A p-adic Birch and Swinnerton-Dyer conjecture for modular abelian varieties

Transcription

1 A p-adic Birch and Swinnerton-Dyer conjecture for modular abelian varieties Steffen Müller Universität Hamburg joint with Jennifer Balakrishnan (Harvard) and William Stein (U Washington) Forschungsseminar Arithmetische Geometrie Humboldt-Universität zu Berlin Tuesday, November 27, 2012

2 Elliptic Curves Let N 1 be an integer and let J 0 (N) be the Jacobian of the modular curve X 0 (N). Let f(z) = n=1 a ne 2πinz S 2 (Γ 0 (N)) be a newform such that all a n Q. Let Ann T (f) be the annihilator of f in the Hecke algebra T = Z[...,T n,...] generated by the Hecke operators on J 0 (N). Then A f = J 0 (N)/Ann T (f)j 0 (N) is an elliptic curve defined over Q. Wiles et al. have shown: Every elliptic curve A f over Q arises in this way. Consequence: The L-function L(A f,s) = L(f,s) of A f can be continued analytically to C. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 2 / 39

3 Modular abelian varieties f(z) = n=1 a ne 2πinz S 2 (Γ 0 (N)): newform Then K f = Q(...,a n,...) is totally real. A f = J 0 (N)/Ann T (f)j 0 (N): abelian variety /Q associated to f, g = [K f : Q]: dimension of A f, G f = {σ : K f R}, f σ (z) = n=1 σ(a n)e 2πinz for σ G f, L(A f,s) = σ G f L(f σ,s): L-function of A f, can be continued analytically to C, Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 3 / 39

4 Néron differentials and periods on A f Let A denote the Néron model of A f over Spec(Z). A Néron differential on A f is a generator of the global relative differential g-forms on A, pulled back to A f. Example. If A f is an elliptic curve in minimal Weierstraß form y 2 +a 1 xy +a 3 y = x 3 +a 2 x 2 +a 4 x+a 6, then dx 2y+a 1 x+a 3 is a Néron differential. We define the real period (resp. the minus period) of A f by Ω ± A f := ω Af, A f (C) ± where ω Af is a Néron differential and A f (C) ± is the set of elements of A f (C) fixed by ± complex conjugation. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 4 / 39

5 Tamagawa numbers and regulators Let v be a prime number, Let A v be the special fiber of A above v and let A 0 v denote its connected component. Then Φ v = A v /A 0 v is a finite group scheme defined over F v. The Tamagawa number c v (A f ) is the number of F v -rational points on Φ v. Let, NT denote the Néron-Tate (or canonical) height pairing on A f. The regulator Reg(A f /Q) is defined by Reg(A f /Q) := det( P i,p j NT ) i,j, where P 1,...,P r generate the free part of A f (Q). Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 5 / 39

6 BSD conjecture The Shafarevich-Tate group (A f /Q) is defined using Galois cohomology We will assume that it is finite throughout this talk. Let L (A f,1) be the leading term of the series expansion of L(A f,s) in s = 1. Let A f (Q) tors denote the group of rational points on A f of finite order, likewise for the dual abelian variety A f of A f. Conjecture (Birch-Swinnerton-Dyer, Tate) We have rk(a f (Q)) = ord s=1 L(A f,s) and L (A f,1) = Reg(A f/q) (A f /Q) v c v(a f ) A f (Q) tors A f (Q). tors Ω + A f Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 6 / 39

7 p-adic analogues? Let p > 2 be a prime such that A f has good ordinary reduction at p, that is, p a p. Question. Is there a p-adic analogue of the BSD conjecture? Idea. Define a p-adic analytic L-function associated to A f which interpolates L(A f,s) p-adically at special values (e.g. at s = 1). Problem: Need to make L(A f,1) algebraic. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 7 / 39

8 p-adic analogues? Let p > 2 be a prime such that A f has good ordinary reduction at p, that is, p a p. Question. Is there a p-adic analogue of the BSD conjecture? Idea. Define a p-adic analytic L-function associated to A f which interpolates L(A f,s) p-adically at special values (e.g. at s = 1). Problem: Need to make L(A f,1) algebraic. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 7 / 39

