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Let E be a rational elliptic curve and let p be an odd prime of additive reduction. Let K be an imaginary quadratic field and fix a positive integer c prime to the conductor of E. The main goal of the present article is to define an anticyclotomic p-adic L-function L attached to E / K when E / Q p attains semistable reduction over an abelian extension. We prove that L satisfies the expected interpolation properties; namely, we show that if χ is an anticyclotomic character of conductor c p n , then χ ( L ) is equal (up to explicit constants) to L ( E , χ , 1 ) or L ′ ( E , χ , 1 ) .

Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes