Harder's conjecture for Picard modular forms

Let mm and nn be in the critical range for the algebraic Hecke character ψm\psi_m, and define

Q(m,n)=Λ(ψm,n1)Λ(ψm,n).Q(m,n)=\frac{\Lambda(\psi_m,n-1)}{\Lambda(\psi_m,n)}.

Suppose a prime >m\ell>m divides the denominator of Q(m,n)Q(m,n). Harder's conjecture. There should exist a Picard modular cusp-form Hecke eigenform of weight

(b,a+3)=(mn,2nm+1)(b,a+3)=(m-n,2n-m+1)

whose Hecke eigenvalues λνp\lambda_{\nu_p}, for primes p31p\equiv_3 1, satisfy

λνpνˉpa+b+2+(pa+1+1)νpb+1.\lambda_{\nu_p}\equiv_\ell \bar{\nu}_p^{a+b+2}+(p^{a+1}+1)\nu_p^{b+1}.

This predicts congruences between Picard modular forms and critical-value denominators in the spirit of Harder's conjectures.

Sources & referencesView supporting material

Primary source

Jonas Bergström and Gerard van der Geer, “Picard modular forms and the cohomology of local systems on a Picard modular surface”, arXiv:2012.07673 (2020).

Additional references

6 papers in this index state this conjecture (2006–2020). The statement above is taken from the most recent of them; the others are arXiv:1605.03450, arXiv:1603.07088, arXiv:1409.6090, arXiv:1203.5611, arXiv:math/0605346.

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