GEOMETRY and NUMBERS Ching-Li Chai

GEOMETRY AND NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a GEOMETRY AND NUMBERS diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Ching-Li Chai Zeta and L-values Sample of geometric structures and symmetries Institute of Mathematics Elliptic curve basics Academia Sinica Modular forms, modular and curves and Hecke symmetry Complex multiplication Department of Mathematics Frobenius symmetry University of Pennsylvania Monodromy Fine structure in characteristic p National Chiao Tung University, July 6, 2012

GEOMETRY AND Outline NUMBERS Ching-Li Chai

Sample arithmetic 1 Sample arithmetic statements statements Diophantine equations Diophantine equations Counting solutions of a diophantine equation Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of Counting congruence solutions prime numbers L-functions and distribution of prime numbers Zeta and L-values Sample of geometric Zeta and L-values structures and symmetries Elliptic curve basics Modular forms, modular 2 Sample of geometric structures and symmetries curves and Hecke symmetry Complex multiplication Elliptic curve basics Frobenius symmetry Monodromy Modular forms, modular curves and Hecke symmetry Fine structure in characteristic p Complex multiplication Frobenius symmetry Monodromy Fine structure in characteristic p GEOMETRY AND The general theme NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a Geometry and symmetry inﬂuences diophantine equation Counting congruence solutions L-functions and distribution of arithmetic through zeta functions and prime numbers Zeta and L-values modular forms Sample of geometric structures and symmetries Elliptic curve basics Modular forms, modular Remark. (i) zeta functions = L-functions; curves and Hecke symmetry Complex multiplication modular forms = automorphic representations. Frobenius symmetry Monodromy Fine structure in characteristic (ii) There are two kinds of L-functions, from harmonic analysis p and arithmetic respectively.

GEOMETRY AND Fermat’s inﬁnite descent NUMBERS Ching-Li Chai

Sample arithmetic statements I. Sample arithmetic questions and results Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions 1. Diophantine equations L-functions and distribution of prime numbers Example. Fermat proved (by his inﬁnite descent) that the Zeta and L-values Sample of geometric diophantine equation structures and symmetries Elliptic curve basics 4 4 2 Modular forms, modular x − y = z curves and Hecke symmetry Complex multiplication Frobenius symmetry does not have any non-trivial integer solution. Monodromy Fine structure in characteristic p Remark. (i) The above equation can be “projectivized” to 4 4 2 2 x − y = x z , which gives an√ elliptic curve E with complex multiplication by Z[ −1]. GEOMETRY AND NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Zeta and L-values

Sample of geometric structures and symmetries Elliptic curve basics Modular forms, modular curves and Hecke symmetry Complex multiplication Frobenius symmetry Monodromy Fine structure in characteristic p

Figure: Fermat

GEOMETRY AND Fermat’s inﬁnite descent continued NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of (ii) Idea: Show that every non-trivial rational point P ∈ E( ) is prime numbers Q Zeta and L-values [ ] the image 2 E of another “smaller” rational point. Sample of geometric structures and (Construct another rational variety X and maps f : E → X and symmetries Elliptic curve basics g : X → E such that g ◦ f = [2]E and descent in two stages. Here Modular forms, modular √ curves and Hecke symmetry X is a twist of E, and f ,g corresponds to [1 + −1] and Complex multiplication √ Frobenius symmetry [1 − −1] respectively.) Monodromy Fine structure in characteristic p GEOMETRY AND Interlude: Euler’s addition formula NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a In 1751, Fagnano’s collection of papers Produzioni diophantine equation Counting congruence solutions Mathematiche reached the Berlin Academy. Euler was asked to L-functions and distribution of prime numbers examine the book and draft a letter to thank Count Fagnano. Zeta and L-values Soon Euler discovered the addition formula Sample of geometric structures and Z r dρ Z u dη Z v dψ symmetries = + , Elliptic curve basics p p p Modular forms, modular 0 1 − ρ4 0 1 − η4 0 1 − ψ4 curves and Hecke symmetry Complex multiplication Frobenius symmetry where Monodromy √ √ Fine structure in characteristic u 1 − v4 + v 1 − u4 p r = . 1 + u2v2

GEOMETRY AND NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Zeta and L-values

