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The theme of this work is the study of the Nekov\'a\v{r}-Selmer groupH^1_f(K,T) attached to a twisted Hida family T of Galois representations and aquadratic number field K. The results that we obtain have the following shape:if a twisted L-function of a suitable modular form in the Hida family has orderof vanishing r at most 1 at the central critical point then the rank ofH^1_f(K,T) as a module over a certain local Hida-Hecke algebra is equal to r.Under the above assumption, we also show that infinitely many twistedL-functions of modular forms in the Hida family have the same order ofvanishing at the central critical point. Our theorems extend to more generalarithmetic situations results obtained by Howard when K is an imaginaryquadratic field and all the primes dividing the tame level of the Hida familysplit in K.

Vanishing of special values and central derivatives in Hida families