9 p-adic analogues? Let p > 2 be a prime such that A f has good ordinary reduction at p, that is, p a p. Question. Is there a p-adic analogue of the BSD conjecture? Idea. Define a p-adic analytic L-function associated to A f which interpolates L(A f,s) p-adically at special values (e.g. at s = 1). Problem: Need to make L(A f,1) algebraic. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 7 / 39

10 p-adic analogues? Let p > 2 be a prime such that A f has good ordinary reduction at p, that is, p a p. Question. Is there a p-adic analogue of the BSD conjecture? Idea. Define a p-adic analytic L-function associated to f which interpolates L(f,s) p-adically at special values (e.g. at s = 1). Problem: Need to make L(f, 1) algebraic. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 7 / 39

11 p-adic analogues? Let p > 2 be a prime such that A f has good ordinary reduction at p, that is, p a p. Question. Is there a p-adic analogue of the BSD conjecture? Idea. Define a p-adic analytic L-function associated to f which interpolates L(f,s) p-adically at special values (e.g. at s = 1). Problem: Need to make L(f, 1) algebraic. In fact, we need to look at L(f σ,1) for all σ G f. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 7 / 39

12 Dirichlet characters If ψ : Z C is a Dirichlet character mod k, we use the following notation: ψ is the conjugate character to ψ. f ψ (z) = n=1 ψ(n) a n e 2πinz, K ψ is the field generated over Q by the values of ψ, τ(ψ) is the Gauß sum of ψ. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 8 / 39

13 Shimura periods Theorem. (Shimura) For all σ G f there exist Ω + f σ R and Ω f σ i R such that the following properties are satisfied: (i) We have πi Ω ± f σ ( i r f σ (z)dz ± i r ) f σ (z)dz K f for all r Q. (ii) If ψ is a Dirichlet character of sign ±, then In particular, L(f ψ,1) τ(ψ) Ω ± f L(f, 1) Ω + f K f K ψ. K f. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 9 / 39

14 Shimura periods cont d Theorem. (Shimura) For all σ G f there exist Ω + f σ R and Ω f σ i R such that the following properties are satisfied: (iii) If ψ is a Dirichlet character of sign ±, then ( ) L(f ψ,1) σ τ(ψ) Ω ± = L(fσ ψσ,1) f τ(ψ σ ) Ω ±. f σ We call a set {Ω ± f σ } σ Gf as in the theorem a set of Shimura periods for f. Shimura periods are not uniquely determined by the theorem. There is always a Dirichlet character ψ such that L(f ψ,1) 0. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 10 / 39

15 Modular symbols Fix a set of Shimura periods {Ω ± f σ } σ Gf. 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]. The plus (resp. minus) modular symbol map associated to f (and α) maps r Q to [r] ± f := πi Ω + f ( i r f(z)dz + i r ) f(z)dz K f. In particular, we have [0] + f = L(f,1). Ω + f Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 11 / 39

16 Mazur-Swinnerton-Dyer p-adic L-function Define measures on Z p : µ ± f (a+pn Z p ) = 1 α n [ ] a ± p n f 1 α n+1 [ ] a ± p n 1 f We can integrate continuous characters χ : Z p C p against µ ± f. Write x Z p as ω(x) x where ω(x) p 1 = 1 and x 1+pZ p. This yields two continuous characters Z p C p. Define L p (f,s) := x s 1 dµ + f (x) for all s Z p, Z p where x s 1 = exp p ((s 1) log p ( x )). Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 12 / 39

17 Interpolation Fix a topological generator γ of 1+pZ 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): L p (f,0) ǫ p (f) = L p(f,1) ǫ p (f) = [0] + f = L(f,1) Ω +. f Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 13 / 39

18 The case of elliptic curves All of this depends on the choice of Ω + f! If A f = E is an elliptic curve, then the real period Ω + E satisfies the assertions of Shimura s theorem, so we can take Ω + f = Ω+ E. This gives a canonical p-adic L-function L p (E,s) associated to E. Let L p (E,T) be the corresponding p-adic power series. By the interpolation property, the classical BSD conjecture in rank 0 is equivalent to L p (E,0) ǫ p (f) = (E/Q) v c v(e) E(Q) tors 2. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 14 / 39

19 Mazur-Tate-Teitelbaum conjecture Conjecture. (Mazur-Tate-Teitelbaum) If A f = E is an elliptic curve such that rk(e/q) = 0, then ord T=0 (L p (f,t)) = 0 and L p(e,0) ǫ p (f) = (E/Q) v c v(e) E(Q) tors 2, where L p(e,0) is the leading coefficient of L p (E,T). Question. How can this be extended to higher rank? Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 15 / 39

20 Mazur-Tate-Teitelbaum conjecture 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(e,0) ǫ p (f) = Reg(E/Q) (E/Q) v c v(e) E(Q) tors 2, where L p(e,0) is the leading coefficient of L p (E,T). Can this be correct? Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 15 / 39

21 Mazur-Tate-Teitelbaum conjecture 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(e,0) ǫ p (f) = Reg(E/Q) (E/Q) v c v(e) E(Q) tors 2, where L p(e,0) is the leading coefficient of L p (E,T). Problem: The left hand side is p-adic, the right hand side is real!. We will modify the right hand side. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 15 / 39

22 Mazur-Tate-Teitelbaum conjecture 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(e,0) = Reg γ(e/q) (E/Q) v c v(e) ǫ p (f) E(Q) tors 2, where L p(e,0) is the leading coefficient of L p (E,T). Here Reg γ (E/Q) = Reg p (E/Q)/log p (γ) r, where Reg p (E/Q) is the p-adic regulator, defined using the p-adic height pairing (more on this later), a p-adic analogue of the real-valued Néron-Tate height pairing. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 15 / 39

23 Extending Mazur-Tate-Teitelbaum An extension of the Mazur-Tate-Teitelbaum conjecture to arbitrary dimension g > 1 should be equivalent to BSD in rank 0, reduce to Mazur-Tate-Teitelbaum 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! Idea: Define L p (A f,s) := σ G f L p (f σ,s) (similar to L(A f,s)). But to pin down L p (f σ,s), first need to fix a set {Ω ± f σ } σ Gf of Shimura periods. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 16 / 39

24 p-adic L-function associated to A f Theorem. (Balakrishnan, Stein, M.) If {Ω ± f } σ σ Gf are Shimura periods, then there exist c Q such that Ω ± A f = c Ω ± f. σ σ G f For the proof, we compare volumes of certain related complex tori. By the theorem, we can fix Shimura periods {Ω ± f σ } σ Gf such that Ω ± A f = σ G f Ω ± f σ. (1) With this choice, define L p (A f,s) := σ G f L p (f σ,s). Then L p (A f,s) does not depend on the choice of Shimura periods, as long as (1) holds. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 17 / 39

25 Interpolation 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. Let ǫ p (A f ) := σ ǫ p(f σ ) be the p-adic multiplier. Then we have the following interpolation property L p (A f,0) ǫ p (A f ) = L(A f,1) Ω + A f. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 18 / 39

26 p-adic heights Let A f be the dual abelian variety of A f. The p-adic height pairing h : A f (Q) A f (Q) Q p is a bilinear pairing with some additional properties (more on this later). Conjecture. (Schneider) The p-adic height pairing is nondegenerate. There are several different constructions of h, due to Néron, Bernardi, Perrin-Riou, Schneider, Mazur-Tate, Nekovář. Can define h for arbitrary abelian varieties over number fields and other types of reduction at p. For p good ordinary, the constructions are all known to be equivalent (due to Mazur-Tate, Nekovář, Besser). If A f is principally polarized, get a pairing h : A f (Q) A f (Q) Q p. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 19 / 39