Figure: Euler GEOMETRY AND Counting sums of squares NUMBERS Ching-Li Chai

2. Counting solutions of a diophantine equation Sample arithmetic statements Diophantine equations Example. Counting sums of squares. Counting solutions of a diophantine equation For n,k ∈ N, let Counting congruence solutions L-functions and distribution of prime numbers n 2 2 Zeta and L-values rk(n) := #{(x1,...,xk) ∈ Z : x1 + ··· + xk = n} Sample of geometric structures and be the number of ways to represent n as a sum of k squares. symmetries Elliptic curve basics Modular forms, modular curves and Hecke symmetry (d−1)/2 0 if n2 6= Complex multiplication (i) r2(n) = 4 · ∑ (−1) = Frobenius symmetry 1 if n = Monodromy d|n, n odd ∑d|n1 2 Fine structure in characteristic f p where n = 2 · n1 · n2, and every prime divisor of n1 (resp. n2) is ≡ 1 (mod 4) (resp. ≡ 3 (mod 4)). 8 · ∑d|n d if n is odd (ii) r4(n) = 24 · ∑d|n,d odd d if n is even

GEOMETRY AND How to count number of sum of squares NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of Method. Explicitly identify the theta series prime numbers Zeta and L-values √ 2 k Sample of geometric k( ) = qm where q = e2π −1τ structures and θ τ ∑ symmetries m∈N Elliptic curve basics Modular forms, modular curves and Hecke symmetry with modular forms obtained in a different way, such as Complex multiplication Frobenius symmetry Eisenstein series. Monodromy Fine structure in characteristic p GEOMETRY AND Counting congruence solutions NUMBERS Ching-Li Chai

Sample arithmetic statements 3. Counting congruence solutions and L-functions Diophantine equations Counting solutions of a diophantine equation (a) Count the number of congruence solutions of a given Counting congruence solutions L-functions and distribution of diophantine equation modulo a (ﬁxed) prime number p prime numbers Zeta and L-values

Sample of geometric structures and (b) Identify the L-function for a given diophantine equation symmetries (basically the generating function for the number of Elliptic curve basics Modular forms, modular congruence solutions modulo p as p varies) curves and Hecke symmetry Complex multiplication with Frobenius symmetry Monodromy an L-function coming from harmonic analysis. (The latter is Fine structure in characteristic p associated to a modular form).

Remark. (b) is an essential aspect of the Langlands program.

GEOMETRY AND The Riemann zeta function NUMBERS Ching-Li Chai

Sample arithmetic statements 4. L-functions and the the distribution of prime Diophantine equations Counting solutions of a numbers for a given diophantine problem diophantine equation Counting congruence solutions L-functions and distribution of Examples. (i) The Riemann zeta function ζ(s) is a prime numbers Zeta and L-values meromorphic function on C with only a simple pole at s = 0, Sample of geometric structures and −s −s −1 symmetries ζ(s) = ∑ n = ∏(1 − p ) for Re(s) > 1, Elliptic curve basics Modular forms, modular n≥1 p curves and Hecke symmetry Complex multiplication −s/2 Frobenius symmetry such that the function ξ(s) = π · Γ(s/2) · ζ(s) satisﬁes Monodromy Fine structure in characteristic p ξ(1 − s) = ξ(s). GEOMETRY AND NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Zeta and L-values

Figure: Riemann

GEOMETRY AND Dirichlet L-functions NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers (ii) Similar properties hold for the Dirichlet L-function Zeta and L-values Sample of geometric −s structures and L(χ,s) = ∑ χ(n) · n Re(s) > 1 symmetries n∈N,(n,N)=1 Elliptic curve basics Modular forms, modular curves and Hecke symmetry × × Complex multiplication for a primitive Dirichlet character χ : (Z/NZ) → C . Frobenius symmetry Monodromy Fine structure in characteristic p GEOMETRY AND NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Zeta and L-values

Figure: Dirichlet

GEOMETRY AND L-functions and distribution of prime numbers NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of (a) Dirichlet’s theorem for primes in arithmetic progression prime numbers ↔ L(χ,1) 6= 0 ∀ Dirichlet character χ. Zeta and L-values Sample of geometric structures and (b) The prime number theorem symmetries Elliptic curve basics ↔ zero free region of ζ(s) near {Re(s) = 1}. Modular forms, modular curves and Hecke symmetry Complex multiplication (c) Riemann’s hypothesis ↔ the ﬁrst term after the main term Frobenius symmetry (s) Monodromy in the asymptotic expansion of ζ . Fine structure in characteristic p GEOMETRY AND Bernoulli numbers and zeta values NUMBERS Ching-Li Chai

Sample arithmetic 5. Special values of L-functions statements Diophantine equations Counting solutions of a Examples. (a) zeta and L-values for Q. diophantine equation Counting congruence solutions Recall that the Bernoulli numbers Bn are deﬁned by L-functions and distribution of prime numbers Zeta and L-values x Bn = · xn Sample of geometric ex − 1 ∑ n! structures and n∈N symmetries Elliptic curve basics Modular forms, modular B0 = 1, B1 = −1/2, B2 = 1/6, B4 = −1/30, B6 = 1/42, curves and Hecke symmetry Complex multiplication B8 = −1/30, B10 = 5/66, B12 = −691/2730. Frobenius symmetry Monodromy Fine structure in characteristic p (i) (Euler) ζ(1 − k) = −Bk/k ∀ even integer k > 0.

(ii) (Leibniz’s formula, 1678; Madhava, ∼ 1400) 1 1 1 π 1 − 3 + 5 − 7 + ··· = 4

GEOMETRY AND Content of L-values NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of (b) L-values often contain deep arithmetic/geometric prime numbers information. Zeta and L-values Sample of geometric √ structures and (i) Leibniz’s formula: Z[ −1] is a√ PID (because the formula symmetries Elliptic curve basics implies that the class number h(Q( −1)) is 1). Modular forms, modular curves and Hecke symmetry Complex multiplication (ii) Bk/k appears in the formula for the number of Frobenius symmetry ( k − ) Monodromy (isomorphism classes of) exotic 4 1 -spheres. Fine structure in characteristic p GEOMETRY AND Kummer congruence NUMBERS Ching-Li Chai

Sample arithmetic (c) (Kummer congruence) statements Diophantine equations (i) (m) ∈ for m ≤ 0 with m 6≡ 1 (mod p − 1) Counting solutions of a ζ Zp diophantine equation 0 0 Counting congruence solutions (ii) ζ(m) ≡ ζ(m )(mod p) for all m,m ≤ 0 with L-functions and distribution of 0 prime numbers m ≡ m 6≡ 1 (mod p − 1). Zeta and L-values Sample of geometric structures and Examples. symmetries Elliptic curve basics 1 Modular forms, modular ζ(−1) = − 2 2 ; −1 ≡ 1 (mod p − 1) only for p = 2,3. curves and Hecke symmetry 2 · 3 Complex multiplication 691 Frobenius symmetry (−11) = ; −11 ≡ 1 (mod p − 1) holds Monodromy ζ 3 2 Fine structure in characteristic 2 · 3 · 5 · 7 · 13 p only for p = 2,3,5,7,13. 1 ζ(−5) = − ≡ ζ(−1)(mod 5). 22 · 32 · 7 Note that 3 · 7 ≡ 1 (mod 5).

GEOMETRY AND NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Zeta and L-values

Figure: Kummer GEOMETRY AND Elliptic curves basics NUMBERS Ching-Li Chai

II. Sample of geometric structures and symmetries Sample arithmetic statements Diophantine equations Counting solutions of a 1. Review of elliptic curves diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Equivalent definitions of an elliptic curve E : Zeta and L-values Sample of geometric a projective curve with an algebraic group law; structures and symmetries Elliptic curve basics a projective curve of genus one together with a rational Modular forms, modular curves and Hecke symmetry point (= the origin); Complex multiplication Frobenius symmetry over C: a complex torus of the form Eτ = C/Zτ + Z, Monodromy Fine structure in characteristic where τ ∈ H := upper-half plane; p over a field F with 6 ∈ F×: given by an affine equation

2 3 y = 4x − g2 x − g3, g2,g3 ∈ F .

