Arithmetic of the moduli of semistable elliptic surfaces

We prove a new sharp asymptotic with the lower order term of zeroth order on Z F q ( t ) ( B ) for counting the semistable elliptic curves over F q ( t ) by the bounded height of discriminant Δ ( X ) . The precise count is acquired by considering the moduli of nonsingular semistable elliptic fibrations over P 1 , also known as semistable elliptic surfaces, with 12 n nodal singular fibers and a distinguished section. We establish a bijection of K -points between the moduli functor of semistable elliptic surfaces and the stack of morphisms L 1 , 12 n ≅ Hom n ( P 1 , M ¯ 1 , 1 ) where M ¯ 1 , 1 is the Deligne–Mumford stack of stable elliptic curves and K is any field of characteristic ≠ 2 , 3 . For char ( K ) = 0 , we show that the class of Hom n ( P 1 , P ( a , b ) ) in the Grothendieck ring of K –stacks, where P ( a , b ) is a 1-dimensional ( a , b ) weighted projective stack, is equal to L ( a + b ) n + 1 - L ( a + b ) n - 1 . Consequently, we find that the motive of the moduli L 1 , 12 n is L 10 n + 1 - L 10 n - 1 and the cardinality of the set of weighted F q -points to be # q ( L 1 , 12 n ) = q 10 n + 1 - q 10 n - 1 . In the end, we formulate an analogous heuristic on Z Q ( B ) for counting the semistable elliptic curves over Q by the bounded height of discriminant Δ through the global fields analogy.