Siegel modular forms of degree two attached to Hilbert modular forms

Abstract

Let E/Q be a real quadratic field and π0 a cuspidal, irreducible, automorphic representation of GL(2, AE) with trivial central character and infinity type (2, 2n + 2) for some non-negative integer n. We show that there exists a non-zero Siegel paramodular newform F : H2 → C with weight, level, Hecke eigenvalues, epsilon factor and L-function determined explicitly by π0. We tabulate these invariants in terms of those of π0 for every prime p of Q.

Theta lifts of Bianchi modular forms and applications to paramodularity

“…(As noted in [23], T (1, p, p, p 2 ) agrees with the classical T (p 2 ) for p N . We also note that our Hecke operators are scaled so as to match the definition in [1, p. 164]. )…”

“…For val p (N ) 1, these definitions are motivated by the results of [31] and the definitions in [23].…”

“…(ii) If v is non-split and finite, then as in [23] [23] (6) (definition extended to the archimedean case). This L-parameter L(Π ∞ ) : W R = C × ∪ C × j → GSp 4 (C) (where j 2 = −1 and jzj −1 = z for z ∈ C × ) can also be described explicitly as follows:…”

“…We also note that one can, in fact, lift any non-Galois-invariant π with cyclotomic central character to an irreducible cuspidal representation of GSp 4 (A Q ) (as discussed in [21,35]). However, the local calculations of [23] and the paramodular newform theory of [31] apply only for trivial central character, which is the reason why we exclude odd weights in the theorem and impose the condition of a trivial central character.…”

“…When K is real quadratic, recent work of Freitas, Le Hung and Siksek in [14] shows that there exists a Hilbert newform f of weight 2 such that L(E, s) = L(f, s). The form f can be lifted to a paramodular Siegel cusp form g of genus 2 and weight 2 thanks to work of Johnson-Leung and Roberts [23]. A direct consequence of the construction of their theta lift is that L(B E , s) = L (g, s).…”