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L and epsilon functions for GSp(4) x GL(2).
I I Piatetski-Shapiro1, D Soudry
1Tel-Aviv University, Tel-Aviv, Israel.
Summary
This study investigates L functions for automorphic representations of GSp(4, A) x GL(2, A). We prove that nontrivial poles of these L functions are linked to representations of GSp(4, A) being a lifting from split O(4).
Area of Science:
- Number Theory
- Representation Theory
- Automorphic Forms
Background:
- Novodvorsky constructed L functions for automorphic cuspidal representations of GSp(4, A) x GL(2, A) with Whittaker models.
- The properties of these L functions, particularly their poles, are crucial for understanding the underlying number-theoretic objects.
Purpose of the Study:
- To determine the conditions under which the L function for GSp(4, A) x GL(2, A) has nontrivial poles.
- To introduce a new construction of L functions for GSp(4) x GL(2) applicable to a broader class of representations, including those without Whittaker models.
Main Methods:
- Proving a characterization for nontrivial poles of Novodvorsky's L function.
- Developing a new construction of L functions via lifting automorphic forms to Sp(4), thereby ensuring the existence of a Whittaker model.
Main Results:
- The L function has nontrivial poles if and only if the GSp(4, A) representation is a lifting from split O(4).
- A new method for constructing L functions for GSp(4) x GL(2) is established, applicable to representations like holomorphic modular forms.
Conclusions:
- The condition for nontrivial poles provides a deep connection between L functions and the theory of liftings.
- The new construction extends the study of L functions to representations previously not covered, utilizing integral expressions derived from Whittaker models of lifted forms.