–Harris), we can associate to Π a Galois representation ρΠ : GK → GLn(Qp). 0 Now consider Π on GLn(AK ) and Π on GLn+1(AK ), which give the Rankin–Selberg 1 ∨ ∨ 0 motive Hf (K, ρΠ ⊗ ρΠ0 ) and the Rankin–Selberg L-function L(Π × Π , s). The Bloch-Kato conjecture in this case asserts that

1 ∨ ∨ 0 Conjecture 1.15 (Bloch–Kato). dim H (K, ρ ⊗ ρ 0 ) = ord 1 L(Π × Π , s). f Π Π s= 2 Theorem 1.16 (Liu–Tian–Xiao–Zhang–Zhu). Assume Π and Π0 have trivial inﬁnitesimal character and p satisﬁes certain assumptions. 0 1 ∨ ∨ (1) If ord 1 L(Π × Π , s) = 0, then H (K, ρ ⊗ ρ 0 ) = 0. s= 2 f Π Π 0 1 ∨ ∨ 1 ∨ ∨ (2) Assume clGGP(Π × Π ) ∈ Hf (K, ρΠ ⊗ ρΠ0 ) is nonzero. Then dim Hf (K, ρΠ ⊗ ρΠ0 ) = 1. 0 0 Remark. By the Beilinson conjecture, clGGP(Π × Π ) 6= 0 ⇐⇒ ord 1 L(Π × Π , s) = 1. s= 2

2. Lecture 2 (September 12, 2018) 2.1. L-functions of elliptic curves. Recall: for an elliptic curve E/Q, we deﬁne

X an Y L(E, s) = = L (E, s) ns p n≥1 p where the Euler factor at p is  (1 − a p−s + p · p−2s)−1 if p N = N ,  p - E −s −1 Lp(E, s) = (1 ± p ) if p k N, 1 if p2 | N. 7 ( 1 if p N, A uniform way to write this is as follows: deﬁne χ(p) = - , so then 0 if p | N,

−s −2s −1 Lp(E, s) = (1 − app + χ(p)p · p ) . Some easy consequences are:

(1) a1 = 1. (2) amn = aman if (m, n) = 1. (3) apr = apapr−1 − χ(p)papr−2 if r ≥ 2.

This exactly matches the relations for Hecke operators Tn. Remark. L(E, s) is an example of a “motivic L-function”, i.e., one associated to Galois representations coming from geometry. Recall the `-adic Tate module T E = lim E[`n], ` ←−n which is a Z`-module of rank 2. The action of GQ on V`E = T`E ⊗ Q` gives rise to a 2-dimensional Galois representation

ρE : GQ → Aut(V`E) ' GL2(Q`). Then for p 6= `, −1 −s Ip Lp(E, s) = det 1 − p Frp | (V`E)

where Ip ⊂ GQp is the inertia subgroup. Notice that  2 if p N,  - Ip dim(V`E) = 1 if p k N, 0 if p2 | N.

Ip In particular, ap = tr(Frp | (V`E) ). 3 Proposition 2.1. L(E, s) converges when Re s > 2 . Proof. By counting the number of solutions to y2 = x3 + Ax + B, we have the trivial bound |E(Fp)| ≤ 2p + 1. However, 50% chance of being a square in Fp gives the heuristic ? |E(Fp)| ∼ p + 1. This is made precise by √ Theorem 2.2 (Hasse). |p + 1 − |E(Fp)|| ≤ 2 p. This implies a 1 n = O s Re s− 1 n n 2 P an 1 and so ns converges if Re s − 2 > 1. Remark. Hasse’s theorem can be seen in a high-brow manner using the Weil conjectures, 1/2 since ap = α + β with |α| = |β| = p . Expected properties of L(E, s) (more generally any motivic L-function): (1) Euler product. (2) L(E, s) has analytic continuation to all of C. (3) L(E, s) has a functional equation. 8 (2) and (3) are more diﬃcult and is one of the motivations of the Langlands program.

L(E, s) o / L(fE, s) motivic L-function automorphic L-function Automorphic L-functions are associated to modular forms or automorphic forms, for which (2) and (3) are easier.

a b 2.2. L-functions of modular forms. Let Γ (N) = : c ≡ 0 (mod N) ⊆ SL (Z). 0 c d 2 Let Sk(N) be the space of cusp forms of level Γ0(N) and weight k. Recall f ∈ Sk(N) if: (1) f : H → C is holomorphic. a b (2) f(γτ) = (cτ + d)kf(τ) if γ = ∈ Γ = Γ (N). c d 0 (3) f is holomorphic and vanishes at all cusps. P n Every f ∈ Sk(N) has a Fourier expansion f(τ) = n≥1 an(f)q .

Deﬁnition 2.3. The L-function of a cusp form f ∈ Sk(N) is deﬁned as

X an L(f, s) := , ns n≥1

where an = an(f). Remark. In this generality, L(f, s) may not have an Euler product. It has an Euler product if and only if f is eigenform for all Tn’s.

k k/2 Proposition 2.4. L(f, s) converges when Re s > 2 + 1. In fact, |an| = O(n ). Remark. • This is the trivial bound for E/Q (for weight k = 2). k−1 • If f is an eigenform, there is a sharper bound |ap| = O(p 2 ) (due to Deligne). Proof. Notice Z 1 −2πin(x+iy) an = e f(x + iy) dx 0 1 for any y. Taking y = n , we have Z 1 2π 1 |an| = e f x + i · dx . 0 n

k On the other hand, |f(τ)|(im τ) 2 is invariant under Γ. Since f is a cusp form, the function k ∗ − k |f(τ)|(im τ) 2 is extended to Γ\H , so it is bounded on H. This implies |f(x+iy)| = O(y 2 ). 1 1 k k 2 2 For y = n , we have |f(x + i · n )| = O(n ) and so an = O(n ). The next goal is to prove (2) and (3) for L(f, s). 9 2.3. Proofs of analytic continuation and functional equation. Starting point: L(f, s) has an explicit integral representation (Mellin transform of f). Deﬁnition 2.5. The Mellin transform of f is Z ∞ dt g(s) = f(it)ts . 0 t −s k Proposition 2.6. g(s) = (2π) Γ(s)L(f, s) when Re s > 2 + 1. R ∞ −t s dt Proof. Recall Γ(s) = 0 e t t . Changing t by 2πnt, Z ∞ dt Γ(s) = (2πn)s e−2πntts , 0 t so Z ∞ dt n−s = (2π)sΓ(s)−1 e−2πntts . 0 t Summing over n gives Z ∞ X an X dt L(f, s) = = (2π)sΓ(s)−1 a e−2πntts ns n t n≥1 n≥1 0 Z ∞ X dt = (2π)sΓ(s)−1 a e−2πntts n t 0 n≥1 Z ∞ = (2π)sΓ(s)−1 f(it)ts dt 0 = (2π)sΓ(s)−1g(s). −s Therefore g(s) = (2π) Γ(s)L(f, s). Deﬁnition 2.7. The complete L-function of f is s −s s Λ(f, s) := N 2 (2π) Γ(s)L(f, s) = N 2 g(s). To get a functional equation, we need some extra symmetry on f.

Deﬁnition 2.8 (Atkin–Lehner involution). Deﬁne an operator WN : Sk(N) → Sk(N) by i k 1 (WN f)(τ) = √ f − . Nτ Nτ 2 One can check that WN = 1.

Theorem 2.9 (Hecke). Let f ∈ Sk(N). Assume WN f = f, where ∈ {±1}. Then (1) Λ(f, s) has analytic continuation to all of C. (2) There is a function equation Λ(f, s) = Λ(f, k − s). is called the sign of the functional equation, and plays an important role in the BSD s −s conjecture. Since N 2 (2π) Γ(s) doesn’t have any zeros, we have the Corollary 2.10. L(f, s) has analytic continuation to all of C.

Remark. This theorem applies to any newform f ∈ Sk(N), so Λ(f, s) = ±Λ(f, k − s). 10 Proof of Theorem 2.9. Z ∞ s s dt Λ(f, s) = N 2 f(it)t 0 t Z ∞ it dt t = f √ ts by t 7→ √ 0 N t N Z 1 it dt Z ∞ it dt = f √ ts + f √ ts . 0 N t 1 N t | {z } | {z } (∗) (∗∗) The term (∗∗) clearly converges, whereas Z 1 i s−k dt (∗) = (WN f) √ t [by deﬁnition of WN ] 0 Nt t Z ∞ it k−s dt −1 = (WN f) √ t [by t 7→ t ]. 1 N t Therefore, Z ∞ Z ∞ it s dt it k−2 dt Λ(f, s) = f √ t + (WN f) √ t . 1 N t 1 N t The theorem follows since WN f = f.

3. Lecture 3 (September 17, 2018) 3.1. Rank part of BSD. Recall: For an elliptic curve E/Q, Y L(E, s) = Lp(E, s). p By the modularity theorem, we have shown: (1) L(E, s) has analytic continuation to C. (2) L(E, s) satisﬁes a functional equation under s ↔ 2 − s. The complete L-function satisﬁes Λ(E, s) = Λ(E, 2 − s) where ∈ {±1}. At the center of the functional equation s = 1, we have ( even if = +1, ord Λ(E, s) = s=1 odd if = −1.

Since ords=1 Λ(E, s) = ords=1 L(E, s), ( even if = +1, r (E) = an odd if = −1.

Question. What is the relation between ran(E) and ralg(E)? Q Heuristically, we can try plugging s = 1 into p Lp(E, s), although this doesn’t converge! Note that −1 −2 −1 Lp(E, 1) = (1 − ap · p + χ(p)p · p ) 11 p − a + χ(p)−1 = p p p = p − ap + χ(p)  p if p is good,  p+1−ap  p = p±1 if p is multiplicative,  p p if p is additive, p = |Ee(Fp)| where Ee is the smooth part of the reduction of E mod p, so Y p L(E, 1)“=” . p |Ee(Fp)|

If E(Q) has “more” points (i.e., ralg(E) is “large”), then |Ee(Fp)| is “large” for each p, and hence L(E, 1) becomes “more” zero (i.e., ran(E) is “large”). In fact, Birch and Swinnerton-Dyer did numerical computation for {y2 = x3 − n2x} in 1958. They computed

Y |Ee(Fp)| “looks like” ∼ c(log X)r p p

where r = ralg(E). This leads to:

Conjecture 3.1 (BSD: rank part). ran(E) = ralg(E). However, all these are heuristics.

(r) Question. How to compute ran(E)? L (E, 1)? 3.2. Computing the leading term L(r)(E, 1). Recall: Z ∞ it dt Λ(f, s) = f √ (ts + tk−s) 1 N t

where WN f = f for the Atkin–Lehner involution. Let us apply this to f = fE ∈ S2(N). 3.2.1. Rank zero. For r = 0 (hence = +1), Z ∞ it dt Λ(E, 1) = 2 f √ t · 1 N t Z ∞ X − 2√πnt = 2 ane N dt 1 n≥1 Z ∞ X − 2√πnt = 2 an e N dt n≥1 1 √ X N an − 2√πn = 2 · · e N . 2π n n≥1 12 Then the incomplete L-function is 2π L(f, 1) = √ Λ(f, 1) N X an − 2√πn = 2 e N . n n≥1 This is a formula we can use to compute whether an elliptic curve has analytic rank 0. Remark. L(f, 1) can be viewed as a “weighted sum” of the divergent series X an L(f, 1)“=” . n 3.2.2. Rank one. For r = 1 (hence = −1), Z ∞ it Λ0(f, 1) = 2 f √ log t dt 1 N Z ∞ X − 2√πnt = 2 ane N log t dt 1 n≥1 Z ∞ X − 2√πnt = 2 an e N log t dt n≥1 1 √ Z ∞ X N an − 2√πnt dt = 2 · e N , 2π n t n≥1 1 where we have used integration by parts to obtain the exponential integral Z ∞ 1 Z ∞ dt e−ct log t dt = e−ct . 1 c 1 t Then the incomplete L-function has derivative 2π L0(f, 1) = √ Λ0(f, 1) N Z ∞ X an − 2√πnt dt = 2 e N . n t n≥1 1 3.2.3. General rank. More generally, for any r ≥ 1, we deﬁne Z ∞ 1 −xt r−1 dt Er(x) := e (log r) . (r − 1)! 1 t Then the leading term is X an 2πn L(r)(f, 1) = 2r! E √ . n r n≥1 N This shows that the analytic rank is easy to compute in practice. R ∞ R ∞ Remark. is easy to compute. Breaking the integral into A + A−1 gives X an − 2√πn − 2√πnA L(f, 1) = e A N + e N . n n≥1 13 Now plugging in diﬀerent values of A (e.g. A = 1 and A = 1.01) gives the only choice for . 3.3. Rank zero: What is L(E, 1)? Example 3.2. Consider E : y2 = x3 − x, which has conductor N = 32 (and complex multiplication by Q(i)). The corresponding modular form is Y f = q (1 − q4n)2(1 − q8n)2 n≥1 = q − 2q5 − 3q9 + 6q13 + 2q17 + ··· .

In fact, ap 6= 0 ⇐⇒ p ≡ 1 (mod 4). Then

X an − √2πn L(E, 1) = 2 e 32 = 0.65551438837 ··· . n 2 3 Example 3.3. Consider√ E : y = x + 1, which has conductor N = 36 (and complex multiplication by Q( −3)). The corresponding modular form is Y f = q (1 − q6n)4 n≥1 = q − 4q7 + 2q13 + 8q19 − 5q25 + ··· .

In fact, ap 6= 0 ⇐⇒ p ≡ 1 (mod 3). Then

X an − πn L(E, 1) = 2 e 3 = 0.701091052663 ··· . n Question. What are these transcendental numbers? Answer. There are natural transcendental numbers associated to elliptic curves: elliptic integrals, or periods of elliptic curves in modern language. Definition 3.4. Suppose E has minimal Weierstrass equation 2 3 2 y + a1xy + a3y = x + a2x + a4x + a6. Define the Néron differential

dx 0 1 ωE := ∈ H (E, ΩE) 2y + a1x + a3 and the Néron period is Z Ω(E) := ωE ∈ R. E(R) Example 3.5. For E : y2 = x3 − x, this equation is already minimal and the Néron differential is dx dx ωE = = √ . 2y 2 x3 − x The real locus E(R) consists of two circles (including the point at infinity), which correspond to two homologous circles in the complex torus E(C). Then Z Z dx Z ∞ dx Ω(E) = ωE = 2 √ = 4 √ = 5.24411510858 ··· . 3 3 E(R) circle 2 x − x 1 2 x − x 14 Observe the miraculous coincidence: L(E, 1) 1 ≈ ∈ Q! Ω(E) 8 Similarly,

Example 3.6. For E : y2 = x3 + 1, the N´erondiﬀerential is ω = √dx . The real locus E 2 x3+1 E(R) is only one circle, which corresponds to a circle in E(C). Then Z ∞ dx Ω(E) = √ = 4.20654631 ··· 3 −1 x + 1 and L(E, 1) 1 ≈ ∈ Q. Ω(E) 6 The upshot is the following Conjecture 3.7. For every elliptic curve E/Q, L(E, 1) ∈ Q. Ω(E) We will prove this next time and explain what happens for L(r)(E, 1).

4. Lecture 4 (September 19, 2018) L(E,1) L(E,1) 4.1. Rationality of Ω(E) . To prove Ω(E) ∈ Q, we need to relate L(E, 1) to a “period”. In general, a period of an algebraic variety X/Q is a quantity of the form Z period = closed i-form, i-cycle i where an i-cycle lives in Hi(X(C), Q) and a closed i-form lives in HdR(X/Q). dt Example 4.1. For X = Gm and ω = t , Z dt = 2πi. S1 t Note that 2πi is the period of the exponential function (inverse of log). Example 4.2. For an elliptic curve, Z ωE = period of the Weierstrass ℘-function (inverse of elliptic integral). E(R)

Our goal is to show that L(E, 1) = L(fE, 1) is related to periods of the (compactiﬁed) ∗ modular curve X0(N) = Γ0(N)\H . Recall the Mellin transform interpretation of L(f, s): (2π)s Z ∞ dt L(f, s) = f(it)ts . Γ(s) 0 t Applying this to s = 1 (without worrying about convergence), Z ∞ Z i∞ L(f, 1) = 2π f(it) dt τ=it −2πi f(τ) dτ. 0 0 15 Then ωf = 2πif(τ) dτ can be interpreted as a 1-form on X0(N), and Z i∞ L(f, 1) = − ωf 0

is an integral along the path 0 → i∞ on X0(N). Problem: 0 → i∞ is not a closed path. But it is not too far from being one, as shown in the following

Theorem 4.3 (Manin–Drinfeld). Take any cusps α, β ∈ X0(N). Then α → β belongs to H1(X0(N)(C), Q), i.e., there exist closed loops γi ∈ H1(X0(N)(C), Z) and ci ∈ Q such that Z β X Z ωf = ci ωf α i γi

for all f ∈ S2(N). Remark. In other words, a rational multiple of L(f, 1) is indeed a period. Remark. This theorem implies (in fact, is equivalent to saying) α − β gives a torsion point on J0(N) = Jac(X0(N)).

