Central critical values of modular L -functions and coefficients of half-integral weight modular forms modulo ℓ

Abstract

Abstract. If F (z) is a newform of weight 2λ and D is a fundamental discriminant, then let L(F ⊗ χ D , s) be the usual twisted L-series. We study the algebraic parts of the central critical values of these twisted L-series modulo primes . We show that if there are two D (subject to some local conditions) for which the algebraic part of L(F ⊗ χ D , λ) is not 0 (mod ), then there are infinitely many such D. These results depend on precise nonvanishing results for the Fourier coefficients of half-integral weight modular forms modulo , which are of independent interest.

Mock modular forms as $p$-adic modular forms

“…(1) It is worth remarking that the proof of Theorem 1.3 shows more generally that there exists exactly one pair (α, δ) ∈ A M × Q p such that F α,δ is a p-adic modular form whenever p ∤ N and a g (p) = 0, but the properties of the CM form allow us to relate δ and α in this case. (2) It is an interesting question whether α = 0 ever occurs in Theorem 1.3. In a particular example considered in [15], it was shown that this does not happen for every prime p < 32500.…”

p-Adic limit of the Fourier coefficients of weakly holomorphic modular forms of half integral weight

“…Remark 4.7: Proposition 4.6 was proved for p ≥ 5 in [1]. One can check that this holds also for p = 3.…”

Critical L-values of Level p Newforms (mod p)

“…If f (z) is an eigenform for the Hecke operators T p 2 , then famous theorems of Kohnen [20], Kohnen and Zagier [17], and Waldspurger [42] relate the numbers a(|D|) 2 and L(F ⊗ χ D , k) for fundamental discriminants D (see below for precise statements). Using these results, a number of recent papers have studied the Fourier coefficients of such forms in connection with values of modular L-functions ( [8], [32], [31], [3]), ranks and Tate-Shafarevich groups of elliptic curves ( [9], [22]), ternary quadratic forms ( [33], [16]), and combinatorial generating functions ( [7], [1], [2]).…”