For an elliptic curve E/Q, we define an extremal prime for E to be a prime p of good reduction such that the trace of Frobenius of E at p is ± 2 √ p , i.e., maximal or minimal in the Hasse interval. Conditional on the Riemann Hypothesis for certain Hecke L-functions, we prove that if End(E) = O K , where K is an imaginary quadratic field of discriminant = −3, −4, then the number of extremal primes ≤ X for E is asymptotic to X 3/4 / log X. We give heuristics for related conjectures.
Abstract. In this article we construct Saito-Kurokawa lifts of mixed level. These are constructed via representation theoretic arguments originally used by Schmidt to construct congruence level and paramodular Saito-Kurokawa lifts.
We define a global linear operator that projects holomorphic modular forms defined on the Siegel upper half space of genus [Formula: see text] to all the rational boundaries of lower degrees. This global operator reduces to Siegel's [Formula: see text] operator when considering only the maximal standard cusps of degree [Formula: see text]. One advantage of this generalization is that it allows us to give a general notion of cusp forms in genus [Formula: see text] and to bridge this new notion with the classical one found in the literature.
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