Idea of proof. Use Hecke operators Tp for p - N: p−1 Z i∞ Z i∞ X Z i∞ Tp = + 0 0 k k=0 p p−1 Z i∞ X Z 0 = (1 + p) − . 0 k k=0 p k Note that for p - N, p → 0 is a closed loop on X0(N). Then Z i∞ X Z (1 + p − Tp) = 0 γi

with 1 + p − Tp acting as an integer (the number of points mod p). It remains to choose p such that this is nonzero, which is possible because a modular form with Tp acting by 1 + p for all p must be an Eisenstein series. Now we are ready to prove the L(E, 1) Theorem 4.4 (Birch). ∈ Q. Ω(E)

Proof. The modularity theorem gives a nontrivial map ϕ : X0(N) → E (deﬁned over Q) ∗ × such that ϕ (ωE) = c · 2πif(τ) dτ for some c ∈ Q . Then L(f, 1) ∼ period of 2πif(τ) dτ, Q×

Ω(E) =period of ωE. Both are real periods, so we conclude L(f, 1) ∼ Ω(E). Q× Conjecture 4.5. c = 1. 16 L(E,1) L(r)(E,1) 4.2. What is Ω(E) ? More generally, what is Ω(E) for r = ran(E)? Example 4.6. Consider E : y2 = x3 − 25x, with N = 800 = 32 · 25, r = 1 and E(Q) ∼= Z × Z/2 × Z/2 generated by (−4, 6), (0, 0), (5, 0) respectively. The corresponding modular form is f = q − 3q9 − 6q13 − 2q17 + ··· . We can use the formula from last time to compute that 0 X an 2πn L (f, 1) = 2 E1 √ = 2.22737037954 ··· . n 800 However, Z ∞ dx Ω(E) = 2 √ = 2.34523957 ··· 3 5 x − 25x and L0(f, 1) = 0.949741 ··· ∈/ Q! Ω(E) L(f,1) Of course, Theorem 4.4 is trivially true, since Ω(E) = 0. Idea: Use arithmetic complexity of the rational points of infinite order. a × Definition 4.7. For x = b ∈ Q with (a, b) = 1, define its height by h(x) := log max{|a|, |b|}. Definition 4.8. For P ∈ E(Q), define its naive height by ( h(x(P )) if x(P ) 6= 0, h(P ) := −∞ if x(P ) = 0. Problem: This depends on the Weierstrass equation of E. Definition 4.9 (Tate). Define the canonical height (or Néron–Tate height) by n ˆ h([2 ]P ) h(P ) := lim ∈ R≥0. n→∞ 4n Example 4.10. Let us return to the previous example E : y2 = x3 − 25x. Then P = (−4, 6) h(P ) = 1.856786 ··· 1681 h(2P ) 2P = ( 144 , ··· ) 4 = 1.8778407 ··· 11183412793921 h(4P ) 4P = ( 223416132416 , ··· ) 16 = 1.89946579 ··· In fact, hˆ(P ) = 1.89948217253 ··· . Observe that L0(f, 1) 1 = ∈ Q! Ω(E)hˆ(P ) 2 Remark. hˆ(P ) does not depend on the Weierstrass equation and is uniquely characterized by the following: ˆ (1) h : E(Q) → R≥0. (2) hˆ(2P ) = 4hˆ(P ). (3) hˆ(P ) − h(P ) is bounded for P ∈ E(Q). 17 Definition 4.11. Let r ≥ 2. Define the Néron–Tate height pairing h−, −i : E(Q)×E(Q) → R≥0 by 1 hP,Qi := hˆ(P + Q) − hˆ(P ) − hˆ(Q) . 2 In particular, hP,P i = hˆ(P ). Definition 4.12. The regulator of E is defined to be

R(E) := det (hPi,Pjir×r) , where {Pi}1≤i≤r are generators of E(Q)/E(Q)tors. Remark. ˆ (1) h(P ) = 0 ⇐⇒ P ∈ E(Q)tors, so h−, −i is perfect on E(Q)/E(Q)tors and R(E) 6= 0. (2) When r = 0, R(E) = 1. (3) When r = 1, R(E) = hˆ(P ). L(r)(E, 1) Conjecture 4.13. ∈ Q×. Ω(E)R(E) This is the ﬁrst step towards the full BSD conjecture Remark. • When r = 0, this is true (proved). • When r = 1, this is true by the r = 0 case and Gross–Zagier formula. • When r ≥ 2, no example is known! But there is enormous numerical evidence. L(r)(E, 1) 4.3. Full BSD conjecture. Our goal is to ﬁnd a formula for . Recall the three Ω(E)R(E) examples we did earlier: L(r)(E,1) E N r Ω(E)R(E) 2 3 1 y = x − x 32 0 8 2 3 1 y = x + 1 36 0 6 2 3 1 y = x − 25x 800 1 2 Idea: Instead of looking at R ω only, we should look at R |ω |. E(R) E E(Qp) E Fact. Z 0 |Ee(Fp)| |ωE| = [E(Qp): E (Qp)] · , E(Qp) p

0 |Ee(Fp)| −1 where E (Qp) = {P ∈ E(Qp): P mod p ∈ Ee(Fp)}. (Recall that p = Lp(E, 1) .) Deﬁnition 4.14. The local Tamagawa number is deﬁned to be 0 cp(E) := [E(Qp): E (Qp)]. Remark.

• cp(E) = Φ(Fp) where ΦFp is the component group scheme of the N´eronmodel of E at p. • cp = 1 if p - N. 18 This is easily computed using Tate’s algorithm. Now we can complete the table above: L(r)(E,1) Q L(r)(E,1) E N r cp Q E(Q)tors Ω(E)R(E) Ω(E)· p cp·R(E) 2 3 1 1 y = x − x 32 0 8 c2 = 2 16 Z/2 × Z/2 2 3 1 1 y = x + 1 36 0 6 c2 · c3 = 3 · 2 = 6 36 Z/2 × Z/3 2 3 1 1 y = x − 25x 800 1 2 c2 · c5 = 2 · 4 = 8 16 Z/2 × Z/2 Next time, we will complete the full formula using the data we see.

5. Lecture 5 (September 24, 2018) 5.1. Full BSD conjecture. Last time we saw from a few examples with r = 0, 1 that

L(r)(E, 1) 1 =? . Q 2 Ω(E) · p cp(E) · R(E) |E(Q)tor| Let’s rewrite this as (r) L (E, 1) ? Y R(E) = Ω(E) · cp(E) · 2 . r! |E(Q)tor| p | {z } | {z } regulator period

R(E) R(Λ) Remark. 2 is more canonical: for any Λ ⊆ E(Q) of rank r, 2 is independent of |E(Q)tor| [E(Q):Λ] the choice of Λ.

Notice the similarity with the class number formula: for any number ﬁeld K/Q,

2r1 (2π)r2 R(K) ress=1 ζK (s) = 1/2 · × · h(K) . |dK | |OK,tor| | {z } | {z } | {z } class number period regulator

Missing in the formula of BSD above is an analogue of h(K) = |Cl(K)|; this will be provided by the size of the Tate–Shafarevich group X(E). Conjecture 5.1 (Full BSD conjecture). Let E/Q be an elliptic curve. Then

(1) (rank part) ralg(E) = ran(E). (2) (formula part)

L(r)(E, 1) Y R(E) = Ω(E) · c (E) · · |X(E)|. r! p |E(Q) |2 p tor This can be seen as a vast generalization of the class formula formula.

Remark. (1) is known when ran(E) ≤ 1 (Gross–Zagier, Kolyvagin). (2) is known for many (but not all) E/Q with ran(E) ≤ 1; the main ingredient is Skinner–Urban’s work on the Iwasawa main conjecture and Kato’s Euler systems. 19 5.2. Tate–Shafarevich group. Deﬁnition 5.2. The Tate–Shafarevich group of E is deﬁned to be ! 1 Y 1 X(E) := ker H (Q,E) → H (Qv,E) . v In other words, an element of X(E) is represented by a genus 1 curve C/Q such that Jac(C) ' E and C(Qv) 6= ∅ for all places v of Q. X(E) measures the failure of local-global principle for Q-points (for C).

2 4 Example 5.3 (Lind, 1940). The curve C : 2y = x − 17 has points over all Qp and R, but has no Q-points. This corresponds to the elliptic curve E = Jac(C): y2 = x3 + 17x which has r = 0 and X(E) ' (Z/2)2. To check the BSD formula, we compute that L(E, 1) = 3.6523 ··· , Ω(E) = 1.82618 ··· , Q p cp(E) = c2 · c17 = 1 · 2 = 2,

E(Q)tor = Z/2Z, and hence L(E, 1) 2 |X(E)| 4 = = 1 and = = 1. Q 2 Ω(E) p cp 1 · 2 |E(Q)tor| 4 3 3 3 Example 5.4 (Selmer, 1951). The curve C : 3x + 4y + 5z = 0 has points over all Qp and R, but no Q-points. This corresponds to the elliptic curve E = Jac(C): x3 + y3 + 60z3 = 0, (with Weierstrass equation y2 = x3 − 24300), which has X(E) = (Z/3)2. Remark. In general, there is no algorithm to compute X(E). The full BSD formula helps to compute |X(E)|. Remark. To see why |X(E)| is an analogue of the class number, note that the class group

Cl(K) can also be described in terms of Galois cohomology. Setting S = ResOK /Z Gm, we have ! ∼ 1 Y 1 Cl(K) = ker H (Q,S) → H (Qv,S) v measuring the failure of any ideal that is locally principal but not globally principal. We know Cl(K) is finite (using geometry of numbers), but we don’t know the finiteness of X(E). Conjecture 5.5. X(E) is finite. Remark.

(1) This conjecture is true if ran(E) ≤ 1 (Gross–Zagier, Kolyvagin). (2) No example is known where ran(E) ≥ 2! 20 Remark (Cassels–Tate pairing). There exists an alternating pairing X(E) × X(E) → Q/Z whose kernal is the divisible part of X(E), so X(E) is ﬁnite =⇒ X(E) ' G × G =⇒ |X(E)| is a square! Remark. Delaunay (2001) developed Cohen–Lenstra heuristics for X(E) as “random” ﬁ- nite abelian groups with nondegenerate alternating pairing (H, (−, −)) (with frequency 1 Aut(H,(−,−)) ). It predicts that the probability of p | |X(E)| is given by: • r = 0: Y 1 f (p) = 1 − 1 − . 0 p2k−1 k≥1 • r = 1: Y 1 f (p) = 1 − 1 − . 1 p2k k≥1 For example,

p f0(p) f1(p) 2 0.58 0.31 3 0.36 0.12 5 0.21 0.04 None of this is support by actual data. However, we don’t even know if given p there exists E/Q such that p | |X(E)| (not to mention positive probability!). Remark. It is known that for p = 2, 3, 5, 7, 13, X(E)[p∞] can be arbitrarily large. The proof uses the fact that X0(p) has genus 0, so p = 11 is not allowed here. 5.3. Tamagawa numbers. There is a reformulation of the BSD formula, due to Bloch, in terms of Tamagawa numbers, which is similar to the case of linear algebraic groups. Definition 5.6. Let G be a semisimple algebraic group over Q. Take a top differential ω (defined over Q) of G. Then |ω|v (volume form) gives a Haar measure µv on G(Qv). Then Q the Tamagawa measure is µ = v µv on G(A), and the Tamagawa number of G is τ(G) := µ(G(Q)\G(A)). Theorem 5.7 (Weil’s conjecture). If G is simply-connected, then τ(G) = 1. This is proved by Langlands, Lai, Kottwitz, etc.

Example 5.8. Let G = SL2/Q. Then Y τ(G) = µ∞(SL2(Z)\SL2(R)) · µp(SL2(Zp)). p But |SL (F )| µ (SL (Z )) = 2 p = 1 − p−2 = ζ (2)−1, p 2 p p3 p so π2 τ(G) = 1 =⇒ µ (SL (Z)\SL (R)) = ζ(2) = ∞ 2 2 6 π =⇒ Vol(SL (Z)\H) = . 2 3 21 Example 5.9. Let G = SLn/Q. Then

τ(G) = 1 ⇐⇒ µ∞(SLn(Z)\SLn(R)) = ζ(2) ··· ζ(n) ζ(2) ··· ζ(n) ⇐⇒ Vol(SL (Z)\Hn) = n2n−1 · . n Vol(S1) ··· Vol(Sn−1) We can do this for E too (next time).

6. Lecture 6 (September 26, 2018)

6.1. Bloch’s reformulation of BSD formula. Last time we saw that for G = SLn, the Tamagawa number is τ(G) = 1; this special case is due to Minkowski and Siegel using geometry of numbers, before the formulation of Weil’s conjecture for semisimple algebraic groups. But for more general algebraic groups, τ(G) may not be well-deﬁned.

Example 6.1. Let G = Gm. Then |G (F )| p − 1 1 µ (G(Z )) = m p = = 1 − . p p p p p Q 1 Notice p(1 − p ) doesn’t converge! To fix this, the idea is to regularize this product by changing the local measure to be 1−1 1 − µ = ζ (1)µ , p p p p and take µ(G(Q)\G(A)) τ(G) = , ζ∗(1) ∗ where ζ (1) is the leading coefficient of ζ(s) at s = 1. Then τ(Gm) = 1. Fro a general linear algebraic group, there exists an L-function L(G, s) such that for all p∈ / S (a finite set of bad primes), |G(F )| p = L (G, 1)−1. pdim G p

Example 6.2. Let G = Gm. Then L(G, s) = ζ(s).

Example 6.3. Let G = GLn. Then L(G, s) = ζ(s)ζ(s + 1) ··· ζ(s + n − 1). Definition 6.4. For any algebraic group G, define the modified Tamagawa measure ∗ Y Y µ := Lv(G, 1)µv · µv v∈ /S v∈S and the Tamagawa number µ∗(G(Q)\G(A)) τ(G) := ∗ . LS(G, 1) Then we have the following generalization of Weil’s conjecture. Theorem 6.5 (T. Ono, Oesterlé,Kottwitz). For any connected linear algebraic group G/Q, |Pic(G) | τ(G) = tor . |X(G)| 22 1 Q 1 Here X(G) = ker (H (Q,G) → v H (Qv,G)) is the Tate–Shafarevich set. Remark. If G is linear, X(G) is known to be finite (Borel–Serre, 1966).

Remark. Suppose G is simply connected and semisimple. Then indeed |Pic(G)tor| = 1. The veriﬁcation that |X(G)| = 1 is a hard theorem due to Harder (1965), Kneser (1966), and Chernousov (1989) for E8, which completed Kottwitz’s proof of the Tamagawa number conjecture.

Bloch’s idea is to look at elliptic curves. Suppose ralg(E) = r. Take a generator Pi ∈ E(Q). 1 Using the fact that E ' Eb = Ext (E, Gm), Pi gives rise to an extension of groups

0 → Gm → XPi → E → 0.

If {Pi} is a basis of E(Q), then r 0 → Gm → X → E → 0. Theorem 6.6 (Bloch). The Tamagawa number conjecture for X is equivalent to the BSD formula for E. In fact, X(X) = X(E), and Bloch’s theorem gives an interpretation of the regulator R(E) also as a volume.

6.2. p-Selmer groups. Selmer groups are more accessible than the Mordell–Weil group E(Q) and the Tate–Shafarevich group X(E), and will be used to prove the BSD conjecture. Start with p 0 → E[p] → E → E → 0. Taking H∗(Q, −), we get a long exact sequence p p 0 → E(Q)[p] → E(Q) → E(Q) → H1(Q,E[p]) → H1(Q,E) → H1(Q,E) → · · · and hence E(Q) 0 → → H1(Q,E[p]) → H1(Q,E)[p] → 0. pE(Q) 1 We want to understand E(Q), but H (Q,E[p]) is an inﬁnite-dimensional Fp-space. Instead we look at the local picture: E(Q) 0 / δ / H1(Q,E[p]) / H1(Q,E)[p] / 0 pE(Q)

locv Y E(Qv) δv Y Y 0 / / H1(Q ,E[p]) / H1(Q ,E)[p] / 0 pE(Q ) v v v v v v

1 Here H (Qv,E[p]) is ﬁnite for each v! Deﬁnition 6.7. The p-Selmer group is 1 Selp(E) := {x ∈ H (Q,E[p]) : locv(x) ∈ im(δv) for all v}. 23 In other words, we have a pullback diagram

1 Selp(E) / H (Q,E[p])

Y E(Qv) Y / H1(Q ,E[p]) pE(Q ) v v v v E(Q) which induces a natural map → Sel (E). pE(Q) p By construction, we have a “p-descent exact sequence” E(Q) 0 → → Sel (E) → X(E)[p] → 0. pE(Q) p

Selp(E) is described locally, and hence more accessible than E(Q) and X(E) which contain global information.

