Abstract

Let f be a primitive form of weight k > 1 with nebentypus ε and χ be a primitive Dirichlet character. Then, we consider the twist of f by χ and its Dirichlet L-series denoted by L(f, s, χ). Those central L-values (or even vanishings and nonvanishings of them) are believed to encode important arithmetic invariants of algebraic objects over various fields.

In this thesis, we mainly study vanishings and nonvanishings of the central Lvalues of primitive form f of weight k ≥ 2 twisted by χ of a prime order l. More precisely, firstly, assume that k > 2, 2 | k and l > 2. Then, as a generalisation of the nonvanishing theorem for k = 2 of Fearnley, Kisilevsky and Kuwata, for the case that L(f, k/2) 6= 0 we prove that for all but finitely many primes l there exist infinitely
many of twists of order l such that L(f, k/2, χ) 6= 0. We also present numerical results on vanishings of twists of l = 3, 5, and 7 for some primitive forms of k > 2, and based on the random matrix theory, make a conjecture that for a primitive form of even weight k > 2, there exist only a finite number of vanishings of twists of order l > 2.

Secondly, assume that k = 2 and l = 3. Then, inspired by the work of Fiorilli for quadratic twists, we estimate the average analytic rank of twists for the group family under some hypotheses including the generalised Riemann hypothesis, which implies that there exist infinitely many cubic twists such that L(f, 1, χ) 6= 0.

Lastly, consider quadratic twists of an elliptic curve E over Q of conductor N associated with quadratic characters χd of fundamental discriminants d with a prime |d|. Then, by controlling 2-Selmer groups of E and its quadratic twist by χd using the method of Mazur and Rubin, we show that for some E satisfying some conditions there exist a set of residue classes |d| mod N such that L(E, 1, χd) 6= 0 under the Birch and Swinnerton-Dyer conjecture.