27 p-adic regulator ϕ : A f A f : polarization. P 1,...,P r : generators of the free part of A f (Q). We define Reg p (A f /Q) := 1 [A f (Q) : ϕ(a f(q))] ) (det(h(p i,ϕ(p j ))) i,j. This is independent of the choice of ϕ. Using this, define Reg γ (A f /Q) := Reg p (A f /Q)/log p (γ) r. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 20 / 39

28 The conjecture We make the following p-adic BSD conjecture: Conjecture. (Balakrishnan, Stein, M.) The Mordell-Weil rank r of A f /Q equals ord T=0 (L p (A f,t)) and L p(a f,0) = Reg γ(a f /Q) (A f /Q) v c v(a f ) ǫ p (A f ) A f (Q) tors A f (Q). tors This conjecture is equivalent to BSD in rank 0, reduces to Mazur-Tate-Teitelbaum if g = 1, is consistent with the main conjecture of Iwasawa theory for abelian varieties, via work of Perrin-Riou and Schneider. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 21 / 39

29 Computing the p-adic L-function 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 Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 22 / 39

30 Normalization To find the correct normalization of the modular symbols, can use the interpolation property σ [0]+ f = L(A,1). σ Ω + A 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(n,d) = 1. Can express [r] + B := σ [r]+ f σ ψ in terms of modular symbols [r] + f σ. We have Ω + B η ψ = D g/2 Ω + A f for some η ψ Q. The correct normalization factor is L(B, 1) Ω + B [0]+ B = η ψ L(B,1) D g/2 Ω + A f [0] +. B Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 23 / 39

31 Computing the p-adic regulator if g = 1 Question. How can we compute p-adic heights? The construction of Mazur-Tate relies on the p-adic σ-function. If g = 1, then this leads to a practical algorithm (Mazur-Stein-Tate), which was heavily optimized in the PhD thesis of Harvey. Problem. It s not clear how to generalize this algorithm to g > 1. Instead, we use a different, but equivalent construction of p-adic heights due to Coleman-Gross. From now on, suppose that A f = Jac(C), where C/Q is a curve of genus g. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 24 / 39

32 Coleman-Gross height pairing The Coleman-Gross height pairing is a symmetric bilinear pairing h : Div 0 (C) Div 0 (C) Q p, where h can be written as a sum of local height pairings h = v h v over all finite places v of Q. We have h(d,div(β)) = 0 for β k(c), so h is well-defined on A f A f. The construction of h v depends on whether v = p or v p. All h v are invariant under changes of models of C Q v. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 25 / 39

33 Local heights away from p Let D,E Div 0 (C) with disjoint support. Suppose v p, X /Spec(Z v ): proper 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). This is completely analogous to the decomposition of the Néron-Tate height on A f in terms of arithmetic intersection theory on X due to Faltings and Hriljac. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 26 / 39

34 Computing local heights away from p Proper regular models can be computed in practice in many cases using Magma. If C is hyperelliptic, divisors on C and extensions to X can be represented using Mumford representation. Intersection multiplicities of divisors on X can be computed algorithmically using linear algebra and Gröbner bases (M.) or resultants (Holmes). All of this is implemented in Magma. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 27 / 39

35 Local heights at p h p (D,E) is defined in terms of Coleman integration on X := C Q p. Suppose X is hyperelliptic, given by a model y 2 = g(x), where deg(g) is odd. Let ω D denote a differential of the third kind on X such that Res(ω D ) = D, ω D is normalized with respect to a certain splitting HdR 1 (X) = H1,0 dr (X) W, where H1,0 dr (X) is the set of holomorphic 1-forms on X. The local height pairing at p is given by the Coleman integral h p (D,E) = ω D. E Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 28 / 39

36 Coleman integration If P, Q X(Q p ) such that P Q (mod p) and ω is a holomorphic 1-form, then it is easy to define and compute Q P ω. Coleman extended this to the rigid analytic space X an C p. We get a well-defined integral Q P ω whenever P,Q X(Q p) and ω is a meromorphic 1-form which is holomorphic in P and Q. Properties of the Coleman-integral include Q P (a 1ω 1 +a 2 ω 2 ) = a 1 Q P ω 1 +a 2 Q P ω 2, R P ω = Q P ω = R Q ω, Q P φ ω = φ(q) φ(p) Q P df = f(q) f(p). ω if φ is a rigid analytic map, Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 29 / 39