Theorem 6.8. Selp(E) is ﬁnite.

1 Idea of proof. Selp(E) ⊆ H (Q,E[p]) can be thought of as a subspace “cut out” by local 1 1 conditions im(δv) ⊆ H (Qv,E[p]) for all v. View a class x ∈ H (Q,E[p]) as a number field L/Q. Then the local condition at v∈ / S (finite set of bad primes) imply that L is unramified at v (control on the ramification). The theorem follows because there are only finitely many number fields L/Q with exponent p and unramified outside S!

Since dimFp E(Q)/pE(Q) = ralg(E) + dimFp E(Q)[p], we obtain the Corollary 6.9 (p-descent).

dimFp Selp(E) = ralg(E) + dimFp E(Q)[p] + dimFp X(E)[p].

In particular, dimFp Selp(E) ≥ ralg(E) + dimFp E(Q)[p].

Remark. Selp(E) is most computable when p = 2. Example 6.10 (Heath-Brown, 1994). For the family {y2 = x3 − n2x}, Heath-Brown proved that the probability that dim Sel2(E) = d + dim E(Q)[2] = d + 2 is given by

−1 d Y 1 Y 2 P (d) := 1 + . 2k 2k − 1 k≥0 k=1 For example, d P (d) 0 0.2097 1 0.4194 2 0.2796 3 0.0799 4 0.0107 24 Remark. Poonen–Rains developed Cohen–Lenstra heuristics (for all E/Q): −1 d ? Y 1 Y p Prob(dim Sel = d) = 1 + . (6.1) p pk pk − 1 k≥0 k=1 This has the following consequences: (1) Average(|Selp(E)|) = p + 1. This is proved by Bhargava–Shankar for p = 2, 3, 5. (2) 1 1 Prob(r (E) ≥ 2) ≤ + =⇒ Prob(r (E) ≥ 2) = 0. alg p p2 alg This is consistent with Goldfeld’s conjecture; in fact, (6.1) implies that 50% of E have ralg = 0 and 50% have ralg = 1. Remark.

(1) Selp(E) = 0 =⇒ ralg(E) = 0. ∞ (2) Assume X(E)[p ] < ∞. Then dim Selp(E) = 1 =⇒ ralg(E) = 1. The implication in (2) is still open in general, but Skinner–Zhang proved many cases, and combining with Bhargava they get that at least 20% of E/Q have ralg = 1. Next time we will state a reformulation of BSD in terms of Selmer groups.

7. Lecture 7 (October 8, 2018) 1 Last time we defined the p-Selmer group Selp(E) ⊆ H (Q,E[p]), which is a finite-dimensional 1 Fp-vector space inside an infinite-dimensional global H . It fits into a p-descent sequence E(Q) 0 → → Sel (E) → X(E)[p] → 0. pE(Q) p

Thus bounds on Selp(E) lead to bounds on the algebraic rank of E. However, one disadvan- tage with this sequence is that only the p-torsion of X(E) is present. Taking limit over all p-powers will allow us to see the p-primary part of X(E). 7.1. p∞-Selmer group. Consider the pn-descent sequence

E(Q) n 0 → → Sel n (E) → X(E)[p ] → 0. pnE(Q) p Taking lim with respect to the maps Z/pn ,→ Z/pn+1 and E[pn] ,→ E[pn+1], we obtain −→n

Qp ∞ 0 → E(Q) ⊗ → Selp∞ (E) → X(E)[p ] → 0. Zp ∞ Definition 7.1. The p -Selmer group is Sel ∞ (E) := lim Sel n (E). p −→n p Remark. We have an exact sequence ∞ E(Q)[p ] n 0 → → Sel n (E) → Sel ∞ (E)[p ] → 0. pnE(Q)[p∞] p p ∞ In particular, if E(Q)[p] = 0, then Selp∞ (E) ' Selp∞ (E)[p ]. 25 r Notice that Selp∞ (E) ' (Qp/Zp) × (finite group). Definition 7.2.

rp(E) := corank(Selp∞ (E)) Q = number of copies of p . Zp Corollary 7.3.

(1) rp(E) ≥ ralg(E). ∞ (2) rp(E) = ralg(E) ⇐⇒ X(E)[p ] is ﬁnite. In particular, BSD and the ﬁniteness of X imply

Conjecture 7.4 (Bloch–Kato). rp(E) = ran(E) for all p. Remark.

(1) rp(E) is more accessible than ralg(E) due to its local nature. (2) By a theorem of Bloch–Kato, Selp∞ (E) can be deﬁned in terms of the p-divisible ∞ group E[p ]/Q or Vp(E)/Q. Warning: In general Selp(E) may not only depend E[p]. Bloch–Kato generalizes this conjecture to any p-adic Galois representation V coming from geometry. Our target theorem is Theorem 7.5 (Gross–Zagier, Kolyvagin).

(1) ran(E) = 0 =⇒ rp(E) = 0. (2) ran(E) = 1 =⇒ rp(E) ≤ 1 and ralg(E) ≥ 1.

Remark. (1) and (2) imply that BSD and Bloch–Kato are true when ran(E) ≤ 1, and that X(E)[p∞] is ﬁnite for all p.

The goal today is to illustrate how to bound Selp∞ (E) from above in the simplest case. 7.2. Tate’s local duality. In rough terms, local duality allows us to quantify different local 1 ∞ conditions in H (Qv,E[p ]). n Definition 7.6. Let M be a finite GQ-module (e.g. M = E[p ]). Define the Cartier dual of M ∨ M (1) := Homab.gp(M, Gm). ∨ The natural pairing M × M (1) → Gm, together with the cup product, gives a pairing i 2−i ∨ 2 h−, −iv : H (Qv,M) × H (Qv,M (1)) → H (Qv, Gm) ' Q/Z. Theorem 7.7 (Tate).

E(Qv) δv 1 n n → H (Qv,E[p ]). p E(Qv) Then 1 n Selpn (E) = {x ∈ H (Q,E[p ]) : locv(x) ∈ Lv for all v}

Theorem 7.12 (Tate). Lv is its own annihilator under h−, −iv.

It would be helpful if we could characterize the local conditions Lv without reference to the local points E(Qv). This is possible in some cases. n 7.3. Local conditions at good primes. The goal is to pin down Lv in terms of only E[p ]. 7.3.1. Case v 6= p.

Definition 7.13 (“Unramified” or “finite” condition). Assume |M| is invertible in Zv. Define

1 1 1 Iv 1 Hf (Qv,M) = Hur(Qv,M) := H (Fv,M ) ⊆ H (Qv,M), i.e., the extension classes which are unramiﬁed. The inﬂation-restriction sequence gives

1 Iv 1 1 Frobv=1 0 → H (Fv,M ) → H (Qv,M) → H (Iv,M) → 0. This leads to a diﬀerent kind of condition. Deﬁnition 7.14 (“Singular” condition).

1 ! 1 1 Frobv=1 H (Qv,M) Hsing(Qv,M) := H (Iv,M) = 1 . Hf (Qv,M) 1 1 ∨ Proposition 7.15. Hf (Qv,M) and Hf (Qv,M (1)) are annihilators of each other under h−, −iv. 1 1 ∨ Corollary 7.16. Hf (Qv,M) × Hsing(Qv,M (1)) → Q/Z is a perfect pairing. Apply this to M = E[pn]. Proposition 7.17. If v 6= p and E has good reduction at v, then 1 n 1 n Lv = Hf (Qv,E[p ]) = H´et(Zv, E[p ]) .

where E is the N´eron model of E, so that E(Zv) = E(Qv). 27 7.3.2. Case v = p. The situation is more involved, but the last proposition suggests the following

Definition 7.18 (Flat condition). Let M be a finite flat group scheme over Zp. Define 1 1 1 Hf (Qp, M) := Hfppf (Zp, M) ⊆ H (Qp, M).

1 1 ∨ Proposition 7.19. Hf (Qp, M) and Hf (Qp, M (1)) are annihilators of each other under h−, −ip. Analogously with the case v 6= p, we have the Proposition 7.20. If v = p and E has good reduction at p, then 1 n Lv = Hf (Qp, E[p ]).

Remark. If n = 1 and p > 2, Raynaud shows that E[p] has a unique extension to Zp, so 1 Hf (Qp, E[p]) is determined by E[p]. 7.4. Kolyvagin’s method. The starting point is a simple application of global class ﬁeld 1 P theory: for s1, s2 ∈ H (Q,E[p]), we have vhs1, s2iv = 0. This condition can be used to bound Selmer ranks. Theorem 7.21. Let S = {bad primes for E} ∪ {p}. Assume we can construct c(`) ∈ H1(Q,E[p]) for all `∈ / S satisfying: 1 (1) For v∈ / S ∪ {`}, locv(c(`)) ∈ Hf (Qv,E[p]). 1 (2) For v = p, locp(c(`)) ∈ Hf (Qp,E[p]). (3) For v ∈ S − {p}, locv(c(`)) = 0 (e.g. when E(Qv)[p] = 0). 1 1 (4) For v = `, Hf (Q`,E[p]) = Fp and 0 6= loc`(c(`)) ∈ Hsing(Q`,E[p]).

Then Selp(E) = 0. Next time we will explain this.

8. Lecture 8 (October 10, 2018) 8.1. Kolyvagin’s method. Last time we deﬁned, for good primes v, the unramiﬁed conditions 1 1 Hf (Qv,E[p]) ⊆ H (Qv,E[p]) 1 1 1 and for v 6= p, the singular conditions Hsing(Qv,E[p]). We have hHf ,Hf iv = 0, and that 1 1 h−, −iv is perfect on Hf × Hsing. Theorem 8.1. Let S = {bad primes} ∪ {p}. Assume we can construct classes c(`) ∈ H1(Q,E[p]) for all `∈ / S such that: 1 (1) For v∈ / S ∪ {`}, locv(c(`)) ∈ Hf (Qv,E[p]). 1 (2) For v = p, locp(c(`)) ∈ Hf (Qp,E[p]). (3) For v ∈ S − {p}, locv(c(`)) = 0. 1 1 (4) For v = `, Hf (Qv,E[p]) = Fp and loc`(c(`)) 6= 0 ∈ Hsing(Qv,E[p]).

Then Selp(E) = 0. 28 Proof. Assume Selp(E) 6= 0, and take 0 6= s ∈ Selp(E). By Chebotarev density, we can ﬁnd 1 `∈ / S such that 0 6= loc`(s) ∈ Hf (Q`,E[p]). Then by (4) X by (1)-(3) hs, c(`)i = hs, c(`)iv = hs, c(`)i` 6= 0

1 1 since s ∈ Hf and c(`) ∈ Hsing. This is a contradiction, since hs, c(`)i = 0 by global class ﬁeld theory. Thus Selp(E) = 0. Remark. The proof shows that we only need c(`) for “many” (a positive proportion of) `∈ / S.

By the theorem, the key to bound Selmer groups is the construction of the classes {c(`)}`/∈S with controlled ramiﬁcation. Next, we will carry out the construction via Heegner points.

Remark. The existence of {c(`)} should be related to ran(E) = 0.

BSD/BK Selp(E) o / L(E, s) d :

Kolyvagin’s $ z explicit reciprocity law method {c(`)} (Bertolini–Darmon)

8.2. Heegner points on X0(N). Recall the modular curve Y0(N) := Γ0(N)\H, where a b Γ (N) = : N | c ⊆ SL (Z), and the compactiﬁed modular curve X (N) := 0 c d 2 0 ∗ Γ0(N)\H . More precisely, what we are deﬁning here are the complex points Y0(N)(C) and X0(N)(C). In the simplest case, Y0(1) = SL2(Z)\H. For τ ∈ H, consider the elliptic curve Eτ := 0 C/Z + Zτ. It is easy to check that Eτ ' Eτ 0 if and only if τ and τ are in the same SL2(Z)-orbit, so Y0(1)(C) = {elliptic curves E/C}. 1 N Z+Zτ To describe Y0(N), consider pairs Eτ ,Cτ = Z+Zτ ' Z/N . Analogously as above, 0 (Eτ ,Cτ ) ' (Eτ 0 ,Cτ 0 ) if and only if τ and τ are in the same Γ0(N)-orbit, and hence

Y0(N)(C) = {(E,C): E elliptic curve over C, C ⊆ E cyclic group of order N} = {E → E0 = E/C: cyclic N-isogeny}.

This moduli interpretation gives a model of X0(N) (and Y0(N)) over Q. We want to construct points on X0(N) and use the modular parametrization X0(N)/Q → E/Q. However, X0(N)(Q) tends to be small (as a curve of genus at least 2 except for small value of N). To overcome this problem, we instead construct algebraic points on X0(N) lying in small degree number fields. ∗ ∗ As a naive try, we consider the uniformization H → X0(N)(C). Although H contains many algebraic points, this map is highly transcendental and will not send algebraic points to algebraic points! The miracle, though, is that this may still work in special cases, namely for imaginary quadratic points. To see this, we use the moduli interpretation. Recall elliptic curves E over C can be divided into two possibilities: (1) End(E) = Z. 29 ∼ p (2) End(E) = O ⊆ OK , where K = Q( −|dK |) is an imaginary quadratic field and O ⊆ OK is a finite index subring. Definition 8.2. An elliptic curve E in case (2) is said to have complex multiplication (CM). Example 8.3. The elliptic curve E : y2 = x3 + nx has CM by O = Z[i], where [i](x, y) = (−x, iy).

2 3 Example 8.4. The elliptic curve E : y = x + n has CM by O = Z[ζ3], where

[ζ3](x, y) = (ζ3x, y). The theory of complex multiplication tells us that CM elliptic curves are defined over number fields. Recall that for a number field K, the Hilbert class field HK is the maximal unramified abelian extension of K, with Gal(HK /K) ' Cl(K). Theorem 8.5 (Main Theorem of CM). There is a bijection ∼ {elliptic curves with CM by OK } → {C/a : a ∈ Cl(K)}.

Moreover, each such elliptic curve can be deﬁned over the Hilbert class ﬁeld HK .

φ 0 Deﬁnition 8.6. A Heegner point is a point xK = (E → E ) ∈ X0(N)(C), where φ is a 0 cyclic N-isogeny, such that End(E) = End(E ) = OK .

We know that xK ∈ X0(N)(HK ).

Remark. A Heegner point xK ∈ X0(N) exists ⇐⇒ there exists a, b ∈ Cl(K) such that −1 C/a → C/b is a cyclic N-isogeny ⇐⇒ b/a 'OK /ab is isomorphic to Z/N ⇐⇒ there exists an ideal N ⊆ OK such that OK /N' Z/N.

Deﬁnition 8.7. We say that K satisﬁes the Heegner hypothesis for X0(N) if every prime dividing N splits in K.

If this is true, then taking a prime p above every p | N and N = Q pordp(N) gives O/N = Z/N, and so xK exists. By Chebotarev density, there are inﬁnitely many K satisfying the Heegner hypothesis for a given N.

8.3. Heegner points on elliptic curves.

Deﬁnition 8.8. Fix a modular parametrization ϕ : X0(N) → E sending ∞ ∈ X0(N) to 0 ∈ E. Deﬁne a Heegner point on E to be X yK := σ(ϕ(xK )) ∈ E(K).

σ∈Gal(HK /K) It is not necessary to further trace down to Q. Using the moduli interpretation, one can check the

Proposition 8.9. yK = −(E)yK in E(K)/E(K)tor, where (E) ∈ {±1} is the sign of the functional equation.

In particular, yK ∈ E(Q) + E(K)tor if and only if (E) = −1. 30 √ 2 3 Example 8.10. For the elliptic curve E = X0(32) : y = x√ + 4x, K = Q( −7) satisﬁes the Heegner hypothesis for X0(32). Then HK = K = Q( −7) since Cl(K) = 0. Using ∗ ∗ H → E = Γ0(32)\H , we ﬁnd √ √ −7 − 1 −7 + 3 x = y = , ∈ E(K). K K 2 2

Moreover, yK has infinite order, and (E) = +1. Heegner used this construction to prove the following 2 3 2 Theorem 8.11. The elliptic curve E : y = x − n x has ralg ≥ 1 when n is a prime ≡ 5 (mod 8). 2 3 Example√ 8.12. For the elliptic curve E : y + y = x − x with conductor N = 37, K = Q( −7) satisfies the Heegner hypothesis for X0(37). Then yK = (0, 0) ∈ E(K) has infinite order, and this agrees with (E) = −1.

