Twisting of Siegel paramodular forms. (English) Zbl 1376.11034
Authors’ abstract: Let \(S_k(\Gamma^{\mathrm{para}}(N))\) be the space of Siegel paramodular forms of level \(N\) and weight \(k\). Fix an odd prime \(p\nmid N\) and let \(\chi\) be a nontrivial quadratic Dirichlet character \(\mod p\). Based on [Int. J. Number Theory 10, No. 4, 1043–1065 (2014; Zbl 1297.11035)], we define a linear twisting map \(\mathcal{T}_\chi:S_k(\Gamma^{\mathrm{para}}(N))\to S_k(\Gamma^{\mathrm{para}}(Np^4))\). We calculate an explicit expression for this twist, give the commutation relations of this map with the Hecke operators and Atkin-Lehner involution for primes \(\ell\neq p\), and prove that the \(L\)-function of the twist has the expected form.
Reviewer: Gabriele Nebe (Aachen)
MSC:
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
| 14G35 | Modular and Shimura varieties |
Citations:
Zbl 1297.11035Software:
ecdataReferences:
| [1] | Asgari, M. and Schmidt, R., Siegel modular forms and representations, Manuscripta Math.104 (2001) 173-200. · Zbl 0987.11037 |
| [2] | Brumer, A. and Kramer, K., Paramodular abelian varieties of odd conductor, Trans. Amer. Math. Soc.366 (2014) 2463-2516. · Zbl 1285.11087 |
| [3] | Cremona, J., Algorithms for Modular Elliptic Curves (Cambridge University Press, Cambridge, England, 1992). · Zbl 0758.14042 |
| [4] | Johnson-Leung, J. and Roberts, B., Siegel modular forms of degree two attached to Hilbert modular forms, J. Number Theory132 (2012) 543-564. · Zbl 1272.11063 |
| [5] | Johnson-Leung, J. and Roberts, B., Twisting of paramodular vectors, Int. J. Number Theory10 (2014) 1043-1065. · Zbl 1297.11035 |
| [6] | J. Johnson-Leung and B. Roberts, Fourier coefficients for twists of Siegel paramodular forms, preprint (2015); arXiv:1505.05463. · Zbl 1425.11080 |
| [7] | Krieg, A. and Raum, M., The functional equation for the twisted spinor \(L\)-series of genus 2, Abh. Math. Semin. Univ. Hamburg83 (2013) 29-52. · Zbl 1297.11042 |
| [8] | Poor, C. and Yuen, D., Paramodular cusp forms, Math. Comp.84 (2015) 1401-1438. · Zbl 1392.11028 |
| [9] | Roberts, B. and Schmidt, R., Local Newforms for GSp(4), , Vol. 1918 (Springer, Berlin, 2007). · Zbl 1126.11027 |
| [10] | Ryan, N. and Tornaría, G., A Böcherer-type conjecture for paramodular forms, Int. J. Number Theory7 (2011) 1395-1411. · Zbl 1233.11051 |
| [11] | Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press, Princeton, New Jersey, 1971). · Zbl 0221.10029 |
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