GEOMETRY AND Weistrass theory NUMBERS Ching-Li Chai For Eτ = C/Zτ + Z, let Sample arithmetic statements xτ (z) = ℘(τ,z) Diophantine equations Counting solutions of a 1 1 1 diophantine equation = + − Counting congruence solutions z2 ∑ (z − m − n)2 (m + n)2 L-functions and distribution of (m,n)6=(0,0) τ τ prime numbers Zeta and L-values

Sample of geometric d yτ (z) = dz℘(τ,z) structures and symmetries Elliptic curve basics Then Eτ satisﬁes the Weistrass equation Modular forms, modular curves and Hecke symmetry 2 3 Complex multiplication y = 4x − g2(τ)xτ − g3(τ) Frobenius symmetry τ τ Monodromy Fine structure in characteristic with p 1 g (τ) = 60 2 ∑ (mτ + n)4 (0,0)6=(m,n)∈Z2 1 g (τ) = 140 3 ∑ (mτ + n)6 (0,0)6=(m,n)∈Z2 GEOMETRY AND The j-invariant NUMBERS Ching-Li Chai Elliptic curves are classiﬁed by their j-invariant Sample arithmetic g3 statements j = 1728 2 Diophantine equations 3 2 Counting solutions of a g2 − 27g3 diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Over C, j(Eτ ) depends only on the lattice Zτ + Z of Eτ . So j(τ) Zeta and L-values is a modular function for SL2(Z): Sample of geometric structures and symmetries aτ + b Elliptic curve basics j = j(τ) Modular forms, modular cτ + d curves and Hecke symmetry Complex multiplication Frobenius symmetry for all a,b,c,d ∈ with ad − bc = 1. Monodromy Z Fine structure in characteristic p We have a Fourier expansion 1 j(τ) = + 744 + 196884q + 21493760q2 + ··· , q √ 2π −1τ where q = qτ = e .

GEOMETRY AND Modular forms, modular curves and Hecke NUMBERS symmetry Ching-Li Chai Sample arithmetic statements Diophantine equations 2. Modular forms and modular curves Counting solutions of a diophantine equation ⊂ ( ) ( ) Counting congruence solutions Let Γ SL2 Z be a congruence subgroup of SL2 Z , i.e. Γ L-functions and distribution of prime numbers contains all elements which are ≡ I2 (mod N) for some N. Zeta and L-values Sample of geometric (a) A holomorphic function f (τ) on the upper half plane H is structures and symmetries said to be a modular form of weight k and level Γ if Elliptic curve basics Modular forms, modular curves and Hecke symmetry −1 k a b Complex multiplication f ((aτ + b)(cτ + d) ) = (cτ + d) · f (τ) ∀γ = ∈ Γ Frobenius symmetry c d Monodromy Fine structure in characteristic p and has moderate growth at all cusps.

(b) The quotient YΓ := Γ\H has a natural structure as an (open) algebraic curve, deﬁnable over a natural number ﬁeld; it parametrizes elliptic curves with suitable level structure. GEOMETRY AND Modular curves and Hecke symmetry NUMBERS Ching-Li Chai

Sample arithmetic statements 0 k (c) Modular forms of weight k for Γ = H (X ,ω ), where X is Diophantine equations Γ Γ Counting solutions of a diophantine equation the natural compactiﬁcation of YΓ, and ω is the Hodge line Counting congruence solutions L-functions and distribution of bundle on XΓ prime numbers Zeta and L-values

∨ Sample of geometric ω|[E] = Lie(E) ∀[E] ∈ XΓ structures and symmetries Elliptic curve basics (d) The action of GL2( )det>0 on “survives” on the modular Modular forms, modular Q H curves and Hecke symmetry curve YΓ = Γ\H and takes a reincarnated form as a family of Complex multiplication Frobenius symmetry algebraic correspondences. Monodromy Fine structure in characteristic The L-function attached to a cusp form which is a common p eigenvector of all Hecke correspondences admits an Euler product.

GEOMETRY AND NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Zeta and L-values

Figure: Hecke GEOMETRY AND The Ramanujan τ function NUMBERS Ching-Li Chai

Example. Weight 12 cusp forms for SL ( ) are constant Sample arithmetic 2 Z statements multiples of Diophantine equations Counting solutions of a diophantine equation = q · ( − qm)24 = (n)qn Counting congruence solutions ∆ ∏ 1 ∑ τ L-functions and distribution of n prime numbers m≥1 Zeta and L-values