Next time we will relate the Heegner point yK with L(E, 1) via the Gross–Zagier formula, and prove that ran(E) = 1 =⇒ ralg(E) ≥ 1. 9. Lecture 9 (October 15, 2018) Last time we introduced the Heegner hypothesis: an imaginary quadratic ﬁeld K satisﬁes the Heegner hypothesis for X0(N) if every p | N splits in K, (Heeg)

and used this to construct Heegner points xK ∈ X0(N)(HK ) and yK ∈ E(K).

9.1. Gross–Zagier formula. Let EK be the base change of E from Q to K. Then L(EK , s) satisﬁes a functional equation

L(EK , s) ↔ L(EK , 2 − s)

with sign of functional equation (EK ). There is a simple formula for computing (EK ).

Proposition 9.1. Assume (dK ,N) = 1. Then (EK ) = χK (−N), where χK : Z/|dK | → {±1} is the quadratic character associated to K/Q.

Corollary 9.2. If K satisﬁes (Heeg), then (EK ) = −1. More generally,

Y ordp N (EK ) = − (−1) . p inert in K To summarize, we have a diagram

(EK ) = −1 +3 ran(EK ) = odd 4< =

(Heeg) ?

"* } yK ∈ E(K)

so it is natural ask to ask if there is a relationship between yK ∈ E(K) and ran(EK ) = odd. 31 Theorem 9.3 (Gross–Zagier). R ω ∧ iω L0(E , 1) = E(C) hy , y i . K 1/2 × 2 K K NT |dK | |OK //{±1}| 0 1 ∗ Here ω ∈ H (EQ, Ω ) is such that ϕ ω = 2πifE(z) dz, where fE is the normalized newform associated to E. Remark. Since Z Z ω ∧ iω · deg ϕ = 8π2f(z)f(z) dx dy = (f, f) E(C) X0(C) is the Petersson inner product, we can rewrite the Gross–Zagier formula as (f, f) hy , y i L0(E , 1) = · K K NT . K 1 × 2 2 deg ϕ |dK | |OK /{±1}|

Remark. The Heegner point yK depends on the choice of N ⊆ OK , but is determined up to hy , y i sign and torsion. It also depends on ϕ : X (N) → E, but K K NT is canonical! 0 deg ϕ The proof of the Gross-Zagier formula is a computation. To understand it conceptually, we need to rephrase the formula. 0 Corollary 9.4. L (EK , 1) = 0 ⇐⇒ hyK , yK iNT = 0 ⇐⇒ yK is of inﬁnite order. Thus ran(EK ) = 1 =⇒ ralg(EK ) ≥ 1. Remark. Comparison with the BSD formula gives

1 ? [E(K): ZyK ] | (E )| 2 = X K Q × 2 p cp(E) · |OK | · c ∗ where c is the Manin constant (conjectured to be 1): ϕ ωE = c · 2πifE(z) dz. In particular, this gives the conjectural implication ( p - yK ? ∞ =⇒ X(E )[p ] = 0. E(K)[p] = 0 K 9.2. Back to E/Q. Now we explain how the result above descends to Q. Deﬁnition 9.5. Let E(K)/Q be the quadratic twist of E by K, i.e., the unique elliptic curve over Q that is isomorphic to E over K but not isomorphic to E over Q. Explicitly, if E has Weierstrass equation y2 = x3 + Ax + B, then (K) 2 3 E : dK y = x + Ax + B.

Remark. Consider ρE : GQ → Aut(VpE). Then ρE(K) = ρE ⊗ χK where χK : GQ GK/Q ' {±1}. (K) Proposition 9.6. L(EK , s) = L(E, s)L(E , s). Proof. Check by deﬁnition, or L(E , s) = L(IndGQ ρ , s) = L(ρ ⊕ ρ ⊗ χ , s) = L(ρ , s)L(ρ ⊗ χ , s). K GK E E E K E E K (K) Corollary 9.7. ran(EK ) = ran(E) + ran(E ). 32 By BSD, this predicts an identity on algebraic ranks, which is also easy to see directly. (K) Proposition 9.8. ralg(EK ) = ralg(E) + ralg(E ). Proof. Looking at the action of complex conjugation, we have E(K) ⊗ Q ' E(Q) ⊗ Q ⊕ E(K)(Q) ⊗ Q into the eigenspaces for c = ±1.

Theorem 9.9. If ran(E) = 1, then ralg(E) ≥ 1. Proof. By a theorem of Waldspurger (next time), one can choose K satisfying (Heeg) such (K) that ran(E ) = 0. Then ran(EK ) = 1 + 0 = 1, hence ralg(EK ) ≥ 1. But (E) = −1, so c c yK = −(E)yK = yK in E(K)/E(K)tor. Therefore P = yK + yK ∈ E(Q) is of inﬁnite order, and ralg(E) ≥ 1.

Corollary 9.10. If ran(E) = 1, then L0(E, 1) ∈ Q×. Ω(E)R(E) L(E(K), 1) Proof. Choose K as before. Then L0(E , 1) = L0(E, 1)L(E(K), 1), where ∈ Q×. K Ω(E(K)) R ω ∧ iω × E(C) Q (K) The result follows from the Gross–Zagier formula using 1 ∼ Ω(E)Ω(E ). |dK | 2 c Corollary 9.11. If (E) = −1 and yK + yK ∈ E(Q) is torsion, then ran(E) ≥ 3.

Using this we can construct E/Q with ran(E) = 3.

Remark. There is no example of E/Q with provably correct ran(E) = 4. Theorem√ 9.12 (Goldfeld, 1976). If there exists EQ with ran(E) ≥ 3, then the class number of Q( −D) has an explicit lower bound 1−δ h(D) Cδ,E · (log |D|)

for all δ > 0, where Cδ,E is an effective constant. Example 9.13 (Gauss elliptic curve). Consider E : y2 + y = x3 − 7x + 6 with N = 5077. c Buhler–Gross–Zagier computed that yK + yK is torsion, hence ran(E) ≥ 3. Together with Goldfeld, this solves Gauss’ class number problem. 9.3. Gross–Zagier formula for Shimura curves. Fix an imaginary quadratic field K/Q. + − + − For (N, dK ) = 1, define N = N N where every p | N splits in K and every p | N is inert in K. Example 9.14. The Heegner hypothesis (Heeg) implies that N = N +. − #{p|N −} Assume N is square-free. Then (EK ) = −(−1) , so ( +1 if #{p | N −} is odd, (E ) = K −1 if #{p | N −} is even. Definition 9.15. K satisfies the generalized Heegner hypothesis (Heeg∗) if N − is a square- free product of an even number of primes.

Next time we will construct yK for K satisfying this. 33 10. Lecture 10 (October 17, 2018) Last time we introduced the generalized Heegner hypothesis:  N = N +N −, with N − a square-free product of an even number of primes,  p | N + =⇒ p split in K, (Heeg*) p | N − =⇒ p inert in K. Our goal is to construct Heegner points in this setting and ﬁll in a similar diagram as last time (Heeg*) +3 (EK ) = −1 +3 ran(EK ) = odd 5 ? ? $, u Heegner points yK 10.1. Gross–Zagier formula for Shimura curves. Since #{p | N −} is even, we can construct the following

Deﬁnition 10.1. Let B = BN be the unique quaternion algebra over Q that is ramiﬁed exactly at {p | N}, i.e.,

( − ∼ M2(Qv) if v - N , Bv = − division quaternion algebra over Qv if v | N . Example 10.2. Every quaternion algebra over Q is of the form B = Q{1, i, j, ij} where i2 = a, j2 = b, ij = −ji. 1 1 • If a = 1, then we recover the matrix algebra B ' M2(Q) via i 7→ ( −1 ), j 7→ ( b ) 1 and ij 7→ ( −b ). q • If a is a prime p ≡ 1 (mod 4) and b is a prime q such that 6= 1, then B is p exactly ramiﬁed at {p, q}. a b As an analogue of M (N) = : N | c , 0 c d

+ Deﬁnition 10.3. An Eichler order of level N is a subring O ⊆ OB (maximal order in B) of ﬁnite index, such that ( + − M0(N )p if p - N , Op = − OBp if p | N , where OBp is the maximal order of Bp.

Now we can define the analogue of Γ0(N). Definition 10.4. Γ(N +,N −) = {γ ∈ O× : γγ = 1}. − + + Example 10.5. If N = 1, then Γ(N , 1) = Γ0(N ). + − × Definition 10.6. Using Γ(N ,N ) ,→ B (R) ' GL2(R), the condition γγ = 1 implies + − that Γ(N ,N ) has image in SL2(R). We define the Shimura curve X(N +,N −) := Γ(N +,N −)\H. 34 Example 10.7. − + − • If N = 1, then X(N ,N ) = Y0(N). • If N − 6= 1, then X(N +,N −) is already compact. Remark. This is an example of a Shimura variety. One can rewrite X(N +,N −) adelically as × × Q follows. Consider the compact open subgroup Kp = Op ⊆ B (Qp). Take Kf = p Kp ⊆ × × + − B (Af ), again a compact open subgroup. Then Kf ∩ B1 (Q) = Γ(N ,N ) and X(N +,N −) = Γ(N +,N −)\H ∼ × × ± = B (Q)\B (Af ) × H /Kf .

Notice H = SL2(R)/SO2(R). Now we are ready to construct Heegner points using the moduli interpretation of Shimura curves, similarly as in the case of modular curves: • In the level 1 case, − X(1,N )(C) = {abelian surface A/C with OB ,→ End(A)} 2 τ by sending τ ∈ H to C /OB · ( 1 ) via OB ,→ M2(C). • In the general case, + − 0 + 2 X(N ,N )(C) = {A → A : cyclic OB-isogeny of degree (N ) }. − Example 10.8. If N = 1, then Aτ = Eτ × Eτ .

Let K be an imaginary quadratic field satisfying (Heeg*), with Hilbert class field HK . 0 + − 2 Definition 10.9. Define Heegner points xK = (A → A ) ∈ X(N ,N )(HK ) if A ∼ E , 0 02 0 A ∼ E and End(E) = End(E ) = OK . Definition 10.10. Using the modular parametrization ϕ : X(N +,N −) → E, define X yK = ϕ(σ(xK )) ∈ E(K).

σ∈Gal(HK /K) Theorem 10.11 (Yuan–Zhang–Zhang). Assume K satisﬁes (Heeg∗). Then (f, f) hy , y i L0(E , 1) = · K K NT . K 1 × 2 2 deg ϕ |dK | |OK /{±1}| Although this looks exactly the same as the original Gross–Zagier formula, the proof is much more complicated.

10.2. Waldspurger’s formula. To study the case (EK ) = +1, we need a formula of Waldspurger which was proved around the same time as Gross–Zagier and concerns the central value L(EK , 1). Assume the hypothesis: N − is a square-free product of an odd number of primes. (Wald)

This implies (EK ) = +1. The goal is to construct “yK ” such that yk ←→ L(EK , 1). But #{p | N −} is odd, so we cannot make a quaternion algebra B that is exactly ramified at {p | N −}. Instead, we consider the 35 Definition 10.12. Let B = BN −∞ be the unique quaternion algebra over Q ramified at {p | N −} ∪ {∞}. In particular, B(R) ' H (where i2 = j2 = −1, ij = −ji), so x −y B×(R) ∼= SU = ∈ M (C): |x|2 + |y|2 = 1 1 2 y x 2 is compact and does not act on H. + Definition 10.13. Let O ⊆ OB be an Eichler order of level N . Define the Shimura set + − × × X(N ,N ) := B (Q)\B (Af )/Kf Q × where Kf = p Op . (This is a 0-dimensional Shimura variety.)

Remark. A quaternion algebra B/Q is called indefinite if B(R) ' M2(R), and definite if B(R) ' H. Remark. Recall that for a number field F , × × Y × ∼ F \AF,f / OF,p → Cl(OF ). p Similarly, for a definite quaternion algebra B, − ∼ X(1,N ) = Cl(OB) = {right ideal classes of OB}. Next we introduce Heegner points on this 0-dimensional Shimura variety.

Definition 10.14. Fix an embedding K,→ B (with K ∩ O = OK ). Then we have × × Y × + − Cl(OK ) = K \AK,f / OK,p → X(N ,N ) = Cl(O) p I 7→ IO. + − A point xK ∈ Cl(OK ) (or its image) is called a CM point (or Gross point) on X(N ,N ). (Cl(OK ) permutes these CM points.) new Definition 10.15. Let f ∈ S2 (N), which transfers under the Jacquet–Langlands correspondence to a function ϕ : X(N +,N −) → C with the same Hecke eigenvalues as f. (This is the analogue of X(N +,N −) → E in the indefinite case.) Define Waldspurger’s toric period X yK = ϕ(xK )(| Aut(xK )|) ∈ C

xK ∈Cl(OK ) where Aut(x) = {γ ∈ B×(Q): γx = x}/{±1}, and X deg ϕ = |ϕ(x)|2(| Aut(x)|). x∈X(N +,N −) Finally, we can state the Theorem 10.16 (Waldspurger). Assume K satisﬁes (Wald). Then (f, f) |y |2 L(E , 1) = · K . K 1 × 2 2 deg ϕ |dK | |OK /{±1}|

Corollary 10.17. L(EK , 1) 6= 0 ⇐⇒ yK 6= 0. 36 11. Lecture 11 (October 22, 2018) Last time we stated Waldspurger’s formula: if K satisﬁes (Wald) for N = N +N −, then (f, f) |y |2 L(E , 1) = · K K 1 × 2 2 deg ϕ |dK | |OK /{±1}|

where yK is the toric period. 11.1. Examples of Waldspurger’s formula.

2 3 2 Example 11.1. Let E = X0(11), with Weierstrass equation y + y = x − x − 10x − 20 and corresponding newform

Y n 2 11n 2 2 2 f = fE = q (1 − q ) (1 − q ) = η(z) η(11z) n≥1 = q − 2q2 − q3 + 2q4 + q5 + ··· . √ Then K = Q( −5) satisﬁes (Wald) (note 11 is inert in K), with N + = 1 and N − = 11. Let + − B = B11,∞ with maximal order O = OB, so that X(N ,N ) = X(1, 11) = Cl(OB). We make use of the following two formulas: X 1 N − − 1 (1) (Eichler’s mass formula) If N − is prime, then = . | Aut(x)| 12 x∈Cl(OB ) Y N − 1 (2) | Aut(x)| = denom . 12 x 10 5 1 1 In our case, Eichler’s mass formula gives 12 = 6 = 2 + 3 , so the second formula conﬁrms that |Cl(OB)| = 2 with | Aut(x1)| = 2 and | Aut(x2)| = 3. Consider the modular parametrization ϕ : X(N +,N −) → C with Hecke eigenvalues equal to those of f. It is easy to check that x1 7→ −1 and x2 7→ 1. Then yK = −2 + 3 = 1 (which implies ran(EK ) = 0) and deg ϕ = 2 + 3 = 5. Now we compute that (K) L(EK , 1) = L(E, 1)L(E , 1) = (0.25384 ··· )(0.65240 ··· ), (f, f) = 3.70309 ··· , L(E , 1) 1 =⇒ K = 0.044721 ··· = √ . (f, f) 10 5 2 1 |yK | 1 1 1 This matches up with 1 · = √ · = √ , as predicted by Waldspurger’s |dK | 2 deg ϕ 20 5 10 5 formula.

Question. How does yK vary when K varies? 3 Answer. They ﬁt into another modular form of weight 2 .

Definition 11.2. Let Cl(O) = {x1, ··· , xn} (right ideal classes of O). For each i, define the left order of xi Ri = {γ ∈ B : γxi = xi}, 37 and Si = {γ ∈ Z + 2Ri : tr(γ) = 0}, which is a rank 3 Z-module together with a quadratic form given by the norm N. 3 Definition 11.3. Define a weight 2 θ-series by 1 X g := qNv. i 2 v∈Si √ Thus the D-th Fourier coefficient of gi encodes the maps xQ( −D) ∈ {CM points} 7→ xi ∈ + − X(N ,N ) under Cl(OK ) → Cl(O). Definition 11.4. Associated to f, define n X X n g := ϕ(xi)gi = bnq . i=1 n≥1

Theorem 11.5. yK = b|dK |.

