Abstract

Introduction 1 2. Special points on Shimura curves and Gross curves 3 3. Admissible primes and raising the level 6 4. p-adic L-functions and special values formulae 7 5. Selmer groups 12 6. Ramified classes and reciprocity laws 20 7. ε-BSD formulae in the definite case 23 8. ε-BSD formulas in the indefinite case 30 9. Proof of Theorems B and C 33 10. Statements and declarations 36 References 37

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Let E be an elliptic curve over Q, and let and be odd two-dimensional Artin representations for which ⊗ is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms f , g, and h of respective weights two, one, and one, giving rise to E, , and via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain p-adic iterated integrals attached to the triple ( f, g, h), which are p-adic avatars of the leading term of the Hasse-Weil-Artin L-series L(E, ⊗ , s) when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on E-referred to as Stark points-which are defined over the number field cut out by ⊗ . This formula can be viewed as an elliptic curve analogue of Stark's conjecture on units attached to weight-one forms. It is proved when g and h are binary theta series attached to a common imaginary quadratic field in which p splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing p-adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell-Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark-Heegner points attached to Shintani-type cycles on H p × H), and extensions of Q with Galois group a central extension of the dihedral group D 2n or of one of the exceptional subgroups A 4 , S 4 , and A 5 of PGL 2 (C).