# P-adic L-function

A **p-adic L-function** is a p-adically valued analytic function that interpolates special values of a classical L-function, such as the [Riemann zeta function](https://www.edgepedia.org/riemann-zeta-function) or the [Dirichlet L-function](https://www.edgepedia.org/dirichlet-l-function) of a character, at integers in an arithmetic progression. The theory originated with Tomio Kubota and Heinrich-Wolfgang Leopoldt in 1964, who constructed p-adic analogs of Dirichlet's L-functions, and [Kenkichi Iwasawa](https://www.edgepedia.org/kenkichi-iwasawa) developed the subject in parallel through his theory of cyclotomic fields.<sup>[1](https://www.degruyterbrill.com/document/doi/10.1515/9781400881703/html)</sup><sup> • </sup><sup>[2](https://www.numdam.org/item/SB_1988-1989__31__33_0.pdf)</sup>

| Key fact | Detail |
|---|---|
| Origin | Introduced by Kubota and Leopoldt in 1964 as p-adic analogs of Dirichlet's L-functions; Iwasawa gave a related construction<sup>[1](https://www.degruyterbrill.com/document/doi/10.1515/9781400881703/html)</sup><sup> • </sup><sup>[2](https://www.numdam.org/item/SB_1988-1989__31__33_0.pdf)</sup> |
| Why interpolation is needed | The Euler product definition of ζ(s) is not applicable in the p-adic setting, and all known approaches involve p-adic interpolation<sup>[2](https://www.numdam.org/item/SB_1988-1989__31__33_0.pdf)</sup> |
| Domain and values | The Kubota-Leopoldt p-adic L-function of a finite-order character χ is a ℂ_p-valued function on ℤ_p<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup> |
| Interpolation property | \( L_{p}(\chi, 1-k) = (1 - \chi\omega^{-k}(p)\,p^{k-1})\,L(\chi\omega^{-k}, 1-k) \) for \( k \ge 1 \), where ω is the Teichmüller character<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup> |
| Three constructions | Measure-theoretic (analytic), Coleman's arithmetic construction via cyclotomic units, and Iwasawa's algebraic construction via Galois modules over the Iwasawa algebra<sup>[4](https://arxiv.org/abs/2309.15692)</sup> |
| Elliptic curves | To an elliptic curve E over ℚ and a prime p, a p-adic L-function can be associated at least when E does not have additive reduction at p<sup>[5](https://doc-gitlab.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/padic_lseries.html)</sup> |
| Trivial zeros | When 1 or \( p^{-1} \) annihilates the p-Euler factor, the p-adic L-function has a zero of multiplicity strictly bigger than that of the complex L-function<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC34502/)</sup> |

## The interpolation problem

The classical Riemann zeta function ζ(s) is defined for Re(s) > 1 by an [Euler product](https://www.edgepedia.org/euler-product) and extends meromorphically to the whole complex plane. Its values at negative integers are rational numbers, trivial in the sense that most of them vanish. The p-adic question asks whether there exists a continuous function of a p-adic variable s in ℤ_p whose values at the odd negative integers are essentially the rational numbers ζ(n).<sup>[2](https://www.numdam.org/item/SB_1988-1989__31__33_0.pdf)</sup> The Euler product itself cannot be used, since it does not converge p-adically, so the construction must proceed by interpolation: one seeks a p-adically continuous function agreeing with prescribed classical values on a [dense set](https://www.edgepedia.org/dense-set) of integers.<sup>[2](https://www.numdam.org/item/SB_1988-1989__31__33_0.pdf)</sup>

The nonvanishing values of ζ at negative integers occur only at odd negative integers, and this parity obstruction is handled by twisting with the Teichmüller character ω, which factors through \( (\mathbb{Z}/p\mathbb{Z})^{\times} \) for odd p.<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup>

## The Kubota-Leopoldt construction

The modern analytic definition treats a p-adic L-function as the [Mellin transform](https://www.edgepedia.org/mellin-transform) of a p-adic measure. A bounded ℂ_p-valued distribution on the inverse system of finite sets \( \mathbb{Z}/mp^{n}\mathbb{Z} \) defines a measure on the profinite space \( \mathbb{Z}_{p,m} \), and integrating continuous functions against such measures yields p-adically continuous functions.<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup> The specific measures used are built from the distributions \( E^{(k)}_{c} \), which arise from power sums and satisfy the distribution relations that make the family coherent as n grows.<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup>

