Constructor for the Carlitz module #40433
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| r""" | ||
| Carlitz module | ||
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| AUTHORS: | ||
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| - Xavier Caruso (2025-07): initial version | ||
| """ | ||
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| # ***************************************************************************** | ||
| # Copyright (C) 2025 Xavier Caruso <[email protected]> | ||
| # | ||
| # This program is free software: you can redistribute it and/or modify | ||
| # it under the terms of the GNU General Public License as published by | ||
| # the Free Software Foundation, either version 2 of the License, or | ||
| # (at your option) any later version. | ||
| # http://www.gnu.org/licenses/ | ||
| # ***************************************************************************** | ||
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| from sage.structure.parent import Parent | ||
| from sage.structure.element import Element | ||
| from sage.categories.finite_fields import FiniteFields | ||
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| from sage.rings.infinity import Infinity | ||
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| from sage.rings.polynomial.polynomial_ring import PolynomialRing_generic | ||
| from sage.rings.function_field.drinfeld_modules.drinfeld_module import DrinfeldModule | ||
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| def CarlitzModule(A, base=None): | ||
| r""" | ||
| Return the Carlitz module over `A`. | ||
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| INPUT: | ||
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| - ``A`` -- a polynomial ring over a finite field | ||
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| - ``base`` -- a field or an element in a field | ||
| (default: the fraction field of ``A``) | ||
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| EXAMPLES:: | ||
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| sage: Fq = GF(7) | ||
| sage: A.<T> = Fq[] | ||
| sage: CarlitzModule(A) | ||
| Drinfeld module defined by T |--> τ + T | ||
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| We can specify a different base. | ||
| This is interesting for instance for having two different variable | ||
| names:: | ||
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| sage: R.<z> = Fq[] | ||
| sage: K = Frac(R) | ||
| sage: CarlitzModule(A, K) | ||
| Drinfeld module defined by T |--> τ + z | ||
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| Using a similar syntax, we can construct the reduction over the | ||
| Carlitz module modulo primes:: | ||
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| sage: F.<a> = Fq.extension(z^2 + 1) | ||
| sage: CarlitzModule(A, F) | ||
| Drinfeld module defined by T |--> τ + a | ||
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| It is also possible to pass in any element in the base field | ||
| (in this case, the result might not be strictly speaking the | ||
| Carlitz module, but it is always a Drinfeld module of rank 1):: | ||
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| sage: CarlitzModule(A, z^2) | ||
| Drinfeld module defined by T |--> τ + z^2 | ||
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| TESTS:: | ||
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| sage: CarlitzModule(Fq) | ||
| Traceback (most recent call last): | ||
| ... | ||
| TypeError: the function ring must be defined over a finite field | ||
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| :: | ||
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| sage: S.<x,y> = QQ[] | ||
| sage: CarlitzModule(A, S) | ||
| Traceback (most recent call last): | ||
| ... | ||
| ValueError: function ring base must coerce into base field | ||
| """ | ||
| if (not isinstance(A, PolynomialRing_generic) | ||
| or A.base_ring() not in FiniteFields()): | ||
| raise TypeError('the function ring must be defined over a finite field') | ||
| if base is None: | ||
| K = A.fraction_field() | ||
| z = K.gen() | ||
| elif isinstance(base, Parent): | ||
| if base.has_coerce_map_from(A): | ||
| z = base(A.gen()) | ||
| else: | ||
| z = base.gen() | ||
| elif isinstance(base, Element): | ||
| z = base | ||
| else: | ||
| raise ValueError("cannot construct a Carlitz module from the given data") | ||
| return DrinfeldModule(A, [z, 1]) | ||
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| def CarlitzExponential(A, prec=+Infinity, name='z'): | ||
| r""" | ||
| Return the Carlitz exponential attached the ring `A`. | ||
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| INPUT: | ||
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| - ``prec`` -- an integer or ``Infinity`` (default: ``Infinity``); | ||
| the precision at which the series is returned; if ``Infinity``, | ||
| a lazy power series in returned, else, a classical power series | ||
| is returned. | ||
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| - ``name`` -- string (default: ``'z'``); the name of the | ||
| generator of the lazy power series ring | ||
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| EXAMPLES:: | ||
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| sage: A.<T> = GF(2)[] | ||
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| When ``prec`` is ``Infinity`` (which is the default), | ||
| the exponential is returned as a lazy power series, meaning | ||
| that any of its coefficients can be computed on demands:: | ||
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| sage: exp = CarlitzExponential(A) | ||
| sage: exp | ||
| z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^8) | ||
| sage: exp[2^4] | ||
| 1/(T^64 + T^56 + T^52 + ... + T^27 + T^23 + T^15) | ||
| sage: exp[2^5] | ||
| 1/(T^160 + T^144 + T^136 + ... + T^55 + T^47 + T^31) | ||
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| On the contrary, when ``prec`` is a finite number, all the | ||
| required coefficients are computed at once:: | ||
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| sage: CarlitzExponential(A, prec=10) | ||
| z + (1/(T^2 + T))*z^2 + (1/(T^8 + T^6 + T^5 + T^3))*z^4 + (1/(T^24 + T^20 + T^18 + T^17 + T^14 + T^13 + T^11 + T^7))*z^8 + O(z^10) | ||
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| We check that the Carlitz exponential is the compositional inverse | ||
| of the Carlitz logarithm:: | ||
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| sage: log = CarlitzLogarithm(A) | ||
| sage: exp(log) | ||
| z + O(z^8) | ||
| sage: log(exp) | ||
| z + O(z^8) | ||
| """ | ||
| C = CarlitzModule(A) | ||
| return C.exponential(prec, name) | ||
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| def CarlitzLogarithm(A, prec=+Infinity, name='z'): | ||
| r""" | ||
| Return the Carlitz exponential attached the ring `A`. | ||
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| INPUT: | ||
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| - ``prec`` -- an integer or ``Infinity`` (default: ``Infinity``); | ||
| the precision at which the series is returned; if ``Infinity``, | ||
| a lazy power series in returned, else, a classical power series | ||
| is returned. | ||
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| - ``name`` -- string (default: ``'z'``); the name of the | ||
| generator of the lazy power series ring | ||
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| EXAMPLES:: | ||
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| sage: A.<T> = GF(2)[] | ||
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| When ``prec`` is ``Infinity`` (which is the default), | ||
| the exponential is returned as a lazy power series, meaning | ||
| that any of its coefficients can be computed on demands:: | ||
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| sage: log = CarlitzLogarithm(A) | ||
| sage: log | ||
| z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^8) | ||
| sage: log[2^4] | ||
| 1/(T^30 + T^29 + T^27 + ... + T^7 + T^5 + T^4) | ||
| sage: log[2^5] | ||
| 1/(T^62 + T^61 + T^59 + ... + T^8 + T^6 + T^5) | ||
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| On the contrary, when ``prec`` is a finite number, all the | ||
| required coefficients are computed at once:: | ||
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| sage: CarlitzLogarithm(A, prec=10) | ||
| z + (1/(T^2 + T))*z^2 + (1/(T^6 + T^5 + T^3 + T^2))*z^4 + (1/(T^14 + T^13 + T^11 + T^10 + T^7 + T^6 + T^4 + T^3))*z^8 + O(z^10) | ||
| """ | ||
| C = CarlitzModule(A) | ||
| return C.logarithm(prec, name) | ||
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