Sample of geometric and structures and symmetries Tp(∆) = τ(p) · ∆ ∀p, Elliptic curve basics Modular forms, modular curves and Hecke symmetry p 0 Complex multiplication where Tp is the Hecke operator represented by . Frobenius symmetry 0 1 Monodromy Fine structure in characteristic p −s Let L(∆,s) = ∑n≥1 an · n . We have L(∆,s) = ∏(1 − τ(p)p−s + p11−2s)−1. p

GEOMETRY AND CM elliptic curves NUMBERS Ching-Li Chai

3. Complex multiplication Sample arithmetic statements An elliptic E over is said to have complex multiplication if its Diophantine equations C Counting solutions of a 0 diophantine equation endomorphism algebra End (E) is an imaginary quadratic ﬁeld. Counting congruence solutions L-functions and distribution of prime numbers Example. Consequences of Zeta and L-values Sample of geometric • j(C/OK) is an algebraic integer structures and • K · j( / ) = the Hilbert class ﬁeld of K. symmetries C OK Elliptic curve basics Modular forms, modular √ curves and Hecke symmetry π 67 Complex multiplication e = 147197952743.9999986624542245068292613··· Frobenius symmetry √ Monodromy −1+ −67 15 3 3 3 j = −147197952000 = −2 · 3 · 5 · 11 Fine structure in characteristic 2 p

√ eπ 163 = 262537412640768743.99999999999925007259719... √ −1+ −163 j 2 = −262537412640768000 = −218 · 33 · 53 · 233 · 293 GEOMETRY AND Mod p points for a CM curve NUMBERS Ching-Li Chai

A typical feature of CM elliptic curves is that there are explicit Sample arithmetic statements formulas: Let E be the elliptic curve Diophantine equations Counting solutions of a diophantine equation 2 3 y = x + x, Counting congruence solutions L-functions and distribution of prime numbers √ Zeta and L-values which has CM by [ −1]. We have Z Sample of geometric structures and symmetries #E(Fp) = 1 + p − ap Elliptic curve basics Modular forms, modular curves and Hecke symmetry and for odd p we have Complex multiplication Frobenius symmetry Monodromy 3 Fine structure in characteristic u + u p a = p ∑ p u∈Fp 0 if p ≡ 3 (mod 4) = −2a if p = a2 + 4b2 with a ≡ 1 (mod 4)

GEOMETRY AND A CM curve and its associated modular form, NUMBERS continued Ching-Li Chai Sample arithmetic The L-function L(E,s) attached to E with statements Diophantine equations Counting solutions of a −s 1−2s −1 −s diophantine equation ∏ (1 − app + p ) = ∑an · n Counting congruence solutions n L-functions and distribution of p odd prime numbers Zeta and L-values is equal to a Hecke L-function L(ψ,s), where the Hecke Sample of geometric structures and character ψ is the given by symmetries Elliptic curve basics Modular forms, modular 0 if 2|N(a) curves and Hecke symmetry ψ(a) = √ Complex multiplication if a = ( ), ∈ 1 + 4 + 2 −1 Frobenius symmetry λ λ λ Z Z Monodromy Fine structure in characteristic n p The function fE(τ) = ∑n an · q is a modular form of weight 2 and level 4, and

N(a) a2+b2 fE(τ) = ∑ψ(a) · q = ∑ a · q a a≡1 (mod 4) b≡0 (mod 2) GEOMETRY AND Frobenius symmetry NUMBERS Ching-Li Chai

Sample arithmetic 4. Frobenius symmetry statements Diophantine equations Every algebraic variety X over a finite field has a map Counting solutions of a Fq diophantine equation q Frq : X → X, induced by the ring endomorphism f 7→ f of the Counting congruence solutions L-functions and distribution of function field of X. prime numbers Zeta and L-values

Sample of geometric Deligne’s proof of Weil’s conjecture implies that structures and symmetries 11/2 Elliptic curve basics τ(p) ≤ 2p ∀p Modular forms, modular curves and Hecke symmetry Complex multiplication Frobenius symmetry Idea: Step 1. Use Hecke symmetry to cut out a 2-dimensional Monodromy 1 10 Fine structure in characteristic Galois representation inside Het X,Sym (H(E /X)) , which p “contains” the cusp form ∆ via the Eichler-Shimura integral. Step 2. Apply the Eichler-Shimura congruence relation, which relates Frp and the Hecke correspondence Tp; invoke the Weil bound.

