Consani–Scholten quintic

In the mathematical fields of algebraic geometry and arithmetic geometry, the Consani–Scholten quintic is an algebraic hypersurface (the set of solutions to a single polynomial equation in multiple variables) studied in 2001 by Caterina Consani and Jasper Scholten.

Its Hodge diamond is[1][2][3] The Consani–Scholton quintic itself is the non-singular hypersurface obtained by blowing up these singularities.

of good reduction, which for this curve means any prime other than 2, 3, or 5) should have the same L-series as an automorphic form.

This was known for "rigid" Calabi–Yau threefolds, for which the family of Galois representations has dimension two, by the proof of Serre's modularity conjecture.

Consani and Scholten constructed a Hilbert modular form and conjectured that its L-series agreed with the Galois representations for their curve; this was proven by Dieulefait, Pacetti & Schütt (2012).