[Paper Review] On the behaviour of root numbers in families of elliptic curves

H. A. Helfgott|arXiv (Cornell University)|Aug 10, 2004

Algebraic Geometry and Number Theory23 references32 citations

TL;DR

This paper investigates the distribution of root numbers in one-parameter families of elliptic curves over ℚ, proving that the average root number is zero for families with at least one place of multiplicative reduction over ℚ(T), assuming two classical arithmetic conjectures. For families without such reduction, it derives an explicit infinite product formula for the average root number, showing non-random, regular behavior governed by local root numbers and arithmetic properties of ℤ.

ABSTRACT

Let E be a one-parameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any family E with at least one point of multiplicative reduction over Q(t) has average root number 0, provided that two classical arithmetical conjectures hold for two polynomials constructed explicitly in terms of E. The behaviour of the root number in any family E without multiplicative reduction over Q(t) is shown to be rather regular and non-random; we give expressions for the average root number in this case.

Motivation & Objective

• To understand the statistical behavior of root numbers W(E) across one-parameter families of elliptic curves over ℚ.
• To determine under what conditions the average root number is zero or non-zero, depending on the presence of multiplicative reduction over ℚ(T).
• To derive explicit formulas for the average root number in families without multiplicative reduction, expressed as infinite products over local root numbers.
• To establish connections between the average root number and deep arithmetic conjectures, such as those on Möbius and Liouville functions over arithmetic progressions.
• To provide unconditional results for specific families and conditional results relying on standard conjectures in number theory.

Proposed method

• Uses the global root number formula W(E) = -∏ₚ Wₚ(E), decomposing the global sign into local contributions at each prime p.
• Applies reciprocity laws and Hilbert symbols to express local root numbers Wₚ(E) in terms of Legendre symbols and quadratic residues modulo p.
• Employs the theory of local constancy of root numbers in p-adic neighborhoods to reduce global averages to p-adic integrals and infinite products.
• Applies Proposition 7.12 to show equidistribution of the root number sign under conditions involving the Möbius and Liouville functions over lattice cosets in ℤ².
• Uses explicit parametrizations of elliptic curves (e.g., y² = x(x+a)(x+b)) to analyze reduction types and compute local root numbers via modular arithmetic.
• Relies on conjectures such as the Möbius disjointness conjecture and the parity problem for degree 3 L-functions to establish unconditional results in special cases.

Experimental results

Research questions

• RQ1 Under what conditions is the average root number of a one-parameter family of elliptic curves over ℚ equal to zero?
• RQ2 How does the absence of multiplicative reduction over ℚ(T) affect the regularity and structure of the average root number?
• RQ3 To what extent can the average root number be expressed as an infinite product over local root numbers?
• RQ4 What is the connection between the average root number and arithmetic functions like μ and λ over arithmetic progressions?
• RQ5 Can unconditional results be obtained for specific families, even when general results depend on unproven conjectures?

Key findings

• The average root number is zero for any one-parameter family of elliptic curves over ℚ(T) that has at least one place of multiplicative reduction, provided two standard conjectures hold.
• For families without multiplicative reduction over ℚ(T), the average root number is given by an infinite product involving local root numbers and arithmetic functions, and is generally non-zero.
• The average root number over ℚ is shown to be -0.15294… for the family y² = x³ - 1/48 f₁(t)f₂(t)(f₁³ - f₂³)²x - 1/864 (f₁³ + f₂³)(f₁³ - f₂³)³ with f₁(t) = -5 - 2t², f₂(t) = 2 + 5t², and this result is unconditional.
• The average root number is zero for families of the form y² = x(x+a)(x+b) with a,b ∈ ℤ, as shown via the Möbius function and Liouville function equidistribution over lattice cosets.
• The result that the average root number is zero for families with multiplicative reduction is conditional on the truth of two classical conjectures, but becomes unconditional for specific families where these conjectures are known to hold.
• The paper establishes that the root number distribution is not random in the absence of multiplicative reduction, but instead follows a predictable, arithmetic-dependent pattern encoded in infinite products of local factors.