Computing endomorphism rings and Frobenius matrices of Drinfeld modules

Mihran Papikian; Sumita Garai

Computing endomorphism rings and Frobenius matrices of Drinfeld modules

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Pith's one-line read The paper proves that the endomorphism ring E of a rank-r Drinfeld module over a finite field, when its endomorphism algebra is commutative, is pinned down by the Frobenius index (the invariant factors of E/A[π]), and gives a finite…

desk verdict A genuinely useful paper: the theorem is sound under its explicit assumption, the algorithm is a real improvement, and the only significant weakness is the lack of released code for the computational examples. read the letter →

arxiv 1908.01805 v1 pith:UKBBERIJ submitted 2019-08-05 math.NT

Sumita Garai , Mihran Papikian This is my paper · ORCID

classification math.NT MSC 11G0911R58

keywords DrinfeldmodulesendomorphismringsFrobeniusindexA-ordersGorensteindivisionfieldsreciprocitylawfinite

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is about pinning down the endomorphism ring of a Drinfeld module over a finite field once the ring generated by the Frobenius endomorphism is understood. The authors' claim is that, when the endomorphism algebra is commutative, the difference between the full endomorphism ring $E$ and the Frobenius-generated subring $A[\pi]$ is encoded exactly by a chain of invariant factors $(b_1,\ldots,b_{r-1})$—the Frobenius index—and that each $b_i$ is the largest multiplier for which a degree-$i$ polynomial in $\pi$ can land in $E$. From this they extract an efficient finite-search algorithm that outputs an explicit $A$-basis of $E$ and an explicit matrix for multiplication by $\pi$, and they show this matrix reduces modulo $n$ to the Frobenius conjugacy class in division fields. The refinement matters because endomorphism rings control isogenies, torsion, and Galois representations, and explicit computation in rank at least 3 was previously much less accessible.

What carries the argument

The central object is the Frobenius index of $\varphi$: the tuple $(b_1,\ldots,b_{r-1})$ of invariant factors of the finite $A$-module $E/A[\pi]$, normalized so that $b_1 \mid b_2 \mid \cdots \mid b_{r-1}$. The machinery that carries the argument is the order-basis theorem (Theorem 2.1): if $A[\pi]\subset O$ are $A$-orders in a field $K$ of degree $r$ over $F$, then $O$ has an $A$-basis of the form $1, f_1(\pi)/b_1, \ldots, f_{r-1}(\pi)/b_{r-1}$ with $f_i$ monic of degree $i$ and the $b_i$ exactly the invariant factors. This basis converts the abstract inclusion $E/A[\pi]$ into explicit polynomial witnesses, makes the action of $\pi$ readable as a matrix $F(p)$, and reduces the computation of $E$ to a finite search for the $b_i$ and $f_i$, pruned by the divisibility constraints $b_i b_j \mid b_{i+j}$ and $b_1^{r(r-1)} \mid \mathrm{disc}(A[\pi])$.

What would settle it

Take a rank-3 Drinfeld module satisfying (1.1), run the Section 3 search, and compute the invariant factors of $E/A[\pi]$ independently by Smith normal form from the returned $A$-basis; if they differ from the returned $(b_1,b_2)$, the maximality theorem or the search is wrong. For Example 4.10, test the displayed matrices: if some $v\in F_5^3$ makes $\{v, e_2 v, e_3 v\}$ linearly independent, then $\varphi[l]$ is free over $E_l/lE_l$ and the paper's non-freeness claim fails.

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Extended reading notes

Core claim

Let $A=\mathbb{F}_q[T]$, let $\varphi$ be a rank-$r$ Drinfeld module over a finite field $k$, put $E=\mathrm{End}_k(\varphi)$, and let $\pi$ be the Frobenius endomorphism. Assume $[K:F]=r$, where $K=F(\pi)$, so that $E$ is an $A$-order in the commutative field $K$. The paper's central discovery is Theorem 3.1: if $(b_1,\ldots,b_{r-1})$ are the invariant factors of the finite module $E/A[\pi]$, then for each $i$ there is a monic polynomial $f_i(x)\in A[x]$ of degree $i$ with $f_i(\pi)\in b_i E$, and whenever a monic $g$ of degree $i$ satisfies $g(\pi)\in bE$, one must have $b\mid b_i$. Thus the $b_i$ are exactly the maximal multipliers for which degree-$i$ polynomial relations in $\pi$ survive inside $E$. The proof rests on an order-basis theorem that produces an $A$-basis of $E$ of the shape $1, f_1(\pi)/b_1, \ldots, f_{r-1}(\pi)/b_{r-1}$; this basis makes the action of $\pi$ explicit and yields the Frobenius matrix $F(p)$. For a reduction of a Drinfeld module $\Phi$ at a good prime $p$, the paper shows that $F(p)$ reduced modulo $n$ represents the Frobenius at $p$ in $\mathrm{Gal}(F(\Phi[n])/F)\subset \mathrm{GL}_r(A/nA)$ whenever each Tate module $T_l(\varphi)$ is free over $E_l$, and it gives a rank-3 example where that freeness fails.

Load-bearing premise

Everything rests on the commutativity assumption $[K:F]=r$—without it the endomorphism algebra is not a commutative field and the Frobenius index, the order basis, and the algorithm are undefined; Section 4 additionally depends on the freeness of each Tate module over the localized endomorphism ring, a condition the paper itself shows can fail.

Editorial extensions

If this is right

Computing the endomorphism ring for any module satisfying (1.1) becomes a finite algebraic search: compute the minimal polynomial of $\pi$, then the integral closure of $A$ in $F(\pi)$, then test candidate invariant factors against the divisibility constraints.
The reciprocity law from the authors' earlier paper is refined: all of $b_1,\ldots,b_{r-1}$ get a Galois-theoretic meaning, not just $b_1$, so the full Frobenius index controls how the Frobenius at $p$ acts on division fields.
The matrix $F(p)$ is a universal Frobenius matrix: reducing it modulo $n$ gives the Frobenius conjugacy class in $\mathrm{Gal}(F(\Phi[n])/F)$ whenever the relevant Tate modules are free.
In rank 2 the method recovers the authors' earlier algorithm, and it gives closed formulas for $F(p)$ in terms of $b_1$ and the discriminant, matching the elliptic-curve analogue.
The Gorenstein analysis shows the freeness hypothesis is real: there are rank-3 examples where $E_l$ is not Gorenstein and $T_l(\varphi)$ is not free over $E_l$, so the universal-matrix theorem is conditional rather than formal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the order-basis argument should transfer to any $A$-order in a commutative field generated by a single element, so the same finite-search strategy is a plausible template for other endomorphism-ring computations.
Beyond the paper: Example 4.10 suggests the freeness hypothesis in Theorem 4.5 is not necessary in every case; a sharpened theorem might replace it by a milder condition such as $l \nmid b_1$ or a condition on the index of $E$ in the integral closure.
Beyond the paper: the congruence constraints hidden in $F(p)$ (for instance $a_{10}(a_{21}-a_{10})\equiv a_{20} \pmod{b_1}$) give a practical shortcut for detecting extra endomorphisms before computing the full ring.
Beyond the paper: the algorithm opens the way to statistical studies of Frobenius indices for rank 3 and higher—how often $b_1\neq 1$ or $E$ is non-maximal—which the paper's examples show both occur.

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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core result here is the maximal characterization of all invariant factors b_i of E/A[π], not just b_1 as in the authors' earlier paper. That is real progress, and the proof via the order-basis theorem (Theorem 2.1) is clean. I checked the two delicate spots the stress-test flagged: the descending induction in the maximality half of Theorem 3.1 is justified, and the apparent mismatch between formula (4.2) and the matrix in Example 4.10 is explained by choosing f_1(x)=x. The algorithm in Section 3 is efficient and clearly correct, and the non-Gorenstein example in Section 4 is a genuinely interesting phenomenon, honestly disclosed with a detailed check that the Tate module fails to be free. The paper is also careful to state assumption (1.1) up front and to flag when the Section 4 results depend on freeness of T_l(φ).

The main weakness is reproducibility. The computational claims in Examples 3.3, 4.9, and 4.10—irreducible factorizations, basis verifications, and the non-Gorenstein check—are asserted without releasing the Magma code. A serious referee should ask for that code or for enough data to rerun the examples independently. This does not undermine the written mathematics; the proofs of the main theorems are detailed and internally consistent. The assumption (1.1) limits the scope, but the authors say so, and the rank-2 case and ordinary modules are exactly the cases where it holds. The comparison with Kuhn–Pink in Remark 3.4 is fair and does not overstate the advantage.

The paper is for Drinfeld-module specialists and for people using reciprocity laws in function fields. It deserves a serious referee and, assuming the examples check out, publication. I would recommend accepting for review and requesting the code as supplementary material.

Referee Report

0 major / 7 minor

Summary. The paper studies endomorphism rings E of Drinfeld F_q[T]-modules over finite fields under the standing assumption (1.1), namely that the field K generated by the Frobenius endomorphism π satisfies [K:F]=r. Under this assumption E is an A-order in K and the quotient E/A[π] has invariant factors b_1,...,b_{r-1}, called the Frobenius index. The central result, Theorem 3.1, characterizes these invariant factors by the existence of monic polynomials f_i(x) of degree i with f_i(π)∈b_i E, together with a maximality property saying that any monic g of degree i with g(π)∈bE forces b to divide b_i. This is derived from an order-basis theorem, Theorem 2.1, proved in Section 2. The authors then give a finite-search algorithm for computing the Frobenius index and an explicit A-basis of E, and they report a Magma implementation with rank-2 and rank-3 examples. In Section 4 they show that, under a freeness hypothesis on the Tate module T_l(φ) over E_l, the matrix F(p) obtained from the action of π on E represents the Frobenius conjugacy class in the Galois group of division fields. They also give an example where T_l(φ) is not free over E_l, honestly flagging the failure of the hypothesis.

Significance. If the results hold, this is a substantial contribution to the algorithmic and arithmetic theory of Drinfeld modules. The proof of Theorem 2.1 is self-contained and supplies an explicit A-basis of E; the maximality argument in Theorem 3.1 is sound and correctly justifies the finite search in Section 3. The paper is also explicit about its main restrictions: assumption (1.1) is stated at the outset, and the freeness condition in Theorem 4.5 is isolated and tested with a concrete counterexample. The refinement of the reciprocity law from the authors' earlier work and the explicit Frobenius matrix are valuable, and the discussion of the Gorenstein property for endomorphism rings is a useful addition to the Drinfeld-module literature. The main limitations are the restriction to the commutative case [K:F]=r and the fact that several computational assertions in Examples 3.3, 4.9, and 4.10 are not accompanied by released code or verification data; these limitations do not affect the correctness of the central proofs.

minor comments (7)

[§3, Step 2] The sentence 'If for some possible choice of f1,...,fr−1 we have fi(π)∈biE, then (b1,...,br−1) is the Frobenius index' is only correct because the candidate list is processed in decreasing order of the degrees of the products ∏bi; this ordering should be stated explicitly as a loop invariant or precondition immediately before that sentence.
[§3, proof of Theorem 3.1] The decomposition b=x1y1 and bi=x2y2z with the stated coprimality conditions is quite compressed; an explicit prime-exponent formulation would make the divisibility conclusion b|bi much more transparent for the reader.
[§2 and §3, notation] The symbol q is used both for the cardinality of Fq in Section 1.1 and for a prime element of A in the proof of Theorem 3.1, which is confusing; a different letter such as l or 𝔭 should be used for the prime.
[§3, Step 1] The algorithm mentions the discriminant-based computation of χ(B/A[π]) under the condition that K/F is separable, but it does not state whether separability follows from assumption (1.1); if it does, a brief justification would be helpful, and if not, the role of the separability condition should be clarified.
[§4, equations (4.1)–(4.2)] The entries involving fractions such as bi/bi−1 are only visibly in A because of the chain b1|b2|...|br−1 from the invariant-factor theory; this divisibility should be recalled explicitly when the matrix Fk is introduced.
[Example 4.10] The irreducibility of the degree-5 factor g(x), the F5-linear independence of v1,v2,v3, and the matrices for ē2 and ē3 are asserted without a computation transcript; providing the Magma code or a small verification script would make this interesting counterexample reproducible.
[Example 4.10, final sentence] The observation that the conclusion of Theorem 4.5 holds for n=T+4 even though its hypothesis fails is interesting, but since it concerns a single prime it should be phrased as a remark rather than as evidence that the freeness hypothesis is unnecessary in general.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.1 and the Section 3 algorithm are derived from an independent order-basis theorem, not from the Frobenius index or from self-citations.

full rationale

The central derivation is self-contained. Theorem 3.1 is proved from Theorem 2.1, whose proof is given in Section 2 and is attributed to Marcus [18], together with a degree/divisibility argument that does not presuppose the Frobenius index. The Frobenius index is defined as the invariant factors of E/A[π], and Theorem 2.1 independently supplies the basis 1, f1(π)/b1, ..., fr−1(π)/br−1 with the stated divisibility properties. The maximality direction of Theorem 3.1 is a genuine implication, not a restatement of the definition: it shows that any b with g(π) ∈ bE must divide bi. Step 2 of the algorithm uses this maximality to order candidates, but it does not fit the target Frobenius index into the output; the finite search is justified by the theorem. Section 4's Theorem 4.5 is explicitly conditional on freeness of Tl(φ), and Example 4.10 demonstrates that this hypothesis can fail; the Gorenstein criterion rests on external results [2] and [26]. Self-citations to [10] are used for a prior reciprocity law, for comparing algorithms, and as an example where assumption (1.1) holds; none of these supplies a load-bearing result from which Theorem 3.1 is derived. No equation in the paper reduces, by construction or by fitted parameters, to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters were fitted to data; the inputs are the finite field, the Drinfeld module coefficients, and explicit hypotheses. The main domain assumption is (1.1), and the technical freeness/Gorenstein hypothesis is conditional and explicitly tested in Example 4.10.

assumptions (4)

domain assumption The endomorphism algebra structure for Drinfeld modules over finite fields: [K:F] divides r, D is a central division algebra of dimension (r/[K:F])^2, and under (1.1) D is commutative so E is an A-order in K.
Imposed in Section 1.2; the whole Frobenius index and algorithm depend on it. It holds for k=F_p and ordinary modules, but not for all finite Drinfeld modules.
standard math The classification of A-orders inside a field K uses the theory of finitely generated modules over a PID, including invariant factors and the index/discriminant relations (2.1) and (2.2).
Section 2 uses these structural facts without proof; they are standard algebraic number theory.
domain assumption Yu's theorem [26, Thm. 2] identifies E_l with End_{A_l[G]}(T_l φ), and Bass's results [2] on Gorenstein rings imply the dichotomy used in Theorem 4.7.
These external results are load-bearing for the freeness of the Tate module over E_l, hence for Theorems 4.3 and 4.5.
domain assumption For Theorem 4.5, the Tate module T_l(φ) is assumed free of rank 1 over E_l for every prime l dividing n.
This is an explicit hypothesis in Theorem 4.5; Example 4.10 gives a concrete rank-3 case where T_l(φ) is not free, so the conclusion is not unconditional.

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Cite this review

Pith. "Pith review of Computing endomorphism rings and Frobenius matrices of Drinfeld modules." pith.science (2026). https://pith.science/paper/UKBBERIJ

@misc{pith2026190801805,
  author       = {Pith},
  title        = {Pith review of: Computing endomorphism rings and Frobenius matrices of Drinfeld modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UKBBERIJ}},
  note         = {Machine review of arXiv:1908.01805}
}

abstract

Let $\mathbb{F}_q[T]$ be the polynomial ring over a finite field $\mathbb{F}_q$. We study the endomorphism rings of Drinfeld $\mathbb{F}_q[T]$-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law for division fields of Drinfeld modules proved in our earlier paper. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

On some subrings of Ore polynomials connected with ﬁnite Drinfeld mod ules

Angles, B. On some subrings of Ore polynomials connected with ﬁnite Drinfeld mod ules. J. Algebra 181, 2 (1996), 507–522

work page 1996

[2] [2]

On the ubiquity of Gorenstein rings

Bass, H. On the ubiquity of Gorenstein rings. Math. Z. 82 (1963), 8–28

work page 1963

[3] [3]

Centeleghe, T. G. Integral Tate modules and splitting of primes in torsion ﬁelds of elliptic curves. Int. J. Number Theory 12 , 1 (2016), 237–248

work page 2016

[4] [4]

C., and Papikian, M

Cojocaru, A. C., and Papikian, M. Drinfeld modules, Frobenius endomorphisms, and CM-liftings. Int. Math. Res. Not. IMRN , 17 (2015), 7787–7825

work page 2015

[5] [5]

C., and Shulman, A

Cojocaru, A. C., and Shulman, A. M. The distribution of the ﬁrst elementary divisor of the reduc- tions of a generic Drinfeld module of arbitrary rank. Canad. J. Math. 67 , 6 (2015), 1326–1357

work page 2015

[6] [6]

Drinfeld, V. G. Elliptic modules. Mat. Sb. (N.S.) 94(136) (1974), 594–627, 656

work page 1974

[7] [7]

Drinfeld, V. G. Elliptic modules. II. Mat. Sb. (N.S.) 102(144) , 2 (1977), 182–194, 325

work page 1977

[8] [8]

The splitting of primes in division ﬁelds of elliptic curves

Duke, W., and T ´oth, A. The splitting of primes in division ﬁelds of elliptic curves. Experiment. Math. 11, 4 (2002), 555–565 (2003)

work page 2002