A p -adic Birch and Swinnerton-Dyer conjecture for modular abelian varieties Steffen M¨ uller Universit¨ at Hamburg joint with Jennifer Balakrishnan and William Stein Rational points on curves: A p -adic and computational perspective Oxford University Tuesday, September 25, 2012
Notation The conjecture Algorithms Evidence n =1 a n e 2 πinz ∈ S 2 (Γ 1 ( N )) newform, ■ f ( z ) = � ∞ ■ K f = Q ( . . . , a n , . . . ) , ■ A f = J 1 ( N ) / Ann T ( f ) J 1 ( N ) abelian variety / Q associated to f , ■ g = [ K f : Q ] dimension of A f , ■ G f = { σ : K f ֒ → C } , n =1 σ ( a n ) e 2 πinz for σ ∈ G f , ■ f σ ( z ) = � ∞ σ ∈ G f L ( f σ , s ) Hasse-Weil L -function of A f . ■ L ( A f , s ) = � ■ L ∗ ( A f , 1) leading coefficient of series expansion of L ( A f , s ) in s = 1 . Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 2 / 23
BSD conjecture The conjecture Algorithms Evidence Conjecture (Birch, Swinnerton-Dyer, Tate) We have r := rk( A f ( Q )) = ord s =1 L ( A f , s ) and L ∗ ( A f , 1) = Reg( A f / Q ) · | X ( A f / Q ) | · � v c v . Ω + | A f ( Q ) tors | · | A ∨ f ( Q ) tors | A f ■ Ω + � A f : real period A f ( R ) | η | , η N´ eron differential, ■ Reg( A f / Q ) : N´ eron-Tate regulator, ■ c v : Tamagawa number at v , v finite prime, X ( A f / Q ) : Shafarevich-Tate group. ■ Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 3 / 23
Shimura periods The conjecture Algorithms Evidence Let p > 2 be a prime such that A f has good ordinary reduction at p . We want to find a p -adic analogue of the BSD conjecture. First need to construct a p -adic L -function. f σ ∈ C × such that we have Theorem. (Shimura) For all σ ∈ G f there exists Ω + (i) L ( f σ , 1) ∈ K f , Ω + fσ � � = L ( f σ , 1) L ( f, 1) (ii) σ , Ω + Ω + fσ f (iii) an analogue of (i) for twists of f by Dirichlet characters. Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 4 / 23
Modular symbols The conjecture Algorithms Evidence ■ Fix a Shimura period Ω + f . ■ Fix a prime p of K f such that p | p . ■ Let α be the unit root of x 2 − a p x + p ∈ ( K f ) p [ x ] . ■ For r ∈ Q , the plus modular symbol is �� i ∞ � i ∞ f := − πi � [ r ] + f ( z ) dz + f ( z ) dz ∈ K f . Ω + r − r f ■ Define a measure on Z × p : � a � + � + � f,α ( a + p n Z p ) = 1 1 a µ + − α n p n α n +1 p n − 1 f f Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 5 / 23
Mazur/Swinnerton-Dyer p -adic L -function The conjecture Algorithms Evidence p as ω ( x ) · � x � where ω ( x ) p − 1 = 1 and � x � ∈ 1 + p Z p . ■ Write x ∈ Z × p � x � s − 1 dµ + � ■ Define L p ( f, s ) := f,α ( x ) for all s ∈ Z p . Z × ■ Fix a topological generator γ of 1 + p Z p . ■ Convert L p ( f, s ) into a p -adic power series L p ( f, T ) in terms of T = γ s − 1 − 1 . ■ Let ǫ p ( f ) := (1 − α − 1 ) 2 be the p -adic multiplier. Then we have the following interpolation property (due to Mazur-Tate-Teitelbaum): f = ǫ p ( f ) · L ( f, 1) L p ( f, 0) = L p ( f, 1) = ǫ p ( f ) · [0] + . Ω + f Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 6 / 23
Mazur-Tate-Teitelbaum conjecture The conjecture Algorithms Evidence All of this depends on the choice of Ω + f ! If A f = E is an elliptic curve, a canonical choice is given by Ω + f = Ω + E . Conjecture. (Mazur-Tate-Teitelbaum) If A f = E is an elliptic curve, then we have r := rk( E/ Q ) = ord T =0 ( L p ( f, T )) and L ∗ p ( f, 0) = Reg γ ( E/ Q ) · | X ( E/ Q ) | · � v c v . | E ( Q ) tors | 2 ǫ p ( f ) ■ L ∗ p ( f, 0) : leading coefficient of L p ( f, T ) , ■ Reg γ ( E/ Q ) = Reg p ( E/ Q ) / log( γ ) r , where Reg p ( E/ Q ) is the p -adic regulator (due to Schneider, N´ eron, Mazur-Tate, Coleman–Gross, Nekov´ aˇ r). Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 7 / 23
Extending Mazur-Tate-Teitelbaum The conjecture Algorithms Evidence An extension of the MTT conjecture to arbitrary dimension g > 1 should ■ be equivalent to BSD in rank 0, ■ reduce to MTT if g = 1 , ■ be consistent with the main conjecture of Iwasawa theory for abelian varieties. Problem. Need to construct a p -adic L -function for A f ! σ ∈ G f L p ( f σ , s ) . ■ Idea: Define L p ( A f , s ) := � ■ But to pin down L p ( f σ , s ) , first need to fix a set { Ω + f σ } σ ∈ G f of Shimura periods. Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 8 / 23