37 Computing local heights at p The work of Balakrishnan-Besser makes Coleman integration on hyperelliptic curves practical. Write ω D = η ω, where η is holomorphic. We only discuss the computation of E η. Suppose P Q (mod p), but P and Q are fixed by Frobenius. Then we can compute Q P η using a system of linear equations if we know the action of Frobenius on basis differentials xi dx 2y, i = 0,...,2g 1. The latter can be computed using Kedlaya s algorithm. Using properties of Coleman integrals, can compute Q P η for arbitrary P, Q. This has been implemented by Balakrishnan in Sage. Further computational tricks (due to Balakrishnan-Besser) can be used to compute E ω. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 30 / 39

38 Computing the p-adic regulator Suppose P 1,...,P r A f (Q) are generators of A f (Q) mod torsion. Suppose P i = [D i ], D i Div(C) 0 pairwise relatively prime and with pointwise Q p -rational support. Recall that Reg p (A f /Q) = det((m ij ) i,j ), where m ij = h(d i,d j ). Problem. Given a subgroup H of A f (Q) mod torsion of finite index, need to saturate it. Currently only possible for g = 1,2 (g = 3 work in progress due to Stoll), so in general only get Reg p (A f /Q) up to a Q-rational square. See also recent work of Holmes. For g = 2, can use generators of H and compute the index using Néron-Tate regulators to get Reg p (A f /Q) exactly. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 31 / 39

39 Empirical evidence for g = r = 2 From Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves (Flynn, Leprevost, Schaefer, Stein, Stoll, Wetherell 01), we considered 16 genus 2 curves C N whose Jacobians A N are optimal quotients of J 0 (N). Each A N has Mordell-Weil rank 2 over Q. N Equation of C N 67 y 2 + (x 3 + x + 1)y = x 5 x 73 y 2 + (x 3 + x 2 + 1)y = x 5 2x 3 + x 85 y 2 + (x 3 + x 2 + x)y = x 4 + x 3 + 3x 2 2x y 2 + (x 3 + x 2 + 1)y = 2x 5 + x 4 + x y 2 + (x 3 + x 2 + 1)y = x 5 + x y 2 + (x 3 + x 2 + 1)y = x 4 x 2 x y 2 + (x 3 + x 2 + 1)y = 2x 3 + x 2 + x 125 y 2 + (x 3 + x + 1)y = x 5 + 2x 4 + 2x 3 + x 2 x y 2 + (x 3 + x 2 + 1)y = x 5 + x 4 2x 3 + 2x 2 2x 147 y 2 + (x 3 + x 2 + x)y = x 5 + 2x 4 + x 3 + x y 2 + (x 3 + x + 1)y = x 3 + 4x 2 + 4x y 2 + (x 3 + x 2 + x)y = x 5 + 2x 4 + 3x 3 + x 2 3x 167 y 2 + (x 3 + x + 1)y = x 5 x 3 x y 2 + (x 3 + x 2 + 1)y = x 5 + x 4 + x y 2 = x 5 x 4 + x 3 + x 2 2x y 2 + (x 3 + x + 1)y = x 3 + x 2 + x Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 32 / 39

40 Empirical evidence for g = r = 2, cont d Tamagawa numbers, A N (Q) tors and (A N /Q)[2] were already computed by Flynn et al. To numerically verify p-adic BSD, need to compute p-adic regulators Reg p (A N /Q) and p-adic special values L p(a N,0). We first used Riemann sums for the p-adic special values, leading to very few digits of precision. We recomputed the special values later using overconvergent modular symbols. All regulators were computed to precision at least p 12. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 33 / 39