For a finite-order character χ on \( \mathbb{Z}_{p,m}^{\times} \), the Kubota-Leopoldt p-adic L-function is defined as

\[ L_{p}(\chi, s) = -(1 - \chi(c)\langle c \rangle_{p}^{\,1-s})^{-1}\, M_{p}(\chi E_{c}^{(1)})(1-s), \]

where \( M_{p} \) is the p-adic Mellin transform \( M_{p}(\nu)(s) = \int_{\mathbb{Z}_{p,m}^{\times}} \langle x \rangle_{p}^{\,s} x_{p}^{-1}\, d\nu(x) \), and \( \langle x \rangle_{p} \) denotes the projection of a p-adic unit to its principal part \( 1 + q\mathbb{Z}_{p} \), with q = p for odd p and q = 4 for p = 2.<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup> The factor \( (1 - \chi(c)\langle c \rangle_{p}^{\,1-s})^{-1} \) removes the dependence of the definition on the auxiliary integer c, and the resulting function is indeed independent of that choice.<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup>

The key interpolation formula evaluates the function at nonnegative integers. For a primitive [Dirichlet character](https://www.edgepedia.org/dirichlet-character) χ of conductor \( p^{n}m \) and \( k \ge 1 \),

\[ L_{p}(\chi, 1-k) = (1 - \chi\omega^{-k}(p)\,p^{k-1})\, L(\chi\omega^{-k}, 1-k), \]

so the p-adic function recovers the classical L-values of the twisted characters at negative integers, up to the removed p-Euler factor \( 1 - \chi\omega^{-k}(p)p^{k-1} \).<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup> Since the nonnegative integers are dense in ℤ_p and \( L_{p}(\chi, s) \) is continuous, these values determine the function completely.<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup>

## Three constructions

The Kubota-Leopoldt p-adic analog of the Riemann zeta function admits three descriptions of different character.<sup>[4](https://arxiv.org/abs/2309.15692)</sup>

1. **Measure-theoretic (analytic).** One constructs a bounded p-adic measure on \( \mathbb{Z}_{p}^{\times} \) whose Mellin transform interpolates the special values of ζ; this is the approach described above.<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup><sup> • </sup><sup>[4](https://arxiv.org/abs/2309.15692)</sup>
2. **Arithmetic (Coleman).** The same function is obtained from cyclotomic units, connecting the analytic object to explicit units in cyclotomic fields.<sup>[4](https://arxiv.org/abs/2309.15692)</sup>
3. **Algebraic (Iwasawa).** The function arises as a power series over the Iwasawa algebra, built from Galois modules; the power series attached to a measure ν on \( 1 + q\mathbb{Z}_{p} \) satisfies \( f(u^{s} - 1) = \int_{1+q\mathbb{Z}_{p}} x^{s}\, d\nu(x) \) for \( s \in \mathbb{Z}_{p} \), where u is a topological generator such as \( 1 + q \).<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup><sup> • </sup><sup>[4](https://arxiv.org/abs/2309.15692)</sup>

The connection with Iwasawa modules is one reason, following Iwasawa and Mazur, to express the results in terms of p-adic measures on Galois groups rather than as functions of a single variable.<sup>[2](https://www.numdam.org/item/SB_1988-1989__31__33_0.pdf)</sup> Most of the ingredients in the original construction were already known to Kummer in the nineteenth century.<sup>[2](https://www.numdam.org/item/SB_1988-1989__31__33_0.pdf)</sup>

## Iwasawa theory and the main conjecture

P-adic L-functions are the analytic side of [Iwasawa theory](https://www.edgepedia.org/iwasawa-theory), which studies cyclotomic ℤ_p-extensions of number fields through modules over the Iwasawa algebra. The p-adic L-function appears there as a characteristic series, and the main conjecture of Iwasawa theory asserts the equality of this analytic object with an algebraic one, namely a characteristic series of a suitable [Selmer group](https://www.edgepedia.org/selmer-group).<sup>[5](https://doc-gitlab.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/padic_lseries.html)</sup>

For elliptic curves, the p-adic L-function is defined by interpolation of L-values of E at quadratic twists, and through the main conjecture it should equal a characteristic series of a Selmer group attached to E.<sup>[5](https://doc-gitlab.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/padic_lseries.html)</sup> The p-adic version of the [Birch and Swinnerton-Dyer conjecture](https://www.edgepedia.org/birch-and-swinnerton-dyer-conjecture) predicts that the order of vanishing of this function at the trivial character equals the rank of the Mordell-[Weil group](https://www.edgepedia.org/weil-group) \( E(\mathbb{Q}) \), or that rank plus one when the reduction at p is split multiplicative.<sup>[5](https://doc-gitlab.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/padic_lseries.html)</sup>

## Trivial and exceptional zeros

A distinctive feature of p-adic L-functions is that they can vanish where the corresponding complex L-function does not. A trivial zero should appear when 1 or \( p^{-1} \) annihilates the p-Euler factor, meaning the p-adic L-function has a zero of multiplicity strictly bigger than the one of the complex L-function.<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC34502/)</sup> This is visible already in the interpolation formula: the Euler factor \( 1 - \chi\omega^{-k}(p)p^{k-1} \) multiplying the classical value can itself vanish, forcing the p-adic value to zero regardless of the classical value.<sup>[3](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)</sup>

