Non-archimedean construction of elliptic curves and abelian surfaces - PowerPoint PPT Presentation

Non-archimedean construction of elliptic curves and abelian surfaces ICERM WORKSHOP Modular Forms and Curves of Low Genus: Computational Aspects Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3 1 Universitat de Barcelona 2 University of Warwick 3 University of Sheffield September 28 th , 2015 September 28 th , 2015 Marc Masdeu Non-archimedean constructions 0 / 34

Modular Forms and Curves of Low Genus: Computational Aspects Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3 1 Universitat de Barcelona 2 University of Warwick 3 University of Sheffield September 28 th , 2015 September 28 th , 2015 Marc Masdeu Non-archimedean constructions 1 / 34

Quaternionic automorphic forms of level N F a number field of signature p r, s q , and fix N Ă O F . Choose factorization N “ Dn , with D square free . Fix embeddings v 1 , . . . , v r : F ã Ñ R , w 1 , . . . , w s : F ã Ñ C . Let B { F be a quaternion algebra such that Ram p B q “ t q : q | D u Y t v n ` 1 , . . . , v r u , p n ď r q . Fix isomorphisms B b F v i – M 2 p R q , i “ 1 , . . . , n ; B b F w j – M 2 p C q , j “ 1 , . . . , s. Ñ PGL 2 p R q n ˆ PGL 2 p C q s H n ˆ H s These yield B ˆ { F ˆ ã 3 . ý R > 0 R > 0 PGL 2 ( R ) H 3 PGL 2 ( C ) H C R September 28 th , 2015 Marc Masdeu Non-archimedean constructions 2 / 34

Quaternionic automorphic forms of level N (II) Fix R D 0 p n q Ă B Eichler order of level n . F acts discretely on H n ˆ H s 0 p n q ˆ { O ˆ Γ D 0 p n q “ R D 3 . Obtain an orbifold of (real) dimension 2 n ` 3 s : 0 p n qz p H n ˆ H s Y D 0 p n q “ Γ D 3 q . The cohomology of Y D 0 p n q can be computed via H ˚ p Y D 0 p n q , C q – H ˚ p Γ D 0 p n q , C q . Hecke algebra T D “ Z r T q : q ∤ D s acts on H ˚ p Γ D 0 p n q , Z q . à χ : T D Ñ C . H n ` s p Γ D H n ` s p Γ N 0 p n q , C q χ , 0 p n q , C q “ χ Each χ cuts out a field K χ , s.t. r K χ : Q s “ dim H n ` s p Γ D 0 p n q , C q χ . September 28 th , 2015 Marc Masdeu Non-archimedean constructions 3 / 34

Abelian varieties from cohomology classes Definition 0 p n q , C q χ eigen for T D is rational if a p p f q P Z , @ p P T D . f P H n ` s p Γ D If r “ 0 , then assume N is not square-full: D p � N . Conjecture (Taylor, ICM 1994) f P H n ` s p Γ D 0 p n q , Z q a new , rational eigenclass . 1 Then D E f { F of conductor N “ Dn attached to f . i.e. such that # E f p O F { p q “ 1 ` | p | ´ a p p f q @ p ∤ N . More generally, if χ : T D Ñ C is nontrivial, cutting out a field K , then 2 D abelian variety A χ , with dim A χ “ r K : F s and multiplication by K . Assumption above avoids “fake abelian varieties”, and it is needed in our construction anyway. September 28 th , 2015 Marc Masdeu Non-archimedean constructions 4 / 34

Goals of this talk In this talk we will: Review known explicit forms of this conjecture. 1 § Cremona’s algorithm for F “ Q . § Generalizations to totally real fields. Propose a new, non-archimedean , conjectural construction. 2 § (joint work with X. Guitart and H. Sengun) Explain some computational details. 3 Illustrate with examples . 4 September 28 th , 2015 Marc Masdeu Non-archimedean constructions 5 / 34