41 Summary of evidence Theorem. (Balakrishnan, Stein, M.) Assume that for all A N the Shafarevich-Tate group over Q is 2-torsion. Then our conjecture is satisfied up to least 4 digits of precision at all good ordinary primes 5 < p < 100 such that C N Q p has an odd degree model over Q p. Typically, we have at least 6 digits of precision. The assertion (A N /Q) = (A N /Q)[2] is equivalent to classical BSD (Flynn et al.). For all N 167 the differences of Q-rational points on C N generate A N (Q). For N = 167, the divisors we used generate finite index subgroups, depending on p. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 34 / 39

42 N = 188 For example, for N = 188, we have: p-adic regulator Reg p (A N /Q) p-adic L-value p-adic multiplier ǫ p (A N ) O(7 8 ) O(7 4 ) O(7 8 ) O(11 7 ) O(11 8 ) O(11 8 ) O(13 8 ) O(13 8 ) O(13 8 ) O(17 8 ) O(17 8 ) O(17 8 ) O(19 8 ) O(19 8 ) O(19 8 ) O(23 8 ) O(23 8 ) O(23 8 ) O(37 8 ) O(37 8 ) O(37 8 ) O(41 8 ) O(41 8 ) O(41 8 ) O(43 8 ) O(43 8 ) O(43 8 ) O(53 8 ) O(53 6 ) O(53 8 ) O(59 8 ) O(59 6 ) O(59 8 ) O(61 8 ) O(61 7 ) O(61 8 ) O(67 8 ) O(67 6 ) O(67 8 ) O(71 8 ) O(71 6 ) O(71 8 ) O(73 8 ) O(73 5 ) O(73 8 ) O(79 8 ) O(79 5 ) O(79 8 ) O(83 8 ) O(83 5 ) O(83 8 ) O(89 8 ) O(89 5 ) O(89 8 ) O(97 8 ) O(97 4 ) O(97 8 ) Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 35 / 39

43 N = 188 normalization The additional BSD quantities for N = 188 are (A N )[2] = 1, A N (Q) tors 2 = 1, c 2 = 9, c 47 = 1. We find that for the quadratic character ψ associated to Q( 233), the twist B of A N by ψ has rank 0 over Q. Algebraic computation yields [0] + B = 144, η ψ = 1, computed by comparing bases for the integral 1-forms on the curve C N and its twist by ψ. η ψ L(B,1) 233 Ω + A N = 36. So the normalization factor for the modular symbol is 1/4. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 36 / 39

44 Rank 4 evidence The Jacobian A of the twist C of X 0 (31) by the Dirichlet character associated to Q( 47) has rank 4 over Q. We checked our conjecture for p = 29,61,79 to 8 digits of precision under the assumption that (A/Q) is 2-torsion. Since the twist is odd, we had to use the minus modular symbol associated to J 0 (31). For the normalization of the minus modular symbol, we used the twist of X 0 (31) by the Dirichlet character associated to Q( 19), whose Jacobian has rank 0 over Q. For the regulator computations, we needed to work with generators of subgroups of finite index, depending on p. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 37 / 39

45 Supersingular reduction Suppose A f has supersingular reduction at p. For elliptic curves, an analogue of the conjecture of Mazur-Tate-Teitelbaum is due to Bernardi-Perrin-Riou. Computation of p-adic special values works analogously. To extend Coleman-Gross, we would need a canonical splitting of H 1 dr (C Q p). It s not known how to do this! Other constructions of the p-adic height don t seem suitable for computations. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 38 / 39

46 Toric reduction Suppose A f has purely toric reduction at p. If g = 1 and the reduction is nonsplit multiplicative, Mazur-Tate-Teitelbaum is analogous to the good ordinary case. If g = 1 and the reduction is split multiplicative, Mazur-Tate-Teitelbaum becomes more interesting. Computation of p-adic special values works similarly. An extension of Coleman-Gross to this case is work in progress of Besser. Work of Werner provides formulas for the p-adic height pairing if the rigid uniformisation of A f is known. Steffen Müller (Universität Hamburg) p-adic BSD for modular abelian varieties 39 / 39