Example 11.6. Let E = X0(11) and Cl(OB) = {x1, x2} as before. Then we can compute that 1 X 2 2 2 1 g = qx +11y +11z = + q4 + q11 + 2q12 + q16 + 2q20 + ··· , 1 2 2 x,y,z∈Z x≡y (mod 2) 2 2 2 1 X x +11y +33z 1 3 12 15 16 20 g = q 3 = + q + q + 3q + 3q + 3q + ··· , 2 2 2 x,y,z∈Z x≡y (mod 3) y≡z (mod 2) 3 4 11 16 20 and hence√ g = −g1 + g2 = q − q − q + 2q + q + ··· . In particular, yK = 1 for K = Q( −5)! Indeed, for E = X0(11), ? 2 BSD formula for EK ⇐⇒ |X(E/K)| = |b|d || . √ K In particular, X(E/Q( −5)) = 0! The first two non-trivial examples are given by g = q3 − q4 − q11 + 2q16 + q20 + ··· + 3q67 + ··· + 4q91 + ··· . √ √ Indeed, X(E/Q( −67)) = (Z/3)2 and X(E/Q( −91)) = (Z/2)4. 3 Remark. By studying Fourier coefficients of weight 2 forms, one can show there are a lot of (K) K such that yK 6= 0, which implies that ran(E ) = 0 for many K. Recall that this is the input for deducing BSD over Q from BSD over K. 11.2. Back to Kolyvagin. Recall that we have constructed Heegner points on modular curves, Shimura curves or Shimura sets. Our next goal is to construct classes c(`) ∈ 1 1 H (K,E[p]) such that loc`(c(`)) is ramified, i.e., its image in Hsing(K`,E[p]) is nonzero. As a naive try, say K satisfies (Heeg*). Then we have Heegner points yK ∈ E(K). Using the connecting homomorphism E(K) δ : → H1(K,E[p]), pE(K) 38 1 we obtain δ(yK ) ∈ H (K,E[p]). Problem: If `∈ / S = {x | Np∞}, then δ(yK ) is unramified at `. The idea is to find another elliptic curve E(`) such that E[p] →∼ E(`)[p] and E(`) has conductor N`. If so, choose K satisfying (Heeg*) for N`. This gives yK (`) ∈ E(`)(K) and 1 ∼ 1 c(`) := δ(yK (`)) ∈ H (K,E(`)[p]) = H (K,E[p]). Then it is possible that c(`) is ramified at `, as ` | N`. To summarize, we can produce ramification by raising the level under congruences mod p.

new new Question. Given f ∈ S2 (N), when can we ﬁnd g ∈ S2 (N`) such that f ≡ g (mod p) (i.e., aq(f) ≡ qq(g) (mod p) for almost all primes q)? In general this is not possible. For example, when the level is small, the number of modular forms is limited. We can reformulate this question in terms in Galois representations. Consider ρf : GQ → GL2(Zp) and the residual representation ρf : GQ → GL2(Fp).

new ∼ Question. When can we find g ∈ S2 (N`) such that ρf = ρg? ∼ Under this reformulation, we immediately see a necessary condition. Since ρf |G = ρg|G Q` Q` and ρf is unramified at `, we want ρg to be unramified (even though ρg is ramified at `), ` ∗ so g has multiplicative reduction at `. By Tate’s uniformization, Frob is sent to ± . ` 0 1 Taking tr(Frob`), we get a`(f) ≡ ±(` + 1) (mod p).

Deﬁnition 11.7. ` - Np is called a level-raising prime for f (mod p) if

a`(f) ≡ ±(` + 1) (mod p). Ribet shows this is almost a suﬃcient condition:

Theorem 11.8 (Ribet, 1990). If ρf is irreducible, and ` is a level-raising prime for f `-new (mod p), then there exists g ∈ S2 (N`) such that ρf ' ρg.

Example 11.9. Let E = X0(11) and p = 2. It is clear that

a`(f) ≡ ±(` + 1) (mod 2) ⇐⇒ a`(f) is even,

new so ` = 7 is level-raising. The corresponding g ∈ S2 (77) is the newform with label 77a: 2 3 5 7 11 13 17 19 f = fE −2 −1 1 −2 1 4 −2 0 g = 77a 0 −3 −1 −1 −1 −4 2 −6

12. Lecture 12 (October 24, 2018) Last time we stated the

Theorem 12.1 (Ribet). Let f ∈ S2(N) such that ρf is irreducible, and ` be a level-raising `-new prime for f (mod p), i.e., a`(f) ≡ ±(` + 1) (mod p). Then there exists g ∈ S2 (N`) such that ρf ' ρg. 39 12.1. Geometric interpretation of Ribet’s theorem: Ihara’s lemma. Recall that the modular curve X0(N)/Q has the moduli interpretation {(E,C): E elliptic curve, C ⊆ E[N] cyclic subgroup of order N}. There is a correspondence

X0(N`)

π1 π2 y % X0(N) X0(N), where

π1 :(E,C) 7→ (E,C[N]), π2 :(E,C) 7→ (E/C[`], C/C[`]) which make sense since C ' Z/N`.

0 Example 12.2. When N = 1, X0(`) parametrizes isogenies (E → E ) of degree `, and 0 0 0 π1 :(E → E ) 7→ E, π2 :(E → E ) 7→ E . The Hecke operator can be deﬁned as ∗ ∗ T` := π2∗π1 = π1∗π2 ∈ End(J0(N)). ∗ ∗ Consider the map π1 + π2, which ﬁts into the diagram

π∗+π∗ 0 1 ⊕2 1 2 0 1 H (X0(N), Ω ) / H (X0(N`), Ω )

⊕2 S2(N) / S2(N`).

∗ ∗ Concretely, π1(f(z)) = f(z) (via z 7→ z) and π2(f(z)) = `f(`z) (via z 7→ `z). These are `-old forms, so `-new ∗ ∗ S2 (N`) = coker(π1 + π2).

To deal with mod p congruences (which require Zp-structures), we look at ∗ ∗ 1 ⊕2 1 π1 + π2 : H´et(X0(N), Zp) → H´et(X0(N`), Zp). ∗ ∗ After inverting p, this is injective and the Hecke eigenvalues on coker(π1 + π2)Qp are exactly `-new given by those on S2 (N`).

Definition 12.3. Let T be the Hecke algebra generated by Tq for all q - N`. Let χf : T → Zp χf be the Hecke eigenvalues associated to f. Let m = mf := ker(T → Zp Fp) be the maximal ideal of T encoding congruences with f. To prove Ribet’s theorem, it suffices to prove that the map ∗ ∗ 1 ⊕2 1 π1 + π2 : Hét(X0(N), Zp)m → Hét(X0(N`), Zp)m ∗ ∗ (localization at m = mf ) has cokernel containing a point in characteristic 0, i.e., coker(π1 +π2) is not torsion. 40 Theorem 12.4 (Ihara’s lemma). Assume ρf is irreducible. Then ∗ ∗ 1 ⊕2 1 π1 + π2 : Hét(X0(N), Fp)m → Hét(X0(N`), Fp)m is injective.

∗ ∗ Proof that Ihara implies Ribet. If coker(π1 +π2)Zp is torsion, then Ihara’s lemma implies that ∗ ∗ ∗ ∗ coker(π1 + π2)Zp is trivial, so (π1 + π2)Zp is an isomorphism. To get a contradiction, look at the dual map 1 1 ⊕2 (π1∗, π2∗): Hét(X0(N`), Zp)m → Hét(X0(N), Zp)m which is also an isomorphism by duality. The composition 1 ⊕2 ∼ 1 ⊕2 Hét(X0(N), Zp)m → Hét(X0(N), Zp)m is an isomorphism, with 2 × 2 matrix given by ` + 1 T ` + 1 ±(` + 1) ` ≡ (mod p). T ` + 1 ±(` + 1) ` + 1 ` m This is a contradiction to the composition being an isomorphism.

12.2. Geometry of modular curves in characteristic p. Recall that X0(N) has moduli interpretation (E,C) with C ⊆ E[N]. The group E[N] still behaves well in characteristic p - N (i.e., is étale).In particular, if p - N, then 2 E[N](Fp) ' (Z/N) . But E[p] behaves differently in characteristic p: ( Z/p if E is ordinary, E[p](F ) ' p 0 if E is supersingular. Equivalently, ( O ⊆ OK if E is ordinary, End(E/Fp ) ' O ⊆ Op,∞ if E is supersingular, where K is an imaginary quadratic field, and Op,∞ is the maximal order in Bp,∞. To define an integral model of X0(N), we need to change the moduli interpretation.

Theorem 12.5 (Igusa, Deligne–Rapoport, Katz–Mazur). X0(N) has a model over Z. It is ﬂat, projective and of relative dimension 1 over Z.

(1) If p - N, then X0(N)Zp has the same moduli interpretation as before; X0(N)Fp is smooth, and has Newton stratum ord ss X0(N)Fp = X0(N)Fp ∪ X0(N)Fp according to whether E is ordinary or supersingular, where the supersingular locus ss X0(N)Fp is a ﬁnite set of points.

(2) If p k N, then X0(N)Zp has moduli interpretation E/S elliptic curve, C ⊆ E[p] ﬁnite ﬂat S/Zp 7→ (E,C): ; group scheme over Zp of rank N

X0(N)Fp is no longer smooth but is semistable, with two irreducible components iso- ss ss morphic to X0(N/p)Fp and intersecting along X0(N/p)Fp = X0(N) . 41 2 (3) If p | N, then X0(N)Zp has moduli interpretation in terms of Drinfeld level structure: M C = [aP ],P ∈ E[p] a∈Z/p

as a Cartier divisor on E; X0(N)Fp is not smooth and not semistable. Remark. If p k N, the picture looks like

Let us describe the gluing datum. Over Fp, we have

X0(N/p) X0(N/p)

α1 α2 % y X0(N)

π1 π2 y % X0(N/p) X0(N/p) where (assuming N/p = 1 for simplicity)

deg p 0 deg p 0 0 π1 :(E → E ) 7→ E, π2 :(E → E ) 7→ E ,

Frobp (p) (p) Verp α1 : E 7→ (E → E ), α2 : E 7→ (E → E). Notice

π1 ◦ α1 = Id, π2 ◦ α2 = Id,

π1 ◦ α2 = Frobp, π2 ◦ α1 = Verp . (p) (p) (p) The two copies of X0(N/p) are glued via E ↔ E , as (E ) ' E if E is supersingular. ss Next time we will describe X0(N)Fp explicitly and prove Ihara’s lemma. 13. Lecture 13 (October 29, 2018)

Last time we saw that the special ﬁber X0(N)Fp is smooth if p - N. Our goal is to describe ss the supersingular locus X0(N)Fp , which is a ﬁnite set. 13.1. The supersingular locus.

ss Proposition 13.1. X0(N)Fp ' X(N, p), the Shimura set associated to Bp,∞, which is in turn isomorphic to Cl(O) for the Eichler order O of level N in Bp,∞. ss Remark. This is an instance of “switching invariants”: X0(N)Fp is indefinite at ∞ and split at p, while X(N, p) is definite at ∞ and ramified at p. 42 ss Idea of proof. Given (E,C) ∈ X0(N)Fp , we have End(E) 'Op,∞ (maximal order in Bp,∞), under which the subring End(E,C) corresponds to O (Eichler order of level N). 0 0 ss 0 0 Given any point (E ,C ) ∈ X0(N)Fp , look at I = Hom((E,C), (E ,C )), which is a rank 4 Z-module and admits a right action of O = End(E,C): for f ∈ I and g ∈ O, f · g := f ◦ g. This construction gives a map ss X0(N)Fp → X(N, p) = Cl(O) (E0,C0) 7→ I = Hom((E,C), (E0,C0)). One can check this is a bijection. Last time we saw that Ihara’s lemma implies Ribet’s theorem for level-raising congruences. Recall

Theorem 13.2 (Ihara’s lemma). Assume ρf is irreducible (so that the maximal ideal m = mf ⊆ T is “non-Eisenstein”). Then 1 ⊕2 1 H´et(X0(N), Fp)m → H´et(X0(N`), Fp)m is injective.

1 1 Idea of proof. Use p-adic comparison theorem between Hét(X0(N), Fp) and HdR(X0(N)/Fp ) and reduce to showing that ∗ ∗ 0 1 ⊕2 0 1 π1 + π2 : H (X0(N)Fp , Ω ) → H (X0(N`)Fp , Ω ) is injective. ∗ ∗ Suppose (ω1, ω2) ∈ ker(π1 + π2). By looking at the Hecke action of T`, we see that if 0 0 (E ,C ) ∈ T`(E,C), then 0 0 0 0 (E ,C ) ∈ div (ωi) ⇐⇒ (E,C) ∈ div (ωi) 0 ss ord for i = 1, 2. Thus div (ωi) is supported on the whole X0(N)Fp (if (E,C) ∈ X0(N)Fp , then n {T` (E,C): n ≥ 1} is infinite). 0 p−1 But there is an explicit differential form (Hasse invariant) A ∈ H (X0(N)Fp , ω ) of ss weight p − 1 with simple zeros at exactly X0(N)Fp . If p − 1 > 2, this is a contradiction! 2 If p = 3 (resp. p = 2), then ωi = c · A (resp. ωi = c · A ). Both cases are impossible, since the Hecke action on A is given by an Eisenstein ideal: Tq ≡ q + 1 (mod p). 13.2. Geometry of Shimura curves. We want to carry over the geometry of modular curves to the case of Shimura curves. Recall the Shimura curve X(N +,N −) has moduli interpretation {(A, ι, C)} where: • A is an abelian surface, • ι : ON − ,→ End(A), + 2 • C ⊆ A[N] of order (N ) and cyclic as ON − -module.

Analogous to X0(N) = X(N, 1), we can extend this moduli interpretation to characteristic p - N = N +N −. But at p | N, changes need to be made (especially more diﬃcult when p | N −)! Let us begin with the classiﬁcation of abelian surfaces with special endomorphisms in

− characteristic p. Let A/Fp be an abelian surface with ON -action. Then A is either: • ordinary: A[p] ' (Z/p)2 ⇐⇒ A is isogenous to E2 for an ordinary elliptic curve E. 43 • supersingular: A[p] = 0 ⇐⇒ A is isogenous to E2 for a supersingular E. If A is supersingular, we say that A is superspecial if A is isomorphic to E2 for a supersingular E. Similarly to the case of elliptic curves, we have

( O ⊆ M (O ) if A is ordinary, End(A) = 2 K O ⊆ M2(Op,∞) if A is supersingular, where K is an imaginary quadratic ﬁeld.

Remark. By deﬁnition, every superspecial A is supersingular. When p - N −, the converse is true: A is superspecial if and only if A is supesingular.

The geometry of Shimura curves is summarized in the

Theorem 13.3 (Buzzard, Drinfeld). Assume p2 - N. Then X(N +,N −) has a model over Zp which is ﬂat and projective.

+ − (1) If p - N, then X(N ,N )Fp is smooth, with a distinguished ﬁnite set of marked + − ss points X(N ,N )Fp . Moreover,

+ − ss + − X(N ,N )Fp ' X(N , pN ),

which is a deﬁnite Shimura set (note that pN − has an odd number of prime factors).

Remark. In this case, X(N +,N −)ss = X(N +,N −)s.sp. Fp Fp

+ + − (2) If p k N , then X(N ,N )Fp is not smooth but is semistable, with two irreducible components each isomorphic to X(N +/p, N −) and intersecting along

+ − ss + − ss (1) + − X(N ,N )Fp = X(N /p, N )Fp = X(N /p, pN ).

Remark. Cases (1) and (2) are analogous to the situation for X0(N).

− + − (3) If p | N , then X(N ,N )Fp is not smooth but is semistable. Its geometry is described by Ceredink–Drinfeld p-adic uniformization:

+ − X(N ,N )Fp = X1 ∪ X2,

1 + − where each Xi is a family of P ’s indexed by the Shimura set X(N ,N /p) (switching 1 1 invariants). Each P in X1 intersects with exactly p + 1 P ’s in X2 (and vice versa), 1 forming P (Fp). Finally,

+ − + − X1 ∩ X2 = (p + 1) · |X(N ,N /p)| = X(pN ,N /p). 44 The geometry of X(N +,N −) is summarized in the picture:

+ − ss + − • For p - N, X(N ,N )Fp = X(N , pN ). − + − + − • For p | N , the irreducible components of X(N ,N )Fp are X(N ,N /p).

