Pellarin Values from Cauchy Coordinates on Arbitrary Curves

Source abstract

Let $A$ be the ring of functions regular outside a closed point of degree $\mathrm{N}$ on a smooth projective curve over $\mathbb{F}_q$. We express the normalized weight-one ideal Pellarin value of a rank-one Drinfeld $A$-module as an ordinary Hilbert-class trace of rational Cauchy differentials. A pairing on projective period lattices identifies the differential associated with each ideal class and computes its coefficients by finite torsion traces. For a nonfree lattice, the formula uses an isogeny with arbitrary nonzero derivative to a module with free period lattice. Each ideal-class subseries extends regularly to the characteristic point with value one, so the full series takes the value $|\operatorname{Cl}(A)|$. For every rank-one module, the quotient of the Anderson generating differential by the Drinfeld exponential is a meromorphic lattice sum whose value at zero is the Cauchy differential. Its Taylor coefficients give the higher-weight lattice differentials. For sign-normalized Hayes modules with free period lattice, an ordinary Hilbert-class trace after projection to powers divisible by $q^{\mathrm{N}}-1$ generates ideal values at $n\equiv1\pmod{q^{\mathrm{N}}-1}$. An algebraic parameter rescaling gives a trace formula for character-twisted values at $n\equiv1\pmod{q-1}$. The exact coefficient fields explain the difference between these weight ranges. For sign-normalized modules, decomposition by ideal classes also relates the element-type and ideal-type series at every positive weight. Explicit computations include the Hu--Huang--Yau module with nonprincipal period lattice and an elliptic curve with an infinite point of degree three, including the holomorphic term of its Cauchy differential. At rational infinity, the elliptic formula recovers the Green--Papanikolas identity.