F “ Q : Cremona’s algorithm for elliptic curves Eichler–Shimura construction ş _ H 0 p X 0 p N q , Ω 1 q Hecke � Jac p X 0 p N qq � � C { Λ f – E f p C q . X 0 p N q – H 1 p X 0 p N q , Z q Compute H 1 p X 0 p N q , Z q (modular symbols). 1 Find the period lattice Λ f by explicitly integrating 2 Cż ¯G ´ ÿ a n p f q e 2 πinz : γ P H 1 Λ f “ X 0 p N q , Z 2 πi . γ n ě 1 Compute c 4 p Λ f q , c 6 p Λ f q P C by evaluating Eistenstein series. 3 Recognize c 4 p Λ f q , c 6 p Λ f q as integers ❀ E f : Y 2 “ X 3 ´ c 4 48 X ´ c 6 864 . 4 September 28 th , 2015 Marc Masdeu Non-archimedean constructions 6 / 34

F ‰ Q : constructions for elliptic curves F totally real . r F : Q s “ n , fix σ : F ã Ñ R . ω f P H n p Γ 0 p N q , C q ❀ Λ f Ď C . S 2 p Γ 0 p N qq Q f ❀ ˜ Conjecture (Oda, Darmon, Gartner) C { Λ f is isogenous to E f ˆ F F σ . Known to hold (when F real quadratic) for base-change of E { Q . Exploited in very restricted cases (Demb´ el´ e, Stein+7). Explicitly computing Λ f is hard. § No quaternionic computations (except for Voight–Willis?). F not totally real : no known algorithms. . . Theorem If F is imaginary quadratic , the lattice Λ f is contained in R . Idea Allow for non-archimedean constructions. September 28 th , 2015 Marc Masdeu Non-archimedean constructions 7 / 34

Non-archimedean construction From now on: fix p � N . Denote by ¯ F p “ alg. closure of the p -completion of F . Theorem (Tate uniformization) There exists a rigid-analytic, Galois-equivariant isomorphism η : ¯ F ˆ p {x q E y Ñ E p ¯ F p q , with q E P F ˆ p satisfying j p E q “ q ´ 1 E ` 744 ` 196884 q E ` ¨ ¨ ¨ . Choose a coprime factorization N “ pDm , with D “ disc p B { F q . Compute q E as a replacement for Λ f . Starting data: f P H n ` s p Γ D 0 p m q , Z q p ´ new , pDm “ N . September 28 th , 2015 Marc Masdeu Non-archimedean constructions 8 / 34

Non-archimedean path integrals on H p Consider H p “ P 1 p C p q � P 1 p F p q . It is a p -adic analogue to H : § It has a rigid-analytic structure. § Action of PGL 2 p F p q by fractional linear transformations . § Rigid-analytic 1 -forms ω P Ω 1 H p . ş τ 2 § Coleman integration ❀ make sense of τ 1 ω P C p . ş H p ˆ Div 0 H p Ñ C p . : Ω 1 Get a PGL 2 p F p q -equivariant pairing For each Γ Ă PGL 2 p F p q , get induced pairing (cap product) ş H p q ˆ H i p Γ , Div 0 H p q � C p H i p Γ , Ω 1 ż ´ ¯ ✤ φ, ř � ř γ γ b D γ φ p γ q . γ D γ Ω 1 H p – space of C p -valued boundary measures Meas 0 p P 1 p F p q , C p q . September 28 th , 2015 Marc Masdeu Non-archimedean constructions 9 / 34

Measures and integrals T Bruhat-Tits tree of GL 2 p F p q , | p | “ 2 . P 1 p F p q – Ends p T q . v ∗ Harmonic cocycles HC p A q “ Ñ A | ř e ∗ f t E p T q o p e q“ v f p e q “ 0 u ˆ v ∗ µ ( U ) U ⊂ P 1 ( F p ) Meas 0 p P 1 p F p q , A q – HC p A q . P 1 ( F p ) So replace ω P Ω 1 H p with µ ω P Meas 0 p P 1 p F p q , Z q – HC p Z q . Coleman integration: if τ 1 , τ 2 P H p , then ż τ 2 ż ˆ t ´ τ 2 ˙ ˆ t U ´ τ 2 ˙ ÿ ω “ dµ ω p t q “ lim µ ω p U q . log p log p Ý Ñ t ´ τ 1 t U ´ τ 1 P 1 p F p q τ 1 U U P U Multiplicative refinement (assume µ ω p U q P Z , @ U ): ż τ 2 ż ˆ t ´ τ 2 ˙ ˆ t U ´ τ 2 ˙ µ ω p U q ź ˆ ω “ ˆ dµ ω p t q “ lim . Ý Ñ t ´ τ 1 t U ´ τ 1 P 1 p F p q τ 1 U U P U September 28 th , 2015 Marc Masdeu Non-archimedean constructions 10 / 34

