Multiplicity one for $L$-functions and applications

TL;DR

The work develops strong multiplicity one theorems for analytic L-functions defined by Dirichlet series with Euler products and a functional equation, under weaker Ramanujan-type hypotheses and with averages of prime coefficients replacing pointwise equalities. It systematizes lifts via symmetric/exterior powers and Rankin–Selberg products, building a framework to compare two L-functions through a ratio of completed L-functions and a control of zeros on the critical line. The main contributions include general theorems ensuring L1(s)=L2(s) from near-coincidence of coefficients on primes, plus variants that handle partial Ramanujan bounds and lifts; the approach also clarifies how information from lifts constrains the analytic behavior. When arithmetic sources are known, the results yield stronger multiplicity-one conclusions, especially under Selberg orthogonality-type hypotheses for automorphic L-functions. The paper also demonstrates concrete applications to automorphic GL(n) representations and Siegel paramodular forms, illustrating that close prime-coefficient data forces equality of the underlying arithmetic objects or their Hecke data.

Abstract

We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of multiplicity one type results for the number-theoretic objects attached to $L$-functions. These results follow from our main result, which has slightly weaker hypotheses than previous multiplicity one theorems for $L$-functions. Significantly stronger results are available when the L-function is known to be automorphic.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5