$P$-adic $L$-functions: $t$-modules and Dirichlet-Goss $L$-series

Abstract: We prove that the order of vanishing of a natural twist of the $P$-adic Carlitz zeta values at the positive "even" integers is always $1$. We also obtain the same result for $P$-adic Pellarin $L$-series, and for almost all $P$-adic Dirichlet-Goss $L$-series. To do so, we demonstrate a $P$-adic version of a reduced variant of the class formula for Anderson $t$-modules proven by Anglès, Ngo Dac and Tavares-Ribeiro in 2020.

• The paper establishes a P-adic reduced class formula linking t-modules to the vanishing behavior of corresponding L-series.
• It proves that for Carlitz, Pellarin, and Dirichlet-Goss L-series, the order of vanishing at specified values is generically one.
• Explicit techniques using Anderson-Thakur polynomials and combinatorial identities drive new insights into special value computations of L-functions.

$P$-adic $L$-functions Associated to $t$-modules and Dirichlet-Goss $L$-series

Overview

The paper "P-adic L-functions: t-modules and Dirichlet-Goss L-series" (2606.08085) presents a rigorous study of $P$-adic $L$-functions in the context of Anderson $t$-modules, including Carlitz modules and their generalizations. The main results concern the vanishing properties and orders of vanishing of natural twists of $P$-adic Carlitz zeta values, Pellarin $L$-series, and Dirichlet-Goss $L$-series at positive "even" integers, establishing that the order is generically $L$0. Central to the analysis is a $L$1-adic variant of the reduced class formula for Anderson $L$2-modules, and the application of these structural results to the behavior of $L$3-series in positive characteristic.

Anderson Modules, $L$4-Values, and Class Formulas

The work starts from the analogy between classical number field zeta values and their function field counterparts, focusing on the Carlitz module over $L$5. The Anderson modules generalize the Carlitz and Drinfeld modules, allowing for higher-dimensional and more flexible algebraic structures over finite field function fields. For such an Anderson module $L$6, the associated $L$7-series is defined via a product over primes in $L$8 reflecting the action on various module-theoretic structures:

$L$9

Taelman's class formula provides a deep connection between this analytic object and lattice-theoretic invariants, specifically a regulator and a class module.

The paper employs a reduced version of the class formula, proven by Anglès, Ngo Dac, and Tavares-Ribeiro, which identifies a suitable subspace $t$0 of the Lie algebra where both the Lie and unit lattices intersect, enabling the class formula to be “reduced” in technical complexity:

$t$1

$t$2-adic Class Formula and Analytic Continuation

The key novelty is the $t$3-adic adaptation of this reduced class formula. The work crucially distinguishes between the "infinite" place and $t$4-adic places, introducing $t$5-adic completions $t$6 and extending module, exponential, and logarithm structures accordingly. The $t$7-adic $t$8-series

$t$9

can now be identically zero, in contrast to the classical ($L$0-adic) case. The main theorem (Theorem 11) states that:

$L$1

where the right-hand side is a $L$2-adic ratio of covolumes, and the equality holds up to a unit.

This formula enables the precise control of vanishing: for rank-$L$3 $L$4-finite Anderson modules, $L$5 if and only if $L$6 is not injective on the $L$7-adic Lie algebra; the order of vanishing at $L$8 of twisted series is independent of $L$9.

Orders of Vanishing and Explicit Results

Carlitz Zeta Values

The $P$0-adic Carlitz zeta values $P$1 are studied in detail. When $P$2 divides $P$3, the classical Carlitz zeta value vanishes, and the paper shows that in this case, the order of vanishing at $P$4 of the twisted $P$5-adic zeta function is exactly one (Theorem 17). The main input is the structure of zeros of the “Riemann hypothesis” for function field zeta functions and explicit calculations using Anderson and Thakur's tools.

Moreover, explicit formulas for the values of derivatives at $P$6 are computed using recently developed techniques, including Pellarin's special points and Anderson-Thakur polynomials (Theorem 23). These formulas connect $P$7-adic derivatives of Carlitz zeta values with Carlitz factorials, Bernoulli-Carlitz numbers, and higher logarithms.

Pellarin $P$8-series

For the multivariable Pellarin $P$9-series, the same vanishing and order of vanishing phenomena are established (Theorems 28 and 30). The $L$0-adic Pellarin $L$1-series vanishes if and only if $L$2, and the order of vanishing is always $L$3 when $L$4. The proof leverages reductions to explicit series using combinatorial identities and an analysis of the zero loci of certain Gauss sums.

Dirichlet-Goss $L$5-series

In the context of Dirichlet-Goss $L$6-series, constructed via Dirichlet characters over $L$7, the main result is that for almost all characters of type $L$8, the $L$9-adic $t$0-function $t$1 vanishes iff $t$2, and the order of vanishing is generically one (Theorem 35). The “almost all” caveat stems from technical properties of the twists, but the result is unconditional when $t$3. The paper raises the natural open question of whether the "almost all" restriction can be completely removed.

Theoretical and Practical Implications

The results tightly link the module-theoretic structure underlying Anderson $t$4-modules with the analytic properties of $t$5-adic $t$6-series in positive characteristic. The $t$7-adic class formula provides a clear algebraic criterion for the vanishing of $t$8-series and relates the $t$9-independence of the order of vanishing to deep properties of the exponents of Anderson modules. For function field arithmetic, this bridges explicit computation and the structural theory of motives in positive characteristic.

On the practical side, the explicit formulas for derivatives and vanishing orders push forward the computational edge of positive characteristic $P$0-values. These can inform algorithms for special value computations and may be relevant for ongoing work on transcendence and algebraicity questions for $P$1-values in characteristic $P$2, as well as for analogues of Iwasawa theory.

Future Directions

The formalism developed potentially supports further advances in several lines:

• Extending the vanishing order result to all Dirichlet characters, eliminating the "almost all" qualification.
• Investigating deeper (higher-rank) Anderson modules and associated Galois representations in the $P$3-adic analytic context.
• Studying the behavior of families of $P$4-adic $P$5-functions, e.g., as $P$6 varies, and possible analogues of Iwasawa invariants in positive characteristic.
• Connecting these analytic class formulas to open conjectures regarding special value transcendence and algebraic independence.

Conclusion

This paper establishes a comprehensive framework for the vanishing and derivative structure of $P$7-adic $P$8-functions attached to Anderson $P$9-modules, the Carlitz module, Pellarin $L$0-series, and Dirichlet-Goss $L$1-series in positive characteristic. By proving a $L$2-adic reduced class formula and leveraging explicit computations, it settles several previously conjectured patterns for vanishing orders, clarifies their independence from the $L$3-adic place, and enables more refined explicit and algebraic investigations into special values of $L$4-functions over function fields.