On the class number of pairs of binary quadratic forms

Biró, András (2025) On the class number of pairs of binary quadratic forms. JOURNAL DE THEORIE DES NOMBRES DE BORDEAUX, 37 (3). pp. 897-924. ISSN 1246-7405

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Abstract

If d 1 , d 2 , t ∈ ℤ , let h ( d 1 , d 2 , t ) be the number of SL 2 ( ℤ ) -equivalence classes of pairs ( Q 1 , Q 2 ) of quadratic forms with integer coefficients satisfying that the discriminant of Q i is d i , and the codiscriminant of Q 1 and Q 2 is t . We give an explicit formula for h ( d 1 , d 2 , t ) assuming that d i is not a square of an integer ( i = 1 , 2 ) , and t 2 - d 1 d 2 ≠ 0 . Previously such formulas were known only under some coprimality conditions.

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