On the integral $2$-adic Tate module of elliptic curves

Abstract: We show that the integral $2$-adic Tate module of an elliptic curve over a complete discretely valued field of odd residue characteristic is determined by its $2$-torsion representation, together with the square class of $c$ and local information about the pairwise differences of the roots of $f$ in a model $E\colon y<sup>2=cf(x)$, where $f$ is monic of degree $3$. The proof uses explicit halving formulae to determine the Galois action on the full tower of $2$-power torsion.

• The paper proves that, over a complete discretely valued field of odd residue characteristic, the integral 2-adic Tate module is determined by the leading coefficient’s square class, Galois action on 2-torsion, and root differences modulo 1+𝔪.
• Its explicit halving-tuple construction lifts compatible Galois-equivariant isomorphisms through every 2-power torsion layer, showing that the curves have identical 2-power torsion fields and actions.
• The paper derives an effective local-constancy criterion—if f₁ ≡ f₂ mod πᴺ with N greater than the maximum root-separation and derivative valuations—while also implying matching 2-primary Néron component groups, including under wild reduction.

Overview

This paper, "On the integral $2$-adic Tate module of elliptic curves" (2608.17725), establishes that for an elliptic curve over a complete discretely valued field $K$ of odd residue characteristic $p$, the integral $2$-adic Tate module $T_2E$ is determined by data visible in a cubic Weierstrass model: the Galois action on the $2$-torsion, the square class of the leading coefficient $c$ in a model $E\colon y^2=cf(x)$ with $f$ monic of degree $3$, and the pairwise differences of the roots of $K$0 up to elements of $K$1. The method is entirely explicit: it uses halving formulae (due to Bekker–Zarhin) to lift a bijection on $K$2 to compatible $K$3-equivariant isomorphisms on every layer $K$4, and then passes to the inverse limit.

The result strengthens an existing rational statement. Dokchitser–Dokchitser–Maistret–Morgan prove, for hyperelliptic curves $K$5 satisfying analogous hypotheses, that $K$6 as $K$7-modules for all $K$8. The present theorem upgrades this to an integral isomorphism at $K$9 in genus $p$0, and consequently implies equality of the $p$1-primary parts of the component groups of the Néron models — including under wild reduction, via SGA 7 Exposé IX §11.

The main theorem

Let $p$2 be complete discretely valued with valuation ring $p$3, uniformiser $p$4, and odd residue characteristic. Write $p$5 for the extended valuation and $p$6. For non-zero $p$7, the condition $p$8 says that $p$9 and $2$0 share valuation and first non-zero residue digit.

Theorem (main). Let $2$1 with $2$2 and $2$3 monic cubics. Suppose:

1. $2$4;
2. there is a $2$5-equivariant bijection $2$6 between the root sets such that $2$7 for all distinct roots $2$8.

Then $2$9 as $T_2E$0-modules.

Two consequences are immediate from the proof rather than stated separately: the curves have identical $T_2E$1-power torsion fields and identical Galois actions on $T_2E$2-power torsion; and their component groups have isomorphic $T_2E$3-primary parts, since the integral representation determines these by SGA 7.

The square-class hypothesis is handled by twisting: after replacing $T_2E$4 by a $T_2E$5-isomorphic equation one may take $T_2E$6, and $T_2E$7 where $T_2E$8 is the quadratic character of $T_2E$9. Thus only the case $2$0 requires work.

A variant over different fields

A second theorem compares two complete discretely valued fields $2$1 of the same odd residue characteristic $2$2 with identified residue fields. After fixing compatible choices of $2$3-th roots of the uniformisers for $2$4 coprime to $2$5, one obtains a canonical isomorphism $2$6 between tame Galois groups. Using a refined equivalence relation $2$7 (equality of valuations and reductions after normalising by powers of $2$8), together with a square-class condition on leading coefficients detected by quadratic characters corresponding under $2$9, the paper proves

$c$0

as $c$1-modules, whenever the root sets lie in the maximal tamely ramified extensions and are matched $c$2-equivariantly up to $c$3. The tameness of the action follows because wild inertia is pro-$c$4 while its image on $c$5 lies in the pro-$c$6 kernel of $c$7. The assumptions $c$8 and $c$9 are automatic unless $E\colon y^2=cf(x)$0; this is the one place where the residue characteristic enters beyond being odd.