The t p u -arithmetic group Γ Choose a factorization N “ pDm . B { F “ quaternion algebra with Ram p B q “ t q | D u Y t v n ` 1 , . . . , v r u . Recall also R D 0 p pm q Ă R D 0 p m q Ă B . Fix ι p : R D 0 p m q ã Ñ M 2 p Z p q . 0 p pm q ˆ { O ˆ 0 p m q ˆ { O ˆ Define Γ D 0 p pm q “ R D F and Γ D 0 p m q “ R D F . 0 p m qr 1 { p s ˆ { O F r 1 { p s ˆ ι p Let Γ “ R D Ñ PGL 2 p F p q . ã Example F “ Q and D “ 1 , so N “ pM . B “ M 2 p Q q . �` a b ˘ ( Γ 0 p pM q “ P GL 2 p Z q : pM | c {t˘ 1 u . c d �` a b ˘ ( Γ “ P GL 2 p Z r 1 { p sq : M | c {t˘ 1 u ã Ñ PGL 2 p Q q Ă PGL 2 p Q p q . c d September 28 th , 2015 Marc Masdeu Non-archimedean constructions 11 / 34

The t p u -arithmetic group Γ Lemma Assume that h ` F “ 1 . Then ι p induces bijections Γ { Γ D Γ { Γ D 0 p m q – V 0 p T q , 0 p pm q – E 0 p T q V 0 “ V 0 p T q (resp. E 0 “ E 0 p T q ) are the even vertices (resp. edges) of T . Proof. Strong approximation ù ñ Γ acts transitively on E 0 and V 0 . 1 Stabilizer of vertex v ˚ (resp. edge e ˚ ) is Γ D 0 p m q (resp. Γ D 0 p pm q ). 2 Corollary ´ ¯ 2 Maps p E 0 p T q , Z q – Ind Γ Ind Γ 0 p pm q Z , Maps p V p T q , Z q – 0 p m q Z . Γ D Γ D September 28 th , 2015 Marc Masdeu Non-archimedean constructions 12 / 34

Cohomology 0 p m qr 1 { p s ˆ { O F r 1 { p s ˆ ι p Γ “ R D Ñ PGL 2 p F p q . ã ´ ¯ 2 Maps p E 0 p T q , Z q – Ind Γ Ind Γ 0 p pm q Z , Maps p V p T q , Z q – 0 p m q Z . Γ D Γ D Want to define a cohomology class in H n ` s p Γ , Ω 1 H p q . Consider the Γ -equivariant exact sequence β � HC p Z q � Maps p E 0 p T q , Z q � Maps p V p T q , Z q � 0 0 � r v ÞÑ ř ϕ ✤ o p e q“ v ϕ p e qs So get: ´ ¯ 2 β 0 Ñ HC p Z q Ñ Ind Γ Ind Γ 0 p pm q Z Ñ 0 p m q Z Ñ 0 Γ D Γ D September 28 th , 2015 Marc Masdeu Non-archimedean constructions 13 / 34

Cohomology (II) ´ ¯ 2 β 0 Ñ HC p Z q Ñ Ind Γ Ind Γ 0 p pm q Z Ñ 0 p m q Z Ñ 0 Γ D Γ D Taking Γ -cohomology, . . . 0 p m q , Z q 2 Ñ ¨ ¨ ¨ β H n ` s p Γ , HC p Z qq Ñ H n ` s p Γ , Ind Γ Ñ H n ` s p Γ , Ind Γ 0 p pm q , Z q Γ D Γ D . . . and using Shapiro’s lemma: 0 p m q , Z q 2 Ñ ¨ ¨ ¨ β H n ` s p Γ , HC p Z qq Ñ H n ` s p Γ D Ñ H n ` s p Γ D 0 p pm q , Z q f P H n ` s p Γ D 0 p pm q , Z q being p -new ô f P Ker p β q . Pulling back get ω f P H n ` s p Γ , HC p Z qq – H n ` s p Γ , Ω 1 H p q . September 28 th , 2015 Marc Masdeu Non-archimedean constructions 14 / 34