In order to apply the ideas of Iwasawa theory to the symmetric square of a newform, we need to be able to define non‐archimedean analogues of its complex L‐series. The interpolated p‐adic L‐function is closely connected via a “Main Conjecture” with certain Selmer groups over the cyclotomic Zp‐extension of Q. In the p‐ordinary case these functions are well understood. In this article we extend the interpolation to an arbitrary set S of good primes (not necessarily satisfying ordinarity conditions). The corresponding S‐adic functions can be characterised in terms of certain admissibility criteria. We also allow interpolation at particular primes dividing the level of the newform. One interesting application is to the symmetric square of a modular elliptic curve E defined over Q. Our constructions yield p‐adic L‐functions at all primes of stable or semi‐stable reduction. If p is ordinary or multiplicative the corresponding analytic function is bounded; if p is supersingular our function behaves like log2(1 + T). 1991 Mathematics Subject Classification: 11F67, 11F66, 11F33, 11F30

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We present the results of our search for the orders of Tate-Shafarevich groups for the quadratic twists of elliptic curves. We formulate a general conjecture, giving for a fixed elliptic curve E over Q and positive integer k, an asymptotic formula for the number of quadratic twists E d , d positive square-free integers less than X, with finite group E d (Q) and |X(E d (Q))| = k 2 . This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of the curve X 0 (49). In section 8 we exhibit 88 examples of rank zero elliptic curves with |X(E)| > 63408 2 , which was the largest previously known value for any explicit curve. Our record is an elliptic curve E with |X(E)| = 1029212 2 .Birch and Swinnerton-Dyer conjecture relates the arithmetic data of E to the behaviour of L(E, s) at s = 1.Conjecture 1 (Birch and Swinnerton-Dyer) (i) L-function L(E, s) has a zero of order r = rank E(Q) at s = 1,If X(E) is finite, the work of Cassels and Tate shows that its order must be a square.The first general result in the direction of this conjecture was proven for elliptic curves E with complex multiplication by Coates and Wiles in 1976 [6], who showed that if L(E, 1) = 0, then the group E(Q) is finite. Gross and Zagier [18] showed that if L(E, s) has a first-order zero at s = 1, then E has a rational point of infinite order. Rubin [26] proves that if E has complex multiplication and L(E, 1) = 0, then X(E) is finite. Let g E be the rank of E(Q) and let r E the order of the zero of L(E, s) at s = 1. Then Kolyvagin [20] proved that, if r E ≤ 1, then r E = g E and X(E) is finite. Very recently, Bhargava, Skinner and Zhang [1] proved that at least 66.48% of all elliptic curves over Q, when ordered by height, satisfy the weak form of the Birch and Swinnerton-Dyer conjecture, and have finite Tate-Shafarevich group.When E has complex multiplication by the ring of integers of an imaginary quadratic field K and L(E, 1) is non-zero, the p-part of the Birch and Swinnerton-Dyer conjecture has been established by Rubin [27] for all primes p which do not divide the order of the group of roots of unity of K. Coates et al. [5] [4], and Gonzalez-Avilés [17] showed that there is a large class of explicit quadratic twists of X 0 (49) whose complex L-series does not vanish at s = 1, and for which the full Birch and Swinnerton-Dyer conjecture is valid (covering the case p = 2 when K = Q( √ −7)). The deep results by Skinner-Urban ([29], Theorem 2) (see also Theorem 7 in section 8.4 below) allow, in specific cases (still assuming L(E, 1) is non-zero), to establish p-part of the Birch and Swinnerton-Dyer conjecture for elliptic curves without complex multiplication for all odd primes p (see examples in section 8.4 below, and section 3 in [10]). The numerical studies and conjectures by Conrey-Keating-Rubinstein-Snaith [7], Delaunay [12][13], Watkins [31], Radziwi l l-Soundararajan [25] (see also the papers [11][10] [9], and references therein) substantially extend the systematic tables given by...

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