14. Lecture 14 (October 31, 2018) 14.1. Inertia action on cohomology. The motivation is to understand

1 1 Fr`=1 Hsing(K`,E[p]) = H (I`,E[p]) ,

where I` is the inertia subgroup at `. Thus we need to understand the action of I` on 1 E[p] = Hét(E, Fp)(1). In general, let us switch notation to the local situation. Suppose X is a smooth projective ur variety over Qp, and K = Qdp is the completion of the maximal unramified extension of Qp 45 ur (so Ip = Gal(K/K)). Let O = OK = Zcp and k = O/p = Fp. Let Λ = Z/` or Q` be the coefficient ring. Recall the wild inertia subgroup P ⊆ I is defined as P = Gal(K/Kt) where Kt = S K(p1/n) is the maximal tamely ramified extension of K. Then I/P ' Q Z (1). p-n `6=p ` I t Qp K K K P

Definition 14.1. Define the tame `-adic character to be the map t : I → Z`(1) given by σ(p1/`m ) σ 7→ ∈ {µ m } = Z (1). p1/`m ` m≥1 ` i Theorem 14.2 (Grothendieck). Let V = H(ét)(XK , Λ) with Λ = Q`, which has an action of I = Gal(K/K). Then there exists an open subgroup J ⊆ I and a nilpotent matrix N : V (1) → V such that for all σ ∈ J, the action of σ on V is given by exp(t(σ)N) (hence unipotent). Definition 14.3. The operator N is called the monodromy operator. Remark. The geometric analogue is _ Z`(1) o I = Gal(K/K) Spec k / Spec O o ? Spec K f O

& {z = 0} D − {0} π (D − {0}) ' Z D = {|z| < 1} 1 point disk punctured disk i After going around a loop in Spec K, we get an automorphism of H (XK , Λ) (monodromy in geometry).

Example 14.4. Let X/Qp be an elliptic curve with multiplicative reduction. Then by Tate’s 1 uniformization theorem, the action of σ ∈ I on V = H (XK , Λ) (or T`X) is given by 1 t(σ) 0 t(σ) = exp , 1 0 0 1 so N = . 0 0

i 14.2. Nearby cycle sheaves and monodromy ﬁltration. Our goal is to describe H (XK , Λ)

and N using the geometry of Xk = XFp . Let XO be a proper integral model of XK .

j i / o _ Xk XO ? XK

Spec k / Spec O o ? _Spec K Definition 14.5. The nearby cycle sheaf is ∗ RΨΛ := i Rj∗Λ ∈ D(Xk). 46 Then i i H (XK , Λ) = H (XO, Rj∗Λ) (for any map j) i ∗ = H (Xk, i Rj∗Λ) (proper base change) i = H (Xk,RΨΛ), so the slogan is cohomology of generic fiber = cohomology of special fiber for nearby cycle sheaf. Thus the nearby cycle sheaf sees the cohomology of “nearby” fibers. ∼ i i Example 14.6. If XO → Spec O is smooth, then RΨΛ = Λ and H (XK , Λ) ' H (Xk, Λ). In this case, N = 0 (unramified). In general, there is an action N : RΨΛ(1) → RΨΛ which is nilpotent. More precisely, N n+1 = 0 for n = dim X. Definition 14.7. i+1 (1) The kernel filtration is given by Fi := ker N :

0 ⊆ F0 ⊆ F1 ⊆ · · · ⊆ Fn = RΨΛ. (2) The image ﬁltration is given by Gj := im N j: RΨΛ ⊇ G1 ⊇ G2 ⊇ · · · ⊇ Gn ⊇ 0. P j (3) The monodromy ﬁltration is given by Mr := i−j=r Fi ∩ G :

0 ⊆ M−n ⊆ · · · ⊆ Mn ⊆ RΨΛ. The monodromy ﬁltration is characterized by the properties:

(1) N(Mr(1)) ⊆ Mr−2. r M ∼ M M (2) N : grr RΨΛ(r) → gr−rRΨΛ is an isomorphism, where grr = Mr/Mr−1. 14.3. Rapoport–Zink weight spectral sequence.

Deﬁnition 14.8. XO is semistable if

(1) XO is regular and ﬂat over O. (2) Xk is a divisor of XO with normal crossings, i.e., its singularities are given by O[x1, ··· , xn]/(x1 ··· xs − p) for some s ≤ n.

We further say that XO is strictly semistable if

(3) all irreducible components of Xk are smooth (“simple” normal crossings). For example, the nodal curve is semistable, but not strictly semistable. The union of two curves intersecting transversally is strictly semistable.

Deﬁnition 14.9. Assume XO is strictly semistable. Then Xk = X1 ∪ · · · ∪ Xm where each Xi is a smooth irreducible component (of codimension 1 in XO). Deﬁne for any subset J ⊆ {1, ··· , m}, \ XJ := Xi. i∈J 47 For any p ≥ 0, let (p) a X := XJ , J⊆{1,··· ,m} |J|=p+1

which is a smooth projective variety over k of codimension p in Xk. For any p ≥ 0, we have • pullback: Hq(X(p), Λ) → Hq(X(p+1), Λ), • pushforward: Hq(X(p), Λ)(−1) → Hq+2(X(p−1), Λ).

Theorem 14.10 (Rapoport–Zink, Saito). Assume XO is strictly semistable. Then we have a spectral sequence p,q p+q M p+q E1 = H (Xk, gr−pRΨΛ) ⇒ H (XK , Λ),

which degenerates at E2 if Λ = Q`. Moreover, p,q M q−2i (p+2i) E1 = H (X , Λ)(−i) i≥0 i≥−p p,q where the diﬀerentials on E1 are sums of pushforwards and pullbacks, and N : E1 → p+2,q−2 E1 is given by ⊗t. Remark. The terminology of “weight” spectral sequence comes from the

M i Conjecture 14.11 (Weight monodromy). Let Λ = Q`. Then gr−rH (XK , Λ) is pure of i+r weight i + r (i.e., the action of Frp has eigenvalues of absolute value p 2 ).

15. Lecture 15 (November 7, 2018) 16. Lecture 16 (November 12, 2018) 17. Lecture 17 (November 14, 2018) 18. Lecture 18 (November 19, 2018)

18.0. Rankin–Selberg L-functions for GLn × GLn−1. As a motivation, recall that the way we proved the rank part of BSD is by going through L(E, s) = L(f, s).

• The left-hand side is a special case of a “motivic L-function” L(ρE, s), where ρE is the motive associated to E. • The right-hand side is an “automorphic L-function” L(π, s), where π is an automorphic representation on GL2(A) associated to f ∈ π. The global Langlands correspondence states that every motivic L-function arises as an automorphic L-function. In this way, arithmetic questions about motives (e.g. counting points on algebraic varieties) can be studied using analytic properties of automorphic representations, which are often easier. Our goal is to study generalizations to the arithmetic of L(s, π × π0), where π (resp. π0) is a cuspidal automorphic representation on GLn(A) (resp. GLn−1(A)), for A = AK the ring of adeles of a number field K/Q. 48 Example 18.1. Let n = 2. Taking π to be the automorphic representation corresponding to f and π0 to be the trivial representation gives L(f, s). Today we will define L(s, π × π0) and state its basic properties. Similarly to the case of the Riemann zeta function, the three basic properties are analytic continuation, functional equation, and Euler product (which relates to the problem of counting points on varieties). The study follows the template of Tate’s thesis. 18.1. Global zeta integrals. Definition 18.2 (Global zeta integrals). Let f ∈ π and ϕ ∈ π0. Define Z g s− 1 Z(f, ϕ, s) := f ϕ(g)| det g| 2 dg, 1 [GLn−1]

where [GLn−1] = GLn−1(K)\GLn−1(A). new Example 18.3. Let n = 2, and f ∈ S2 (N). Recall the L-function X an Y L(f, s) = = L (f, s) ns p n≥1 p and the completed L-function Z ∞ dt Λ(f, s) = (2π)−sΓ(s)L(f, s) = f(it)ts . 0 t y x Let us rewrite this in adelic language. Every point x + iy ∈ H corresponds to ∈ 1 y GL (R), and f corresponds to f = f(iy)y. Then Λ(f, s) generalizes to 2 1 Z y f |y|s−1 dy, 1 [GL1] 1 1 so that L(f, s) = Z(f, , s + 2 ). The basic properties of the global zeta integrals are summarized in the Theorem 18.4 (Properties of global zeta integrals). (1) Z(f, ϕ, s) converges to an analytic function on s ∈ C. (2) Z(f, ϕ, s) satisﬁes a functional equation

Z(f, ϕ, s) = Z(f,e ϕ,e 1 − s), t −1 where fe(g) = f( g ) and similarly for ϕe. N N N 0 N 0 (3) Suppose f = fv ∈ π = v πv and ϕ = ϕv ∈ π = v πv. Then Z(f, ϕ, s) has an Euler product Y Z(f, ϕ, s) = Zv(fv, ϕv, s) v where Zv is to be defined. Sketch of proof. (1) This is clear, since f is a cusp form and hence rapidly-decreasing along cusps. 49 (2) Use the automorphism g 7→ tg−1. (3) Euler product uses the uniqueness of Whittaker models, which will be explained in the following. Before defining the Whittaker models, we need to introduce the Whittaker(–Fourier) expansion. Fix an additive character ψ : K\A → C×. Definition 18.5. For f ∈ π, define Z −1 Wf (g) = f(ng)ψ (n) dn, [Nn] where Nn is the maximal unipotent subgroup of GLn consisting of all the upper-triangular unipotent matrices, and

1 x12 ∗   1 x23  ψ  ..  = ψ(x12 + x23 + ··· + xn−1,n).  . xn−1,x 1 Theorem 18.6 (Jacquet, Piatetski-Shapiro, Shalika). X γ f(g) = W g . f 1 γ∈Nn−1(K)\GLn−1(K) Example 18.7. For n = 2, Z 1 x W (g) = f g ψ−1(x) dx f 1 [Ga] is the familiar Fourier coeﬃcient, and X γ f(g) = W g f 1 γ∈K× X = Wf,ψγ (g) γ∈K× n where ψγ(x) = ψ(γx). Here Wf,ψγ (g) is the generalization of an · q .

Remark. If n > 2, difficulty for the theorem arises because Nn−1 is not abelian. One has to carry out a subtle induction on n. Now we define the Whittaker model. Definition 18.8. Let IndGLn (ψ) = {W ∈ C∞(GL (A)) : W (ng) = ψ(n)W (g), n ∈ N (A), g ∈ GL (A)} , Nn n n n called the space of (global) Whittaker functions. By definition, W (g) ∈ IndGLn (ψ). f Nn Definition 18.9. The (global) Whittaker model is

W (π, ψ) = {Wf (g): f ∈ π}. 50 We have a GLn(A)-equivariant map π → W (π, ψ)

f 7→ Wf . This provides a way of understanding automorphic representations in terms of concrete functions. Now we can use Whittaker functions to prove the Euler product; this technique is called “unfolding”. Z g s− 1 Z(f, ϕ, s) = f ϕ(g)| det g| 2 dg 1 [GLn−1]   Z X γ g s− 1 = W ϕ(g)| det g| 2 dg  f 1 1  [GLn−1] γ∈Nn−1(K)\GLn−1(K) Z g s− 1 = W ϕ(g)| det g| 2 dg f 1 Nn−1(K)\GLn−1(A) Z g s− 1 = W W (g)| det g| 2 dg, f 1 ϕ Nn−1(A)\GLn−1(A) where the last step is by the property of Whittaker functions. Note that Nn−1(A)\GLn−1(A) is factorizable! Analogous to the global situation, we make the following deﬁnitions. Deﬁnition 18.10. Let IndGLn (ψ ) = {W ∈ C∞(GL (K )) : W (ng) = ψ (n)W (g), n ∈ N (K ), g ∈ GL (K )} Nn v n v v n v n v be the space of local Whittaker functions.

Deﬁnition 18.11. A local Whittaker model is a nontrivial GLn(Kv)-equivariant map π → IndGLn (ψ ), v Nn v i.e., a nonzero element in Hom (π , IndGLn (ψ )). GLn(Kv) v Nn v

Theorem 18.12 (Uniqueness of local Whittaker model: Gelfand–Kazhdan). Let πv be a smooth irreducible admissible representation of GLn(Kv). Fix a nontrivial additive character × ψv : Kv → C . Then dim Hom (π , IndGLn (ψ )) ≤ 1. GLn(Kv) v Nn v The proof uses Bessel distributions. N As a consequence, if f = fv is factorizable, then Y Wf (g) = Wfv (gv). v Deﬁnition 18.13 (Local zeta integrals). Z gv s− 1 Z (f , ϕ , s) := W W (g )| det g | 2 dg . v v v fv 1 ϕv v v v Nn−1(Kv)\GLn−1(Kv) 51 Corollary 18.14. Y Z(f, ϕ, s) = Zv(fv, ϕv, s). v 18.2. Relation between Z(f, ϕ, s) and L(s, π × π0).

Theorem 18.15 (Properties of local zeta integrals).

(1) Zv(fv, ϕv, s) converges for Re s 0. −s (2) Zv(fv, ϕv, s) ∈ C(q ) and hence has meromorphic continuation to C. 0 ±s −s (3) {Zv(fv, ϕv, s): fv ∈ πv, ϕv ∈ πv} is a C[q ]-ideal in C(q ) and hence has a gener- −s −1 0 ator Pπv,πv (q ) , where P ∈ C[X] and P (0) = 1. Deﬁnition 18.16. Deﬁne the local L-factor to be

0 −s −1 0 L(πv × πv, s) := Pπv,πv (q ) . Theorem 18.17 (Local functional equation).

n−1 0 Zv(fv, ϕv, s) = ωπ0 (−1) γ(s, πv × πv, ψv) · Zv(fev, ϕfv, 1 − s), 0 where γ(s, πv × πv, ψv) is deﬁned using the uniqueness of local Whittaker models. Deﬁnition 18.18.

0 0 0 L(s, πv × πv) (s, πv × π , ψv) := γ(s, πv × π , ψv) · . v v 0 L(1 − s, πev × πev) 18.3. Global L-functions.

Deﬁnition 18.19.

0 Y 0 L(s, π × π ) = L(s, πv × πv), v-∞ 0 Y 0 Λ(s, π × π ) = L(s, πv × πv), v 0 Y 0 (s, π × π ) = (s, πv × πv, ψv). v By the product formula, (s, π × π0) is independent of the choice of ψ.

Theorem 18.20. (1) L(s, π × π0) has analytic continuation to C. (2) There is a functional equation

0 0 0 Λ(s, π × π ) = (s, π × π )Λ(s, πe × πe ).

The proof uses Z(f, ϕ, s) and Zv(fv, ϕv, s). 52 19. Lecture 19 (November 26, 2018) 19.1. General Waldspurger formula. Recall the Waldspurger hypothesis: let E/Q be an elliptic curve, and K/Q be an imaginary quadratic ﬁeld satisfying #{p | N : p inert in K} is odd, i.e., #{p | N −} is odd. (Wald)

Thus (Wald) implies that (EK ) = +1. The Waldspurger formula states that 2 L(EK , 1) 1 |yK | = 1/2 · , (f, f) |dK | deg ϕ

where yK is a toric period on a certain deﬁnite quaternion algebra and ϕ is the modular parametrization of E. As a consequence, L(EK , 1) 6= 0 if and only if yK 6= 0. This is crucial in proving ran(E) = 0 =⇒ ralg(E) = 0. This formula is actually much more general. Notice that under the association new elliptic curve E newform f ∈ S2 (N) automorphic representation π on GL2(A) (with central character), we have

L(EK , s) ∼ L(πK , s) = L(πK ⊗ 1K , s) × × × where 1K : K \A → C is the trivial character, and ∼ means the L-functions differ by a shift. In general, take a number field F with ring of adeles A = AF , a quadratic extension K/F , a quaternion algebra B/F (including M2(F )), a cuspidal automorphic representation π on G := B×/F ×, and a character χ on H := K×/F ×. Then an embedding K,→ B induces an embedding H,→ G (as algebraic groups over F ). Now we are ready to generalize the toric period yK : Definition 19.1. The automorphic H-period is defined to be Z ℘H (f) := f(h)χ(h) dh ∈ C [H] for f ∈ π, where [H] = H(F )\H(AF ).

Notice ℘H : f 7→ ℘H (f) deﬁnes a linear functional Y0 ℘H ∈ HomH(A)(π ⊗ χ, C) = HomH(Fv)(πv ⊗ χv, C). v Theorem 19.2 (Tunnell, Saito). (1) (Multiplicity one)

dim HomH(Fv)(πv ⊗ χv, C) ≤ 1. 0 1 × × 0 (2) (-dichotomy) Let G (Fv) = PGL2(Fv) and G (Fv) = Bv /Fv . Suppose πv is an irre- 0 1 0 ducible representation of G (Fv), and πv = JL(πv ) is its Jacquet–Langlands transfer 1 0 to G (Fv) (which is nonzero only when πv is discrete series). Then 0 1 dim HomH(Fv)(πv ⊗ χv, C) + dim HomH(Fv)(πv ⊗ χv, C) = 1. 53 0 Moreover, HomH(Fv)(πv ⊗ χv, C) 6= 0 if and only if the local root number 1 , π0 ⊗ χ = χ (−1)η (−1), 2 v v v v

where η = ηK/F . Example 19.3. Let π be an automorphic representation coming from an elliptic curve E/Q, and χ = 1K . Then 1 , π ⊗ χ = (E/K). 2 v v v There are two cases: 1 0 • If v - N, then v(E/K) = +1 and πv = 0. These imply dim HomH(Fv)(πv ⊗χv, C) = 1. • If v | N, then ( +1 if v splits in K, v(E/K) = (−1)ordv N if v is inert in K.