Explicit local constancy

As an application, the main theorem yields an explicit local constancy statement. If $E\colon y^2=cf(x)$1 and $E\colon y^2=cf(x)$2 with

$E\colon y^2=cf(x)$3

then $E\colon y^2=cf(x)$4. The proof constructs the required root bijection via Newton polygons: for each root $E\colon y^2=cf(x)$5 of $E\colon y^2=cf(x)$6, the polynomial $E\colon y^2=cf(x)$7 has a unique root $E\colon y^2=cf(x)$8 with $E\colon y^2=cf(x)$9 for $f$0, and the coefficient estimates force the pairwise-difference condition of the main theorem.

Kisin's local constancy theorem already gives a non-explicit version: for each fixed $f$1 and $f$2 there exists some $f$3 with the analogous property. The corollary supplies an explicit bound when $f$4, which is useful in settings where one needs effective congruence thresholds.

Method: lifting through the halving tower

The technical core is an induction constructing maps $f$5 that restrict to $f$6, commute with multiplication by $f$7, preserve coordinates up to $f$8 (i.e. $f$9 and $3$0), and are $3$1-equivariant group isomorphisms.

Three ingredients drive the induction:

• Halving triples (Bekker–Zarhin). Half-points $3$2 of $3$3 correspond to triples $3$4 with $3$5 and $3$6, with explicit coordinate formulae. Negation flips all signs; translation by $3$7 flips the sign of $3$8 alone.
• An addition formula for halving triples, derived directly from chord-and-tangent formulae and not previously recorded: if $3$9 are half-points of $K$00 with triples $K$01, then $K$02 has triple $K$03 with $K$04.
• Uniqueness and stability lemmas. Two non-zero $K$05-torsion points ($K$06) whose coordinates agree up to $K$07 must coincide; and the relation $K$08 is preserved under sums and differences of square-root parameters, given congruences on both the parameters and their squares.

Given $K$09, each new point $K$10 is mapped by choosing, for each root $K$11, the unique square root $K$12 of $K$13 lying in $K$14; existence and uniqueness hold because $K$15, and the product condition is automatic since any sign correction lies in $K$16. Equivariance follows because $K$17 is exactly the square root attached to $K$18. Injectivity uses the sign-change identities plus induction; additivity in the generic case $K$19 combines the addition formula with the stability lemmas to show $K$20 and $K$21 have $K$22-equivalent coordinates, forcing equality by uniqueness. Passing to the inverse limit gives the Tate module isomorphism.

The same argument transfers verbatim to the two-field setting because $K$23 enjoys the same product, quotient, and square-root formalism as $K$24.

Limitations and open questions

The paper is specific to $K$25 and genus $K$26: the halving-triple machinery relies on the cubic model and on explicit division-by-$K$27 formulae, and no analogue for other primes or higher genera is established here. The author raises the natural question of whether the integral conclusion extends to Jacobians of higher-genus hyperelliptic curves, suggesting Stoll's algorithm for halving points on odd-degree hyperelliptic Jacobians in Mumford representation as a possible route. Whether the explicit congruence bound of the corollary admits analogues for $K$28 is likewise not addressed.

Conclusion

The paper shows that, over a complete discretely valued field of odd residue characteristic, the integral $K$29-adic Tate module of an elliptic curve is controlled by concrete arithmetic data of a Weierstrass model — the square class of the leading coefficient and the pairwise root differences modulo higher-order terms. Beyond the intrinsic interest of an integral refinement of known rational statements, the result yields an explicit local constancy bound sharpening Kisin's qualitative theorem at $K$30, and identifies the $K$31-primary component groups of Néron models along the way. The extension to Jacobians of hyperelliptic curves remains open.