p -adic L -function associated to A f The conjecture Algorithms Evidence f σ } σ ∈ G f are Shimura periods, then there exists c ∈ Q × such Theorem. If { Ω + that Ω + � Ω + A f = c · f σ . σ ∈ G f ■ So we can fix Shimura periods { Ω + f σ } σ ∈ G f such that � Ω + Ω + A f = f σ . (1) σ ∈ G f σ ∈ G f L p ( f σ , s ) . ■ With this choice, define L p ( A f , s ) := � ■ Then L p ( A f , s ) does not depend on the choice of Shimura periods, as long as (1) holds. Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 9 / 23
Interpolation The conjecture Algorithms Evidence ■ Convert L p ( A f , s ) into a p -adic power series L p ( A f , T ) in terms of T = γ s − 1 − 1 . σ ǫ p ( f σ ) be the p -adic multiplier. ■ Let ǫ p ( A f ) := � ■ Then we have the following interpolation property L p ( A f , 0) = ǫ p ( A f ) · L ( A f , 1) . Ω + A f Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 10 / 23
More notation The conjecture Algorithms Evidence ■ L ∗ p ( A f , 0) : leading coefficient of L p ( A f , T ) , ■ Reg γ ( A f / Q ) = Reg p ( A f / Q ) / log ( γ ) r , where r = rk( A f ( Q )) and Reg p ( A f / Q ) is the p -adic regulator. ■ If A f is not principally polarized, then Reg p ( A f / Q ) is only defined up to ± 1 . Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 11 / 23
The conjecture The conjecture Algorithms Evidence We make the following p -adic BSD conjecture (with the obvious sign ambiguity if A f is not principally polarized): Conjecture. The Mordell-Weil rank r of A f / Q equals ord T =0 ( L p ( A f , T )) and L ∗ = Reg γ ( A f / Q ) · | X ( A f / Q ) | · � p ( A f , 0) v c v . | A f ( Q ) tors | · | A ∨ ǫ p ( A f ) f ( Q ) tors | This conjecture ■ is equivalent to BSD in rank 0, ■ reduces to MTT if g = 1 , ■ is consistent with the main conjecture of Iwasawa theory for abelian varieties, via work of Perrin-Riou and Schneider. Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 12 / 23
Computing the p -adic L -function The conjecture Algorithms Evidence To test our conjecture in examples, we need an algorithm to compute L p ( A f , T ) . ■ The modular symbols [ r ] + f σ can be computed efficiently in a purely algebraic way – up to a rational factor (Cremona, Stein), ■ To compute L p ( A f , T ) to n digits of accuracy, can use (i) approximation using Riemann sums (similar to Stein-Wuthrich) – exponential in n or (ii) overconvergent modular symbols (due to Pollack-Stevens) – polynomial in n . ■ Both methods are now implemented in Sage . Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 13 / 23
Normalization The conjecture Algorithms Evidence To find the correct normalization of the modular symbols, can use the interpolation property. ■ Find a Dirichlet character ψ associated to a quadratic number field √ Q ( D ) such that D > 0 and ◆ L ( B, 1) � = 0 , where B is A f twisted by ψ , ◆ gcd( pN, D ) = 1 . B · η ψ = D g/ 2 · Ω + ■ We have Ω + A f for some η ψ ∈ Q × . ■ Can express [ r ] + σ [ r ] + ψ in terms of modular symbols [ r ] + B := � f σ . f σ ■ The correct normalization factor is L ( B, 1) η ψ · L ( B, 1) = . D g/ 2 · Ω + Ω + B · [0] + A f · [0] + B B Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 14 / 23
Coleman-Gross height pairing The conjecture Algorithms Evidence Suppose A f = Jac( C ) , where C/ Q is a hyperelliptic curve of genus g . The Coleman-Gross height pairing is a symmetric bilinear pairing h : Div 0 ( C ) × Div 0 ( C ) → Q p , which can be written as a sum of local height pairings � h = h v v over all finite places v of Q and satisfies h ( D, div( g )) = 0 for g ∈ k ( C ) × . Techniques to compute h v depend on v : ■ v � = p : intersection theory (M., Holmes) ■ v = p : logarithms, normalized differentials, Coleman integration (Balakrishnan-Besser) Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 15 / 23
Local heights away from p The conjecture Algorithms Evidence ■ D, E ∈ Div 0 ( C ) with disjoint support, ■ suppose v � = p , ■ X / Spec( Z v ) : regular model of C , ■ ( . ) v : intersection pairing on X , ■ D , E ∈ Div ( X ) : extensions of D, E to X such that ( D . F ) v = ( E . F ) v = 0 for all vertical divisors F ∈ Div ( X ) . ■ Then we have h v ( D, E ) = − ( D . E ) v · log p ( v ) . Steffen M¨ uller (Universit¨ at Hamburg) p -adic BSD for modular abelian varieties – 16 / 23
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