0 Moreover, χv(−1) = +1 and ηv(−1) = +1 (as v - dK ). So dim HomH(Fv)(πv ⊗χv, C) = + 1 − 1 when v | N , and dim HomH(Fv)(πv ⊗ χv, C) = 1 when v | N . Thus (Wald) implies dim HomH(A)(π ⊗ χ, C) = 1. This is the representation-theoretic interpretation of the Waldspurger condition. Theorem 19.4 (Waldspurger’s nonvanishing criterion). The following are equivalent:

(1) ℘H 6= 0. 1 (2) HomH(A)(π ⊗ χ) 6= 0 and L(π ⊗ χ, 2 ) 6= 0. This is enough for the rank part of BSD. Indeed, Waldspurger also proves a reﬁned formula 1 for L(π ⊗ χ, 2 ) in terms of the period ℘H . One constructs a canonical element α ∈ Hom (π ⊗ χ , C) ⊗ Hom (π ⊗ χ , C) v H(Fv) v v H(Fv) ev ev using matrix coeﬃcients: for f ∈ πv and fe ∈ πev, Z αv(f, fe) := hπv(h)f, feiχv(f) dh. H(Fv)

Remark. If πv is unramiﬁed and f, fe are spherical such that hf, fei = 1, then 1 ζFv (2)L( 2 , πv ⊗ χv) 1 αv(f, fe) = =: Lv , π ⊗ χ . L(1, ηv)L(1, πv, Ad) 2 N N Theorem 19.5 (Waldspurger formula). Let f = fv ∈ π and fe = fev ∈ πe. Then 1 ℘H (f)℘H (fe) 1 ζF (2)L( , π ⊗ χ) Y αv(fv, fev) = · 2 · . 4 L(1, η)L(1, π, Ad) 1 (f, fe) v L( 2 , πv ⊗ χv) · (fv, fev) | 1{z } L( 2 ,π⊗χ) 1 Q Remark. L( 2 , π ⊗ χ) may be thought of as the ratio of ℘H (f)℘H (fe) by “ v αv(fv, fev)” in a 1-dimensional space! 54 19.2. Gan–Gross–Prasad conjectures. In the situation of the Waldspurger formula, G = ∼ × × ∼ PGL2 = SO3 and H = K /F = SO2. The Gan–Gross–Prasad conjectures generalize this to many pairs of classical groups. The case of interest to us is as follows. Let G = Un = U(V ) and H = Un−1 = U(W ), where V (resp. W ) is a hermitian space of dimension n (resp. n − 1) with respect to a quadratic g extension K/F . Assume V = W ⊥ K · u where (u, u) = 1, so that H,→ G via g 7→ . 1 Let π (resp. π0) be a cuspidal tempererd automorphic representation on G (resp. H). Our goal is to relate 1 L , π × π0 ←→ automorphic H-period ℘ , 2 K K H 0 where L(s, πK × πK ) is the Rankin–Selberg L-function for GLn × GLn−1 from last time. Conjecture 19.6 (Gan–Gross–Prasad). Write Π = π × π0. (1) (Multiplicity one)

dim HomH(Fv)(Πv, C) ≤ 1. (2) (-dichotomy) X 0 i dim HomH (Fv)(Πv, C) = 1, i 0 (H ,Πv) i 0 where (H , Πv) runs over the Vogan L-packet of Πv. Moreover, there is a description in terms of (Πv). (3) (Nonvanishing) The following are equivalent: (a) ℘H 6= 0. 1 (b) HomH(A)(Π, C) 6= 0 and L( 2 , Π) 6= 0. (4) (Ichino–Ikeda formula) Let f ∈ Π and f 0 ∈ Πe. Then ℘H (f)℘H (fe) 1 1 Y αv(fv, fev) = ·L , Π · 1 |SΠ| 2 (f, fe) v L( 2 , Πv)(fv, fev)

where SΠ is the component group of the L-parameter of Π. These are all known: (1) by Aizenbud–Gourevitch–Rallis–Schiﬀman for v - ∞, and by Sun–Zhu for v | ∞; (2) by Beuzart-Plessis; (3) by W. Zhang, Jacquet–Rallis, Z. Yun, H. Xue, Chaudouard–Zydor; (4) by W. Zhang, Beuzart-Plessis.

20. Lecture 20 (November 28, 2018) 20.1. Bloch–Kato Selmer groups. As motivation, recall that for an elliptic curve E/Q we have deﬁned the p-Selmer group 1 Selp(E) ⊆ H (Q,E[p]), which sits in a short exact sequence E(Q) 0 → → Sel (E) → X(E)[p] → 0. pE(Q) p 55 Selp(E) is cut by local conditions: at v of Q, E(Q) im δ : → H1(Q ,E[p]) . v pE(Q) v More generally, we have the p∞-Selmer group

rp Sel ∞ (E) := lim Sel n (E) ∼ (Q /Z ) × ﬁnite p −→ p = p p and the Qp-Selmer group

rp Sel (E) := lim Sel n (E) ⊗ Q = Q . Qp ←− p p p

Our goal is to deﬁne an analogue of SelQp (E) for more general Galois representations ρ : GK → Aut(V ) on ﬁnite-dimensional Qp-vector spaces V . This will be the Bloch–Kato 1 Selmer group Hf (K,V ).

Remark. In the case of elliptic curves, V = VpE is the rational Tate module. In particular,

SelQp (E) is invariant under isogenies of E. Let K be a number ﬁeld.

Definition 20.1. A p-adic Galois representation ρ : GK → Aut(V ) comes from geometry if i V is a subquotient of H (XK , Qp)(n) for some i, n, and smooth projective variety X/K. Remark. Let K be a v-adic local field. i (1) If v - p and X/K has good reduction, then H (X, Qp) is unramified as a GK - representation. i (2) If v | p, then H (X, Qp) is de Rham as a GK -representation. To define de Rham representation, we recall the p-adic comparison theorem. Theorem 20.2 (Faltings). i ∼ i H (X, Qp) ⊗Qp BdR → HdR(X/K) ⊗K BdR, where BdR is Fontaine’s p-adic period ring.

GK Using BdR = K, we know that

i GK ∼ i (H (X, Qp) ⊗Qp BdR) → HdR(X/K). Deﬁnition 20.3. Deﬁne GK DdR(V ) := (V ⊗Qp BdR) .

Say V is de Rham if dimK DdR(V ) = dimQp V . Returning to the global situation where K is a number field, we make the Definition 20.4. V is geometric if (1) V is unramified at almost all places v. (2) V is de Rham at v | p. So V comes from geometry =⇒ V is geometric. The converse is conjecturally true, so the condition of V being geometric provides a working definition. Conjecture 20.5 (Fontaine–Mazur). V comes from geometry if and only if V is geometric. 56 i Remark. If v | p and X/K has good reduction, then H (X, Qp) is crystalline at v (defined analogously using Bcrys instead of BdR). Theorem 20.6 (Faltings). i ∼ i H (X, Qp) ⊗Qp Bcrys → Hcrys(Xk/W (k)) ⊗W (k) BdR, where k is the residue field of K and W (k) is the Witt ring of k.

GK 1 Using Bcrys = W (k)[ p ], the notion of crystalline representations is deﬁned similarly. Now we are ready to deﬁne the local conditions for Bloch–Kato Selmer groups.

1 1 Definition 20.7 (Local conditions). Define Hf (Kv,V ) ⊆ H (Kv,V ) as follows: (1) If v - p, define the unramified subspace 1 1 Iv 1 1 Frv=1 Hf (Kv,V ) := H (kv,V ) = ker H (Kv,V ) → H (Iv,V ) . (2) If v | p, define the finite subspace 1 1 1 Hf (Kv,V ) := ker H (Kv,V ) → H (Kv,V ⊗Qp Bcrys) . Similarly to the case of elliptic curves, they satisfy the following properties: (1) If v - p, then 1 0 dim Hf (Kv,V ) = dim H (Kv,V ). Also, local duality works: under the local Tate pairing 1 1 ∗ 2 (·, ·)v : H (Kv,V ) × H (Kv,V (1)) → H (Kv, Qp(1)) ' Qp, 1 1 ∗ Hf (Kv,V ) and Hf (Kv,V (1)) are exact annihilators. (2) If v | p, then 1 0 + dimQp Hf (Kv,V ) = dimQp H (Kv,V ) + dimQp DdR(V )/DdR(V ), where the last term computes the number of negative Hodge–Tate weights of V . Also, local duality works.

Example 20.8. Let E/Q be an elliptic curve and V = Vp(E). Then 1 (1) If v - p, then Hf (Kv,V ) = 0, since the Tate module has no nontrivial Galois invari- 1 ants. This implies H (Kv,V ) = 0. 1 (2) If v | p, then dimQp Hf (Kv,V ) = 1, since HT(V ) = HT(E) = {0, −1}. This implies 1 dim H (Kv,V ) = 2. Finally, we can define the Bloch–Kato Selmer groups. Definition 20.9. The Bloch–Kato Selmer group is defined to be 1 1 1 Hf (K,V ) := {s ∈ H (K,V ) : locv(s) ∈ Hf (Kv,V ) for all v}. It is a deep theorem that this agrees with the original definition for elliptic curves. Theorem 20.10 (Bloch–Kato). Let E/K be an elliptic curve (or abelian variety) and V = Vp(E). Then 1 1 Hf (Kv,V ) = im A(Kv) ⊗ Qp → H (Kv,V ) . 1 Thus Hf (Kv,V ) = SelQp (E). 57 20.2. Bloch–Kato conjecture. Definition 20.11. Define Y L(s, V ) := Lv(s, V ) v-∞ where ( −1 −1 −s Iv det 1 − Frv qv : V at v - p, Lv(s, V ) = −1 −s −1 det (1 − φv qv : Dcrys,v(V )) at v | p. Example 20.12. Let E/Q be an elliptic curve. Then L(s, V ) = L(s + 1,E), with center at s = 0. Conjecture 20.13 (Langlands). Let V be a geometric Galois representation. Then V should come from an automorphic representation. So L(s, V ) has meromorphic continuation to s ∈ C (analytic if V does not contain Qp(1)) and satisfies a functional equation Λ(s, V ) = (s, V )Λ(−s, V ∗(1)). Conjecture 20.14 (Bloch–Kato). 1 ∗ 0 ∗ ords=1 L(s, V ) = dim Hf (K,V (1)) − dim H (K,V (1)). 1 ∗ 0 ∗ Here Hf (K,V (1)) is the Bloch–Kato Selmer group, and H (K,V (1)) = 0 if V is irreducible. The Bloch–Kato conjecture vastly generalizes BSD to any motives.

20.3. Rankin–Selberg motives. Let K/F be a CM extension over a totally real number 0 ﬁeld, and π (resp. π ) be an automorphic representation on GLn,K (resp. GLn−1,K ). Set Π = π × π0. Our goal is to construct a geometric Galois representation VΠ = Vπ ⊗ Vπ0 , which has dimension n(n − 1).

Deﬁnition 20.15. Let π be a cuspidal automorphic representation on GLn(AK ). Say (1) π is cohomological (or regular algebraic in Henniart’s terminology) if the inﬁnitesimal ∗ character χ∞ ∈ X (T ) of π∞ is regular, i.e., ∨ hχ∞, α i= 6 0

for all α ∈ Φ(GLn). (The Hodge–Tate weights are distinct.) c ∼ (2) π is conjugate self-dual if π = πe.

Next time we will construct VΠ under these assumptions.

21. Lecture 21 (December 3, 2018) 21.1. BBK for Rankin–Selberg motives. Recall: Conjecture 21.1 (Bloch–Kato). For a geometric p-adic Galois representation V , 1 ∗ ords=0 L(s, V ) = dim Hf (K,V (1)). 58 Remark. Suppose K/F is a CM extension of a totally real ﬁeld, and V c ∼= V ∗(1). Then L(s, V ) = L(s, V c) = L(s, V ∗(1)), so we have a functional equation relating L(s, V ) ←→ L(−s, V ) with center s = 0.

Now let π be a cuspidal tempered automorphic representation on GLn(AK ). The construction of Vπ is provided by Theorem 21.2 (Chenevier–Harris, Shin and many people). Assume π is cohomological and c conjugate-self-dual (i.e., π ' πe). Then there exists ρπ : GK → Aut(Vπ) (over a p-adic ﬁeld) such that:

(1) Vπ is geometric, i.e., de Rham at v | p with distinct Hodge–Tate weights. (2) If πv is unramiﬁed, then Vπ is unramiﬁed at v if v - p, and cyrstalline at v if v | p. Fur- µ1  .. thermore, suppose πv corresponds to the Satake parameter Cπv =  .  ∈ µn GLn(C) with |µi| = 1. Then the eigenvalues of Frv (at v - p) or φv (at v | p) are 1−n given by {µiq 2 }. In other words, 1−n 2 Vπ|Gv = LLC πv ⊗ | · |v under the local Langlands correspondence.

(3) More generally, Vπ|Gv is compatible with LLC(πv). c ∗ In particular, we have Vπ = Vπ (1 − n).

Idea of proof. When n is odd, or n is even with π∞ satisfying extra regularity conditions, Vπ can be constructed in the cohomology of a certain unitary Shimura variety, by (1) point-counting on Igusa varieties (Shin), (2) comparison of Lefschetz trace formula and Arthur–Selberg trace formula. When n is even, the extra assumption is removed by Chenevier–Harris by congruences (eigenvarieties). Deﬁnition 21.3 (Rankin–Selberg motive). Let π (resp. π0) be an automorphic representa- 0 tion on GLn(AK ) (resp. GLn−1(AK )) as before, and Π = π × π . Deﬁne the Rankin–Selberg motive

VΠ := (Vπ ⊗ Vπ0 )(n − 1). c ∗ It can be checked easily that VΠ = VΠ(1) and 1 −s −1 Y −s 0 −1 L s − ,V = det 1 − q C ⊗ C 0 = 1 − q µ µ v 2 Π v πv πv v i j i,j

0 for the Satake parameters Cπv and Cπv . Now we can state the main theorem, due to Yifeng Liu, Yichao Tian, Liang Xiao, Wei Zhang and Xinwen Zhu. 59 Theorem 21.4 (LTXVV, rank 0). Assume π and π0 have trivial inﬁnitesimal character (with Hodge–Tate weights {0, −1, −2, ··· , −(n − 1)} and {0, −1, −2, ··· , −(n − 2)} respectively). Let VΠ be the Rankin–Selberg motive. Then for almost all p, 1 L , Π 6= 0 =⇒ H1(K,V ∗(1)) = 0, 2 f Π i.e., the Bloch–Kato conjecture holds in rank 0. This is a vast generalization of our previous theorem for elliptic curves.

21.2. Rank one case. We would like to prove the implication

1 ∗ ord 1 L(s, Π) = 1 =⇒ dim H (K,V (1)) = 1. s= 2 f Π In the elliptic curve case, this crucially relies on the Gross–Zagier formula:

0 L (EK , 1) ∼ height of Heegner point yK . A generalization is given by the arithmetic Gan–Gross–Prasad conjecture: 1 L0 , Π ∼ “height” of GGP cycle ∆ . 2 Π Unfortunately the deﬁnition of heights for cycles is conditional on the standard conjectures. Realistically, our goal is to construct ∆Π using the Shimura variety associated to Un×Un−1.

Deﬁnition 21.5. Let V be a hermitian space with respect to K/F of dimension n, and Un = U(V ) be the associated unitary group. Assume Un(R) = U(n−1, 1)×U(n, 0)×· · ·×U(n, 0) (with [F : Q] factors in total). Deﬁne the Hodge cocharacter

× h : C → Un(R) = U(n − 1, 1) × U(n, 0) × · · · × U(n, 0)  z   z  1   z 7→  .  , Id, ··· , Id  ..   1 and

H := orbit of h under Un(R)-conjugation ∼= U(n − 1, 1)/(U(n − 1) × U(1)),

which is a hermitian symmetric domain with dimC H = n − 1.

Write Af = AF,f .

Deﬁnition 21.6. Let K ⊆ Un(Af ) be an open compact subgroup. Deﬁne the Shimura variety

ShK (Un)(C) = Un(F )\Un(Af ) × H/K.