For a motive M attached to an elliptic curve E with multiplicative reduction at p, the Greenberg-Tilouine theorem states, under a mild technical hypothesis, that \( L_{p}(M, s) \) has a simple zero at the trivial zero, with the dominant coefficient governed by a nonzero invariant \( \ell_{p}(E) \) and an explicit complex period \( \Omega_{\infty} \).<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC34502/)</sup> Ralph Greenberg generalized the framework in the ordinary situation, defining the invariant \( \ell_{p}(M) \) in a very general case and conjecturing the behavior of the p-adic L-function at the trivial zero, including the multiplicity of the zero and the dominant coefficient of its expansion there.<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC34502/)</sup>

The related exceptional zero phenomenon concerns the derivative at such zeros. For a modular elliptic curve A over a totally real field F and a totally imaginary quadratic extension E/F, formulas express the derivative of the anticyclotomic p-adic L-function in terms of the Hasse-Weil L-function and p-adic periods of the relevant automorphic forms. Marc Greenberg and Glenn Stevens established such a formula for F = ℚ in 1993; CP Mok proved it for arbitrary totally real fields in 2009; and Matthias Spieß extended it in 2014 under mild assumptions.<sup>[7](https://arxiv.org/abs/1509.08617)</sup>

## Generalizations and computation

The interpolation paradigm extends well beyond Dirichlet characters. For automorphic representations satisfying an ordinarity condition, one modifies the Euler factor at p and constructs p-adic L-functions for families, with the Euler product of the classical L-function converging absolutely for Re(s) sufficiently large and continuing meromorphically with at most simple poles.<sup>[8](https://web.math.ucsb.edu/~zliu/SLF%20IV.pdf)</sup> Constructions exist for Hida families attached to products such as \( \mathrm{GL}_{2} \times \mathrm{Res}_{E/F}\mathrm{GL}_{1} \), via local-global interpolation.<sup>[9](https://arxiv.org/abs/2102.02591)</sup>

The functions are also computable. Sage includes several packages for L-functions and provides a dedicated class for the p-adic L-series of elliptic curves, with implementations based on eclib and on Pollack-Stevens overconvergent modular symbols, and separate classes for ordinary and supersingular situations.<sup>[10](https://doc.sagemath.org/html/en/reference/lfunctions/index.html)</sup><sup> • </sup><sup>[5](https://doc-gitlab.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/padic_lseries.html)</sup>

## References

1. [Kenkichi Iwasawa, *Lectures on P-Adic L-Functions*, Princeton University Press](https://www.degruyterbrill.com/document/doi/10.1515/9781400881703/html)
2. [Pierre Colmez? no, Séminaire Bourbaki, "On p-adic L-functions", Séminaire Bourbaki 1988-1989, exposé 31](https://www.numdam.org/item/SB_1988-1989__31__33_0.pdf)
3. [Romyar Sharifi, *Iwasawa Theory*, Chapter 5: Kubota-Leopoldt p-adic L-functions, UCLA lecture notes](https://math.ucla.edu/~sharifi/notes/iwasawa-ch05.html)
4. [David Loeffler and Sarah Zerbes, "An introduction to p-adic L-functions", arXiv:2309.15692](https://arxiv.org/abs/2309.15692)
5. [Sage 9.4 Reference Manual: p-adic L-functions of elliptic curves](https://doc-gitlab.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/padic_lseries.html)
6. [John Coates and Susan Howson? no, "p-adic L functions and trivial zeroes", PNAS Colloquium Paper](https://pmc.ncbi.nlm.nih.gov/articles/PMC34502/)
7. ["Anticyclotomic p-adic l-functions and the exceptional zero phenomenon", arXiv:1509.08617](https://arxiv.org/abs/1509.08617)
8. [Zheng Liu, "The p-adic L-function for ordinary families and its interpolation properties", UCSB lecture notes](https://web.math.ucsb.edu/~zliu/SLF%20IV.pdf)
9. ["p-adic L-functions via local-global interpolation: the case of GL₂ × GU(1)", arXiv:2102.02591](https://arxiv.org/abs/2102.02591)
10. [Sage Reference Manual: L-Functions](https://doc.sagemath.org/html/en/reference/lfunctions/index.html)


2. ["On p-adic L-functions", Séminaire Bourbaki 1988-1989, exposé 731, Numdam](https://www.numdam.org/item/SB_1988-1989__31__33_0.pdf)
6. ["p-adic L functions and trivial zeroes", PNAS Colloquium Paper](https://pmc.ncbi.nlm.nih.gov/articles/PMC34502/)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › p-adic numbers › p-adic methods in number theory*

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