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arXiv · 2510.10277

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

Jeanine Van Order
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Abstract

We derive new integral presentations for central derivative values of $L$-functions of elliptic curves $E/{\bf{Q}}$ 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 $X_0(N) \times X_0(N)$ along real geodesic cycles. We also relate these sums to Birch-Swinnerton-Dyer constants and periods.

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Jeanine Van Order. 2026-06-08. $L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings. https://arxiv.org/abs/2510.10277

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