Hypergeometric distributions and joint families of elliptic curves

Abstract Recently, the first author as well as the second author with Ono, Pujahari, and Saikia determined the limiting distribution of values of certain finite field 2 $$F_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and 3 $$F_{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> hypergeometric functions. These hypergeometric values are related to Frobenius traces of elliptic curves, and their limiting distribution is determined using connections to the theory of modular forms and harmonic Maass forms. Here, we determine the limiting distribution of values of some 4 $$F_{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> hypergeometric functions which are sums of traces of Frobenius for a pair of elliptic curves. To obtain this result, we generalize Michel’s work on Sato–Tate laws for families of elliptic curves to the setting of pairs of families, and we show that a generic pair admits an independent Sato–Tate distribution as the finite field grows. To this end, we use various results from the theory of étale cohomology, Deligne’s work on the Weil conjectures, and the work of Katz on monodromy groups. In the cases previously studied using modular methods, we elucidate the connection between the modular forms that appear and the machinery of étale cohomology.