Assume K has no torsion elements. Then ShK is a quasi-projective smooth variety over F of dimension n − 1. 60 Remark. Technically it is easier to work with the Shimura variety associated to GU(V ) instead of U(V ), where × GU(V )(R) = {g ∈ GLK⊗R(V ⊗Q R): hgv1, gv2i = χ(g)hv1, v2i, χ(g) ∈ R } for each Q-algebra R. There is an extension χ 1 → U(V ) → GU(V ) → Gm → 1. Consider the Hodge cocharacter C× → GU(V )(R) z    z   z 7→  .  , z · Id, ··· , z · Id .  ..   z

Then ShK (GU(V )) is of PEL type, with

ShK (GU(V ))(C) = {(A, ι, λ, η)} where: • A is an abelian variety of dimension n − 1, • ι : OK → End(A), ∼ ∨ • λ : A → A is an OK -linear polarization, • η is a K-orbit of isomorphisms 1 H (A, Q) ⊗ Af ' V (Af )

of symplectic Af -modules (up to similitude) and of OK -modules.

Remark. If F 6= Q, then ShK (GU(V )) is projective. If F = Q, then ShK (GU(V )) has a canonical toroidal compactiﬁcation. Denote them by XV . Deﬁnition 21.7 (GGP cycle). Let V = W ⊕ K · u with (u, u) = 1. Then the embedding GU(W ) ,→ GU(V ) induces δ : XW ,→ XV (of dimensions n − 2 and n − 1 respectively). Consider

(Id, δ): XW ,→ XW × XV =: X (of dimensions n − 2 and 2n − 3 respectively). The GGP cycle is deﬁned as ∆ := im(Id ×δ) ⊆ CHn−1(X). n−1 Given Π, deﬁne ∆Π = ∆[Πf ] ⊆ CH (X) to be the Πf -isotypic component.

Remark. Local -dichotomy ensures there is a unique choice of (W, V ) such that ∆Π is possibly nonzero. Conjecture 21.8 (Arithmetic GGP). 1 L0 , Π ∼ “height” of ∆ . 2 Π Remark. The arithmetic GGP is a special case of Beilinson’s conjecture: 61 Conjecture 21.9 (Beilinson). Let X/K be a smooth projective variety. Then 2i−1 i ords=center L(s, H (X)) = dim CH (X)0,

i i cl 2i where CH (X)0 := ker(CH (X) → H (X)). Remark. We have the implication 1 L0 , Π 6= 0 =⇒ ∆ 6= 0. 2 Π

Then we can consider its image AJ(∆Π) under the Abel–Jacobi map:

n−1 cl 2n−2 CH (X) / H (X, Qp)(n − 1)[Πf ]

AJ * 1 2n−3 H (K,H (XK , Qp))(n − 1)[Πf ] where the vertical map comes from the Hochschild–Serre spectral sequence and the fact that 2n−2 2n−3 H (XK , Qp)(n − 1)[Πf ] = 0. Note that H is the exact middle dimension.

Remark. When n = 2, AJ(∆Π) is the analogue of yK under the Kummer map E(K) → 1 H (K,Vp(E)). Theorem 21.10 (LTXZZ, rank 1). For almost all p, 1 ∗ AJ(∆Π) 6= 0 =⇒ dim Hf (K,VΠ(1)) = 1. Next time we will sketch the proof of this.

22. Lecture 22 (December 5, 2018) 22.1. Proof strategies. Recall: Theorem 22.1 (LTXZZ). Let Π be the Rankin–Selberg motive as before. 1 1 ∗ • (Rank 0) If L( 2 , Π) 6= 0, then Hf (K,VΠ(1)) = 0. 1 ∗ • (Rank 1) If AJ(∆Π) 6= 0, then dim Hf (K,VΠ(1)) = 1. The strategy is analogous to the elliptic curve case. The key is two explicit reciprocity laws for GGP cycles on unitary Shimura varieties. Roughly speaking, for certain “level-raising primes” v, we can find a congruence Π“≡”Π0 (mod p) where Π is unramified at v and Π0 is ramified at v. The first reciprocity law concerns the case (Π) = +1. Recall that the Ichino–Ikeda 1 formula expresses the central L-value L( 2 , Π) in terms of the automorphic period ℘Un−1 (ϕ) for ϕ ∈ Π. Then the reciprocity law states that

0 ℘Un−1 (ϕ) ≡ ∂v AJ(∆Π ) (mod p), 1 2n−3 1 1 where AJ(∆Π0 ) ∈ H (K,VΠ) with VΠ = H (X)[Πf ], and ∂v : H (Kv,VΠ) → Hsing(Kv,VΠ). Similarly, we can formulate the second reciprocity law for (Π) = −1: 0 locv AJ(∆Π) ≡ ℘Un−1 (Π ) (mod p). 62 1 Then we use Kolyvagin’s method to bound Hf (K,VΠ). New ingredients are needed for the explicit reciprocity laws. For X = XV × XW (of dimensions n − 1 and n − 2 respectively), the K¨unnethformula gives

2n−3 n−1 n−2 H (X)[Πf ] = H (XV ) ⊗ H (VW )[Πf ], since cohomology is concentrated at middle degree. Now we have

1 n−1 n−2 locv AJ(∆Π) ∈ H (kv,H (XV,kv ) ⊗ H (XW,kv ))(n − 1)[Πf ], so we need to understand the geometry of Shimura varieties in characteristic p.

22.2. Geometry of unitary Shimura varieties at good reduction. Recall that for ` N, X (N) has a distinguished set of supersingular points X (N)ss . What makes the - 0 F` 0 F` explicit reciprocity law work is the identiﬁcation X (N)ss ' X(N, `) 0 F`

with the Shimura set associated to B`,∞ (ramiﬁed at ` and ∞). Deﬁnition 22.2 (Switching invariants). Let v be a place of F . Assume v is inert in K. Each hermitian space V gives rise to Vv for the local quadratic extension Kv/Fv. Then the Hasse invariant is ( +1 if disc(Vv) has even valuation, (Vv) := −1 if disc(Vv) has odd valuation.

Given V such that (Vv) = +1 and sign(V∞) = {(n − 1, 1), (n, 0), ··· , (n, 0)}, deﬁne a new hermitian space V • by

• • (Vv ) = −1, sign(V∞) = {(n, 0), (n, 0), ··· , (n, 0)}. Theorem 22.3. Assume v is inert in K, and the level deﬁning the unitary Shimura variety X = XV is given by a hyperspecial subgroup in GU(V )(Fv) (i.e., the stabilizer of a self-dual lattice in Vv).

(1) (Kottwitz, Lan) X has an integral model over OKv , given by the solution to a moduli problem {(A, ι, λ, η)}. The special ﬁber is smooth projective and has dimension n−1. ss (2) (Vollaard–Wedhorn) Let X ⊆ Xk be the supersingular locus (i.e., where A/kv is a supersingular abelian variety). Then the normalization of Xss, denoted by Xgss, is n−1 a disjoint union of copies of a smooth projective variety DLn of dimension b 2 c, indexed by the deﬁnite Shimura set

• • • • X = GU(V )(Q)\GU(V )(Af )/K , where the level at v for X• is given by the stabilizer of

(a self-dual lattice (i.e., Λ∨ = Λ) if n is odd, codim 1 an almost self-dual lattice (i.e., Λ∨ ⊇ Λ) if n is even. 63 As a summary, we have the following picture

X o ? _ Xss o Xgss dim n−1 n−1 dimb 2 c

DLn-ﬁbration X• dim 0 22.3. Tate’s conjecture. Recall that we have a cycle class map

r cl 2r CH (X) → H (X, Qp)(r).

Conjecture 22.4 (Tate). Assume X/kv is smooth projective over the ﬁnite ﬁeld kv. Then

r cl 2r Frv=1 CH (X) ⊗ Qp → H (X, Qp)(r) is surjective. Remark. This is known when X is an abelian variety and r = 1, due to Tate. Tate conjecture says that we can understand H2r(X)(r)Fr=1 via algebraic cycles. As before, let π be an automorphic representation on GLn(AK ), Vπ be the associated Galois representation of dimension n, and X = XV be the associated Shimura variety of dimension n − 1. Deﬁnition 22.5. Assume n is odd. Let v be inert in K. Say v is odd generic for π (mod p) αi if the eigenvalues {α1, ··· , αn} of Frv acting on Vπ (mod p) satisfy 6= 1, qv for all i 6= j. αj Theorem 22.6 (Xiao–Zhu). Assume v is odd generic. Then Tate conjecture holds for Hn−1(X )( n−1 )[π ]Frv=1. In fact, this space is exactly generated by Xss. kv 2 f g In particular, this and Poincar´eduality together imply that ∼ Hn−1(X )( n−1 )[π ]Frv=1 → Hn−1(Xss)( n−1 )[π ] kv 2 f g 2 f 0 • ∗ ' H (X )[πf ] , which is a space of automorphic forms! 22.4. Second reciprocity.

αi Deﬁnition 22.7. Let n be even. Say v is even level-raising for π (mod p) if 6= 1, qv for αj n n 2 2 −1 all i 6= j, except that α n = qv and α n = qv . 2 2 +1

This is analogous to the Steinberg representation for GL2. Deﬁnition 22.8. Say p is nice if p is a good prime for Π = π × π0, p ≥ 5, and:

(1) Vπ and Vπ0 mod p are irreducible and with adequate images; (2) p is isolated for Π. Fact. p is nice for almost all p.

0 Deﬁnition 22.9. Assume n is even. Say v is admissible for Π = π × π (on GLn and GLn−1 respectively) mod nice p if: 64 (1) v is odd generic for π0; (2) v is even level-raising for π; (3) Frv acting on VΠ has eigenvalue 1 with multiplicity 1. Fact. v is admissible for a positive proportion of v. Theorem 22.10 (Second explicit reciprocity). Assume p is nice and v is admissible. If • • • • • locv AJ(∆Π) 6≡ 0 (mod p), then there exists Π on GU(V ) × GU(W ) and ϕ ∈ Π such that • locv AJ(∆Π) ≡ ℘GU(W •)(ϕ ) (mod p). 23. Lecture 23 (December 10, 2018) 23.1. Tate conjecture via geometric Satake. Recall that a new ingredient in proving

explicit reciprocity is the Tate conjecture for unitary Shimura varieties. Let X = XV,k be the special ﬁber associated to GU(1, n − 1). Assume n is odd, so that dim X = n − 1 is even. Last time we saw the diagram

X o Xgss dim n−1 n−1 dim 2

X• dim 0 Shimura set n−1 where the ﬁbers of the vertical map are Deligne–Lusztig varieties of dimension 2 . Theorem 23.1 (Xiao–Zhu). Assume v is odd generic for π. Then

n−1 ss n−1 n−1 n−1 Frv=1 H (Xg)( 2 )[πf ] ' H (X)( 2 ) [πf ].

Remark. This even holds with coefficients Λ = Fp. Let mπ ⊆ T correspond to π. Then ∼ Hn−1(Xss, Λ)( n−1 ) → Hn−1(X, Λ)( n−1 )Frv=1. g 2 mπ 2 mπ This follows from the torsion-free result of Caraiani–Scholze. The proof strategy is: (1) (Injectivity) Show the image of components of Xgss are linearly independent in Hn−1(X). (2) (Compare dimensions) Prove dimensions are equal using comparison of Lefschetz trace formula for X• and X. (1) is more difficult: by looking at the intersection pairing n−1 n−1 2(n−1) (·, ·): H (X) × H (X) → H (X) ' Qp, it suffices to show the matrix under (·, ·) for Hn−1(Xgss) is nondegenerate. This intersection matrix is very difficult to compute, and the idea is to compute the determinant abstractly using the representation theory of Gb. • On the generic fiber, we have an action of Hecke correspondence Hv at v. For G split, the Satake isomorphism gives

∼ W ∼ Gb Hv → C[X∗(T )] → C[Gb] . 65 • On the special ﬁber, we can realize

Xgss

} X• X as an “exotic Hecke correspondence” between X• and X. This has meaning in terms of “spectral operators”, which comes from the representation theory of Gb.

23.2. Geometric Satake. Let F be a local non-archimedean ﬁeld with residue ﬁeld k and ring of integers O = OF ⊆ F . Let G/O be a reductive group.

∞ Deﬁnition 23.2. The spherical Hecke algebra is H := Cc (G(O)\G(F )/G(O)). The Satake isomorphism states that

∼ Gb H → Qp[Gb] with an action by σ-conjugation: for σ the Frobenius, g · h = ghσ(g−1).

∼ Gb Remark. We can rewrite this as H → Qp[Gσb ] , where Gσb ⊆ Gb o hσi. To geometrize the classical Satake isomorphism, we need to introduce: Deﬁnition 23.3. The local Hecke stack is Hkloc := L+G\LG/L+G, where LG(k) = G(F ) and L+G(k) = G(O). For every perfect k-algebra R, deﬁne the disk ∗ 1 DR = Spec WO(R) and the punctured disk DR = Spec WO(R)[ $ ]. Then

loc β Hk (R) = {E → E : E and E are G-bundles on D and β| ∗ is an isomorphism}. 1 2 1 2 R DR Let Perv(Hkloc) be the category of perverse sheaves on Hkloc. Theorem 23.4 (Mirkovic–Vilonen if char F > 0; X. Zhu if char F = 0). There is an equivalence of monoidal categories

∼ Rep(Gb) → Perv(Hkloc)

Vµ 7→ ICµ

which sends the highest weight representation of weight µ to the IC sheaf on Grµ ⊆ Gr. 23.3. Spectral operators. Deﬁnition 23.5. The moduli of local shtukas is

loc β loc ∗ ∼ Sht := {(E1 → E2) ∈ Hk with σ E1 → E2}. By definition, we have a forgetful map Shtloc → Hkloc. 66 Definition 23.6. Define

 β   E / E ∼ / σ∗E   1 2 1  loc   Hk(Sht ) := α σ∗α . 0  β ∼   0 / 0 / ∗ 0   E1 E2 σ E1  Then there is a diagram

Hk(Shtloc) ←− −→ h h y % Shtloc Shtloc

loc loc loc loc Let Shtµ ⊆ Sht be given by the locus where inv(β) ≤ µ, and Shtµ1|µ2 ⊆ Hk(Sht ) be 0 where inv(β) ≤ µ1 and inv(β ) ≤ µ2. Then we have a correspondence

loc Shtµ1|µ2

{ # loc loc Shtµ1 Shtµ2 Example 23.7. loc • Sht0 = [Spec k/G(O)] = [point/G](k) (where G(O) is considered as a discrete group). loc loc • Sht0|0 = [G(O)\G(F )/G(O)] = Hk (k). Deﬁnition 23.8. The category PervCorr(Shtloc) is deﬁned by: (1) objects: perverse sheaves on Shtloc; (2) morphisms: cohomological correspondences supported on Hk(Shtloc), i.e.,

Corr ←−∗ −→! Hom (F1, F2) := Hom( h F1, h F2).

The identiﬁcation Rep(Gb) = Coh([•/Gb]) induces a pullback map Rep(Gb) → Coh([Gσ/b Gb]) V 7→ V := {V ⊗ O } e Gσb

with image inside the subcategory Cohfree([Gσ/b Gb]) of free sheaves. Theorem 23.9 (Xiao–Zhu). There is a functor S such that the diagram

V Rep(G) ∼ / Perv(Hkloc) _ b Satake pullback pullback V = {V ⊗ O } Coh ([Gσ/G]) ∃S / PervCorr(Shtloc) e Gσb free b b commutes. 67 Remark. Let V0 = 1 be the trivial representation of Gb. Then Satake Gb End(V0) = Γ([Gσ/b Gb], O) = Qp[Gσb ] 'H

is the space of twisted conjugation-invariant functions on Gb. More generally, deﬁne J(V ) = Γ([Gσ/b Gb], Ve) Gb = (Qp[Gσb ⊗ V ]) = {f : Gb → V : f(ghσ−1(g)) = g · f(h)}.

Write J := J(V0)('H).

loc 23.4. Computing intersection numbers. Notice there is a map X = Sh(G, µ) → Shtµ and a diagram Xgss

{ • loc # X Sht0|µ X

{ # loc loc Sht0 Shtµ .

Then Xgss can be identiﬁed via 0 • n−1 H (X ) ⊗ Hom(Ve0, Vfµ) → H (X). J

Here Hom(Ve0, Vfµ) = J(Vµ) can be thought of as an “exotic correspondence” via the S- operator. Finally, the intersection matrix is given by ∗ J(Vµ) × J(Vµ ) → J. t   1  Returning to the case G = U(n) for n odd, µ(t) =  .  and Vµ is the standard  ..  1 representation of GLn, the upshot is: Theorem 23.10.

(1) J(Vµ) is a free module of rank 1 over J. (2) The intersection matrix (1 × 1) is given by

n−1 2 Y −1 (Xi + Xi − 2), i=1 −1 −1 where Xi(t) = diag(1, ··· , t, ··· , 1, ··· , 1, ··· , t , ··· , 1) with t (resp. t ) at the i-th (resp. (n − 1 − i)-th) position. 68 Corollary 23.11. After localizing at mπ, we get n−1 2 Y −1 (µi + µi − 2) i=1 where {µi} are the Satake parameters of π. This is nonzero because µi 6= 1!