GEOMETRY AND A hypergeometric differential equation NUMBERS Ching-Li Chai

Sample arithmetic statements 5. Monodromy Diophantine equations Counting solutions of a diophantine equation (a) The hypergeometric differential equation Counting congruence solutions L-functions and distribution of prime numbers d2y dy Zeta and L-values 4x(1 − x) + 4(1 − 2x) − y = 0 Sample of geometric dx2 dx structures and symmetries Elliptic curve basics has a classical solution Modular forms, modular curves and Hecke symmetry Complex multiplication n F(1/2,1/2,1,x) = (−1/2)x Frobenius symmetry ∑ n Monodromy n≥0 Fine structure in characteristic p The global monodromy group of the above differential is the principal congruence subgroup Γ(2). GEOMETRY AND Historic origin NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions Remark. The word “monodromy” means “run around singly”; L-functions and distribution of prime numbers it was (?ﬁrst) used by Riemann in Beiträge zur Theorie der Zeta and L-values Sample of geometric durch die Gauss’sche Reihe F(α,β,γ,x) darstellbaren structures and Functionen, 1857. symmetries Elliptic curve basics Modular forms, modular . . . ; für einen Werth in welchem keine Verzweigung curves and Hecke symmetry Complex multiplication statﬁndet, heist die Function “einändrig order Frobenius symmetry Monodromy monodrom . . . Fine structure in characteristic p

GEOMETRY AND The Legendre family of elliptic curves NUMBERS Ching-Li Chai

Sample arithmetic statements The family of equations Diophantine equations Counting solutions of a 2 1 diophantine equation y = x(x − 1)(x − λ) 0,1,∞ 6= λ ∈ P Counting congruence solutions L-functions and distribution of prime numbers deﬁnes a family π : E → S = 1 − {0,1,∞} of elliptic curves, Zeta and L-values P Sample of geometric with structures and 28 [1 − λ(1 − λ)]3 symmetries j(E ) = Elliptic curve basics λ 2 2 Modular forms, modular λ (1 − λ) curves and Hecke symmetry Complex multiplication This formula exhibits the λ-line as an S -cover of the j-line, Frobenius symmetry 3 Monodromy Fine structure in characteristic such that the 6 conjugates of λ are p 1 1 λ λ − 1 λ, , 1 − λ, , , . λ 1 − λ λ − 1 λ GEOMETRY AND The Legendre family, continued NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of The formula prime numbers Zeta and L-values d d dx y Sample of geometric 4λ (1 − λ) + 4(1 − 2λ) − 1 = −d structures and d 2 d y (x − )2 symmetries λ λ λ Elliptic curve basics Modular forms, modular 1 curves and Hecke symmetry means that the global section [dx/y] of HdR(E /S) satisﬁes the Complex multiplication Frobenius symmetry above hypergeometric ODE. Monodromy Fine structure in characteristic p

GEOMETRY AND Monodromy and symmetry NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation 1. Monodromy can be regarded as attainable symmetries Counting congruence solutions L-functions and distribution of among potential symmetries. prime numbers Zeta and L-values 2. To say that the monodromy is “as large as possible” is an Sample of geometric structures and irreducibility statement. symmetries Elliptic curve basics Modular forms, modular 3. Maximality of monodromy has important consequences. curves and Hecke symmetry Complex multiplication E.g. the key geometric input in Deligne-Ribet’s proof of p-adic Frobenius symmetry Monodromy interpolation for special values of Hecke L-functions attached Fine structure in characteristic p to totally real ﬁelds. GEOMETRY AND Supersingular elliptic curves NUMBERS Ching-Li Chai

Sample arithmetic statements 6. Fine structure in char. p > 0 Diophantine equations Counting solutions of a diophantine equation Example. (ordinary/supersingular dichotomy) Counting congruence solutions L-functions and distribution of Elliptic curves over an algebraically closed field k ⊃ Fp come in prime numbers two flavors. Zeta and L-values Sample of geometric structures and Those with E(k) ' (0) are called supersingular. symmetries Elliptic curve basics There is only a finite number of supersingular j-values. Modular forms, modular curves and Hecke symmetry An elliptic curve E over a finite field Fq is supersingular if Complex multiplication Frobenius symmetry and only if E(Fq) ≡ 0 (mod p). Monodromy Fine structure in characteristic Those with E[p](k) ' Z/pZ are said to be ordinary. p An elliptic curve E over a finite field Fq is supersingular if and only if E(Fq) 6≡ 0 (mod p)

