Implement modularity test for elliptic curves

src/doc/en/reference/references/index.rst

@@ -802,6 +802,10 @@ REFERENCES:
             Cambridge Univ Press, 2001.
             :doi:`10.1017/S0963548301004813`.
 
+.. [Box2022] Josha Box.
+             Elliptic curves over totally real quartic fields not containing `\sqrt{5}` are modular.
+             Trans. Amer. Math. Soc. 375 (2022), 3129-3172.
+
 .. [BF2005] \R.L. Burden and J.D. Faires. *Numerical Analysis*.
             8th edition, Thomson Brooks/Cole, 2005.
 
@@ -1282,6 +1286,10 @@ REFERENCES:
 .. [Bre1997] \T. Breuer "Integral bases for subfields of cyclotomic
              fields" AAECC 8, 279--289 (1997).
 
+.. [Breuil2001] Breuil, Christophe, Brian Conrad, Fred Diamond, and Richard Taylor.
+                *On the modularity of elliptic curves over Q: wild 3-adic exercises*.
+                Journal of the American Mathematical Society 14, no. 4 (2001): 843-940.
+
 .. [Bre2000] Enno Brehm, *3-Orientations and Schnyder 3-Tree-Decompositions*,
              2000.
              https://page.math.tu-berlin.de/~felsner/Diplomarbeiten/brehm.ps.gz
@@ -1457,6 +1465,10 @@ REFERENCES:
 .. [Can1990] \J. Canny. Generalised characteristic polynomials.
              J. Symbolic Comput. Vol. 9, No. 3, 1990, 241--250.
 
+.. [CarNew2023] Ana Caraiani and James Newton.
+                *On the modularity of elliptic curves over imaginary quadratic fields.*
+                :arxiv:`2301.10509`
+
 .. [Car1972] \R. W. Carter. *Simple groups of Lie type*, volume 28 of
              Pure and Applied Mathematics. John Wiley and Sons, 1972.
 
@@ -1945,6 +1957,11 @@ REFERENCES:
             *Counting smaller elements in the tamari and m-tamari lattices*.
             Journal of Combinatorial Theory, Series A. (2015). :arxiv:`1311.3922`.
 
+.. [Chen1996] Imin Chen.
+              *The Jacobian of Modular Curves Associated to Cartan Subgroups*.
+              Oxford DPhil thesis (1996).
+              https://ora.ouls.ox.ac.uk/objects/uuid:46babaa0-c498-4211-a4ad-8dbabb8f05d9
+
 .. [CP2016] \N. Cohen, D. Pasechnik,
             *Implementing Brouwer's database of strongly regular graphs*,
             Designs, Codes, and Cryptography, 2016
@@ -2185,6 +2202,10 @@ REFERENCES:
                     Chapman & Hall/CRC
                     2012
 
+.. [DNS2020] Maarten Derickx, Filip Najman, Samir Siksek
+              *Elliptic curves over totally real cubic fields are modular.*
+              Algebra and Number Theory. (2020) 14 no.7, pp. 1791--1800.
+
 .. [DerZak1980] