Traces of rationality of Darmon points [BD09] Bertolini Darmon, The rationality of Stark Heegner points over genus fields of real quadratic fields Seminari de Teoria de Nombres de Barcelona Barcelona, January 2017 Marc Masdeu University of Warwick Marc Masdeu Traces of rationality 0 / 13
Introduction E{Q an elliptic curve of conductor N pm, with p M. K{Q a real quadratic field, satisfying: p is inert in K l M ùñ l split in K. These conditions imply that ord s 1 LpE{K, sq is odd. BSD predicts DP K P EpKq nontorsion...... but Heegner points are not readily available, since K is real! For example H X K H. Darmon s insight: consider H p P 1 pc p q P 1 pq p q instead. Constructed a point P ψ P EpC p q for each ψ : O K ãñ M 2 pqq. Conjecture (2001): P ψ P EpK ab q, and behave like Heegner points. In particular, he conjectured that ř ψ P ψ P EpKq. This is (a particular case of) what Bertolini Darmon proved in 2009. Marc Masdeu Traces of rationality 1 / 13
Plan 1 Define Darmon points (a.k.a. Stark Heegner points). 2 State the main result of [BD09]. 3 Sketch the proof. Marc Masdeu Traces of rationality 2 / 13
Measure-valued modular symbols and integration Let Γ ` a b c d P M2 pzr1{psq X SL 2 pqq: N c (. Meas 0 pp 1 pq p qq the Γ-module of measures on P 1 pq p q having total measure 0. From previous talks: E f P S 2 pγ 0 pnqq I f P Symb new Γ 0 ppmq pzq`. Proposition D! µ µ f P H 1 pγ, Meas 0 pp 1 pq p qqq satisfying: µ γ pz p q I f t8 Ñ γ8u for all γ P Γ 0 pnq Ă Γ. Recall q q E P pz p the Tate period attached to E{Q p. There is a Γ-equivariant pairing Meas 0 pp 1 pq p qq ˆ Div 0 H p Ñ C p : ż ˆt τ2 pµ, τ 2 τ 1 q ÞÑ log q dµptq. P 1 pq pq t τ 1 Cap product induces a pairing x, y: H 1 pγ, Meas 0 pp 1 pq p qqq ˆ H 1 pγ, Div 0 H p q Ñ C p. Marc Masdeu Traces of rationality 3 / 13
Indefinite integrals x, y: H 1 pγ, Meas 0 pp 1 pq p qqq ˆ H 1 pγ, Div 0 H p q Ñ C p. Consider the short exact sequence of Γ-modules 0 Ñ Div 0 H p ι Ñ Div H p deg Ñ Z Ñ 0. Taking Γ-homology, get a long exact sequence Ñ H 2 pγ, Zq Ñ δ H 1 pγ, Div 0 H p q ι Ñ H 1 pγ, Div H p q deg Ñ H 1 pγ, Zq Ñ Fact: H 1 pγ, Zq Γ ab is a finite abelian group, say of size e Γ. So Θ P H 1 pγ, Div H p q Θ 0 P H 1 pγ, Div 0 H p q such that ι pθ 0 q e Γ Θ. Define xµ f, Θy : 1 e Γ xµ f, Θ 0 y, which is an indefinite integral x, y: H 1 pγ, Meas 0 pp 1 pq p qqq ˆ H 1 pγ, Div H p q Ñ C p. Marc Masdeu Traces of rationality 4 / 13
Darmon points Recall K{Q our real quadratic field, of discriminant D. Cl`pKq the narrow class group of K. Consider optimal embeddings ψ : O K ãñ M 0 pmq t` a b c d P M2 pzq : M cu. Similar to Carlos talk, have action: Cl`pKq ñ EmbpO K q t conjugacy classes of optimal embeddingsu. If σ P Cl`pKq and ψ P EmbpO K q, write ψ σ for the translate. Global class-field theory gives rec: Cl`pKq» Ñ GalpH` K {Kq, H` K narrow Hilbert class field of K. ψ P EmbpO K q Θ K rγ ψ b τ ψ s P H 1 pγ, Div H p q: γ ψ ψpu K q P Γ 0 pmq Ă Γ, where Oˆ K,1 {tors xu Ky. τ ψ is fixed by γ ψ (one can consistently choose one of two options). Define J ψ xµ f, Θ ψ y P C p. Marc Masdeu Traces of rationality 5 / 13
Conjecture (Darmon, 2001) 1 For each ψ P EmbpO K q, there exists P ψ P EpH` K q and t P Qˆ s.t.: J ψ t log E pp ψ q. 2 For each σ P Cl`pKq, the points P ψ satisfy P ψ σ recpσq 1 pp ψ q. Using multiplicative integrals P ψ satisfying J ψ log E pp ψ q. Part 1 of conjecture then says that Pψ P EpH` K q b Q. Darmon Green and Darmon Pollack: numerical evidence. The construction has been generalized to many different settings (Greenberg, Gartner, Trifkovic, Guitart M. Sengun, Longo Vigni, Rotger Seveso,... ). Guitart M.: more numerical evidence supporting these conjectures. Marc Masdeu Traces of rationality 6 / 13
Main Theorem of [BD09] Define: J K ÿ ψpembpo K q Theorem (Bertolini Darmon, 2009) Suppose that: Then: J ψ P C p. E has split multiplicative reduction at p (i.e. a p peq 1). D prime q p of multiplicative reduction. 1 There exists a point P K P EpKq and t P Qˆ such that J K t log E ppq. 2 The point P K is of infinite order if and only if L 1 pe{k, 1q 0. In [BD09] they prove the rationality not only of J K, but also of twisted traces of the points J ψ by genus characters. Analogues of this result: For quaternionic Darmon points (Longo Vigni). For Darmon cycles (Seveso). Strategy of proof inspired by a very famous proof from 1863... Marc Masdeu Traces of rationality 7 / 13
Proof Strategy 1 Construct a p-adic L-function L p pf 8 {K, kq attached to f 8 and K. 2 Relate J K to L p pf 8 {K, kq: J K L 1 ppf 8 {K, kq. 3 Factor L p pf 8 {K, kq 2 in terms of Mazur Kitagawa p-adic L-functions: L p pf 8 {K, kq 2 D k 2 2 Lp pf 8, k, k{2ql p pf K 8, k, k{2q. 4 Deduce the theorem from the results in Carlos talk on the Mazur Kitagawa p-adic L-functions appearing in the RHS. Marc Masdeu Traces of rationality 8 / 13
A p-adic L-function attached to f 8 and K E f P S2 new pγ 0 pnqq I f I` f P H1 pγ 0 pnq, Zq w8 1. I f lives in a family: there exists µ P H 1 pγ 0 pmq, D : q ord such that 1 ρ 2 pµq I f, and 2 For all k P U X Z ě2, Dλpkq P Cˆp such that ρ k pµq λpkqi fk. For each ψ P EmbpO K q, consider γ ψ as before, and τ ψ, τ ψ the fixed points by ψpkˆq acting on Hż p. Very Important Fact: J τ logpx τ ψ yqdµ γτ px, yq. pz 2 pq 1 Define L p pf 8, ψ, kq as ż L p pf 8, ψ, kq Definition The square-root p-adic L-function is: ÿ L p pf 8 {K, kq pz 2 pq 1 ppx τ ψ qpx τ ψ qq k 2 2 dµ γψ px, yq. ψpembpo K q L p pf 8, ψ, kq. Marc Masdeu Traces of rationality 9 / 13
ż L p pf 8, ψ, kq ppx τ ψ qpx τ ψ qq k 2 2 dµ γψ px, yq, pz 2 pq 1 L p pf 8 {K, kq ÿ ψ L p pf 8, ψ, kq. Theorem A - p-adic Gross Zagier We have L p pf 8 {K, 2q 0 and L 1 ppf 8 {K, 2q J K. Proof L p pf 8 {K, ψ, 2q ş pz pq dµ 1 γτ px, yq µ γτ pp 1 pq p qq 0. L 1 ppf 8 {K, ψ, 2q 1 ş 2 pz plogpx τ 2 ψ yq ` logpx τ ψ yqq dµ p q1 γτ px, yq 1 2 pj τ ` Frob p pj τ qq 1 2 pj τ w M J τ σq (for some σ depending only on τ). 1 2 pj τ ` J τ σq (since w M a p 1). Summing over all embeddings we get L 1 ppf 8 {K, 2q 1 2 p1 w ÿ Mq J ψ. σpcl`pkq J τ σ ÿ ψ Marc Masdeu Traces of rationality 10 / 13
Factorization Theorem B - Factorization For all k P U, Proof L p pf 8 {K, kq 2 D k 2 2 Lp pf 8, k, k{2ql p pf K 8, k, k{2q. 1 Interpolation property for L p pf 8 {K, kq 2 (Popa 2006): For all k P U X Z ě2, L p pf 8 {K, kq 2 λpkq 2 p1 a pkq 2 p p k 2 q 2 D k 2 2 L pf 7 k {K, k{2q. λpkq 2 p1 a pkq 2 p p k 2 q 2 D k 2 2 L pf 7 K,7 k, k{2ql pfk, k{2q 2 Use the interpolation property of the MK p-adic L-functions (RHS) to show that the factorization occurs for all k P U X Z ě2. 3 Conclude using that U X Z ě2 is dense in U. Marc Masdeu Traces of rationality 11 / 13
End of proof J K L 1 ppf 8 {K, 2q (A - p-adic GZ) L p pf 8 {K, kq 2 D k 2 2 Lp pf 8, k, k{2ql p pf K 8, k, k{2q (B - Factorization) Easy observations: 1 D k 2 2 1 ` Opk 2q, 2 L p pf 8, k, k{2q Oppk 2q 2 q (exceptional zero case) Deduce that J 2 K 1 2 ˆ d2 dk ˇˇˇk 2 2 L p pf 8, k, k{2q L p pf8 K, 2, 1q. Need to understand the two terms in the RHS. Marc Masdeu Traces of rationality 12 / 13
End of proof (continued) A ` B ùñ JK 2 1 ˆ d2 2 dk ˇˇˇk 2 2 L p pf 8, k, k{2q L p pf8 K, 2, 1q. Theorem C - Bertolini Darmon 2007 1 There exists P P EpQq, and l 1 P Qˆ, such that d 2 dk 2 ˇˇˇk 2 L p pf 8, k, k{2q l 1 log 2 EpPq. 2 P is nontorsion if and only if L 1 pe, 1q 0. a p pe K q 1 ùñ L p pf K 8, 2, 1q 2L pe K, 1q 2l 2 P Qˆ. Get log 2 E pp Kq l 1 l 2 log 2 E ppq, and one sees that l 1l 2 t 2 is a square. Taking square roots yields the theorem: J K t log 2 EpPq, and P is nontorsion iff L 1 pe{k, 1q 0. Marc Masdeu Traces of rationality 13 / 13
Thank you! Bibliography, code and slides at: http://www.warwick.ac.uk/mmasdeu/ Marc Masdeu Traces of rationality 13 / 13