GEOMETRY AND The Hasse invariant NUMBERS Ching-Li Chai

p > Sample arithmetic For the Legendre family, the supersingular locus (for 2) is statements the zero locus of Diophantine equations Counting solutions of a diophantine equation (p−1)/2 2 Counting congruence solutions (p−1)/2 (1/2)j j L-functions and distribution of A(λ) = (−1) · ∑ · λ prime numbers Zeta and L-values j=0 j! Sample of geometric structures and where (c)m := c(c + 1)···(c + m − 1). symmetries Elliptic curve basics Modular forms, modular Remark. The above formula for the coefﬁcients aj satisfy curves and Hecke symmetry Complex multiplication Frobenius symmetry Monodromy a1,...,a(p−1)/2 ∈ Z(p) Fine structure in characteristic p and a(p+1)/2 ≡ ··· ≡ ap−1 ≡ 0 (mod p). GEOMETRY AND Counting supersingular j-values NUMBERS Ching-Li Chai

Sample arithmetic statements Theorem. (Eichler 1938) The number hp of supersingular Diophantine equations Counting solutions of a j-values is diophantine equation Counting congruence solutions L-functions and distribution of  bp/12c if p ≡ 1 (mod 12) prime numbers  Zeta and L-values hp = dp/12e if p ≡ 5 or 7 (mod 12) Sample of geometric  structures and dp/12e + 1 if p ≡ 11 (mod 12) symmetries Elliptic curve basics Modular forms, modular Remark. (i) It is known that hp is the class number for the curves and Hecke symmetry Complex multiplication quaternion division algebra over Q ramiﬁed (exactly) at p and Frobenius symmetry Monodromy ∞. Fine structure in characteristic p (ii) Deuring thought that it is nicht leicht that the above class number formula can be obtained by counting supersingular j-invariants directly.

GEOMETRY AND Igusa’s proof NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation From the hypergeometric equation for F(1/2,1/2,1,x) we Counting congruence solutions L-functions and distribution of conclude that prime numbers Zeta and L-values d2 d Sample of geometric 4λ (1 − λ) + 4(1 − 2λ) − 1 A(λ) ≡ 0 (mod p) structures and dλ 2 dλ symmetries Elliptic curve basics Modular forms, modular curves and Hecke symmetry for all p > 3. It follows immediately that A(λ) has simple Complex multiplication Frobenius symmetry zeroes. The formula for hp is now an easy consequence. (Hint: Monodromy Fine structure in characteristic Use the formula 6-to-1 cover of the j-line by the λ-line.) p Q.E.D. GEOMETRY AND p-adic monodromy for modular curves NUMBERS Ching-Li Chai

Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions For the ordinary locus of the Legendre family L-functions and distribution of prime numbers ord ord Zeta and L-values π : E → S Sample of geometric structures and symmetries the monodromy representation Elliptic curve basics Modular forms, modular curves and Hecke symmetry ord ord ∞ ∼ × Complex multiplication ρ : π1 S → Aut E [p ](Fp) = Zp Frobenius symmetry Monodromy Fine structure in characteristic (deﬁned by Galois theory) is surjective. p

GEOMETRY AND p-adic monodromy for the modular curve NUMBERS Ching-Li Chai

Sample arithmetic statements n × Sketch of a proof: Given any n > 0 and anyu ¯ ∈ (Z/p Z) , Diophantine equations n Counting solutions of a pick a representative u ∈ ofu ¯ with 0 < u < p and let let diophantine equation N Counting congruence solutions L-functions and distribution of 2 4n prime numbers ι : Q[T]/(T − u · T + p ) ,→ Qp Zeta and L-values Sample of geometric × structures and be the embedding such that ι(T) ∈ Zp . Then symmetries Elliptic curve basics Modular forms, modular 2n curves and Hecke symmetry ι(T) ≡ u (mod p ). Complex multiplication Frobenius symmetry Monodromy By a result of Deuring, there exists an elliptic curve E over 2n Fine structure in characteristic Fp p whose Frobenius is the Weil number ι(T). So the image of the monodromy representation contains ι(T), which is congruent n to the given elementu ¯ ∈ Z/p Z. Q.E.D.