L-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings

2025/10/11 by Jeanine Van Order, Van Order, Jeanine
Mathematics · #11F27 #11F41 #11F46 #11G05 #11G18 #11G40 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Primary 11F67 #Secondary 11F32

paper · pdf · doi:10.48550/arxiv.2510.10277

openalex publication_date 2025/10/11 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

We derive new integral presentations for central derivative values of L-functions of elliptic curves E/\bfQ twisted by ring class characters of a real quadratic field K in terms of automorphic Green's functions for certain Hirzebruch-Zagier-like arithmetic divisors on the product of modular curves X₀(N) × X₀(N) along real geodesic cycles. We also relate these sums to Birch-Swinnerton-Dyer constants and periods.

Citations

• On the Eisenstein functoriality in cohomology for maximal parabolic subgroups
• Potential automorphy over CM fields
• Arithmetic of Borcherds products
• On the p-part of the Birch-Swinnerton-Dyer formula for multiplicative\n primes
• The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One
• Height pairings on orthogonal Shimura varieties
• On the Rigid Cohomology of Certain Shimura Varieties
• Indivisibility of Heegner points in the multiplicative case
• An Isomorphism between Scalar-Valued Modular Forms and Modular Forms for Weil Representations
• Faltings heights of CM cycles and derivatives of L-functions
• Hilbert modular forms and their applications
• Borcherds Forms and Generalizations of Singular Moduli
• On Two Geometric Theta Lifts
• On two geometric theta lifts
• Integrals of Borcherds forms

Related

• Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg L-functions
• Elliptic Eisenstein series associated to ideals in real quadratic number fields
• L-Functions of Elliptic Curves Modulo Integers
• Modularity of elliptic curves over cyclotomic ℤ_p-extensions of real quadratic fields
• On Twists of A Family of Elliptic Curves and Their L-Function