This has been proved in [ 29 , Theorem 8. Recently, Mok [ 26 ] has also described how to obtain this lift from Arthur's endoscopic classification, based on work of Chan and Gan which relates Arthur's local correspondence with that of Gan—Takeda. In this section, we use our lifting result to prove the following theorem. Theorem 5. We think that this is the first non-trivial such example over an imaginary quadratic field.

There is a significant amount of numerical data going back to [ 8 , 19 ] which supports this conjecture in the case of elliptic curves. The rest of this section is dedicated to proving Theorem 5. But first, we show how to derive Theorem 5. So by Theorem 4. As we mentioned in Remark 5. It was in fact located via explicit computations of Bianchi modular forms. More specifically, we used the extensive data provided in [ 28 ]. Table 1 provides a summary of these data.

The index entries in Table 1 are based on finite sets of Hecke eigenvalues. Since there is no analogue of Sturm's bound for Bianchi modular forms, the last row entries are not proved to be correct. However, we strongly expect them to reflect the truth.

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For the first pair, the conjectured abelian surface attached to the form is principally polarizable see [ 18 , Corollary 2. So we will only focus on the first pair, for which we found the associated abelian surface.

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Our current approach does not allow us to find the remaining surfaces. We elaborate on this in Remark 5. So, assuming Conjecture 1. So finding these surfaces will require working with more general Humbert surfaces for which no explicit models are yet available. For algorithmic purpose, this set is determined explicitly using class field theory see [ 10 , Lemma 5.

We will use Theorem 5. To check iii , we will show that there is a unique such extension. To see this, consider the following diagram:.

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Since we already know that the traces for these particular primes match, we are done. We are also grateful to Abhinav Kumar for kindly providing us with a preliminary version of his joint work with Elkies [ 11 ], and for many helpful email exchanges. Oxford University Press is a department of the University of Oxford.

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Issues About Advertising and Corporate Services. Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents. Background on Bianchi modular forms.

## Local Newforms for GSp(4) : Brooks Roberts :

Background on Siegel modular forms. Theta lifts of Bianchi modular forms. Application to paramodularity. Theta lifts of Bianchi modular forms and applications to paramodularity Tobias Berger. School of Mathematics and Statistics. Sengun sheffield. Oxford Academic. Google Scholar.

Mathematics Institute. Ariel Pacetti. Cite Citation. Abstract We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. We refer the reader to [ 39 , Section 1.

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We now give precise details for each of these steps. By [ 35 , Proposition 5. Table 1.

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Open in new tab. Table 2. The above discussion already proves Theorem 5. Paramodular Vectors. Zeta Integrals. Non-supercuspidal Representations. Hecke Operators. Proofs of the Main Theorems. Back Matter Pages About this book Introduction Local Newforms for GSp 4 describes a theory of new- and oldforms for representations of GSp 4 over a non-archimedean local field. Eigenvalue Newforms Node Representation theory Siegel algebraic number theory oldforms paramodular representations.

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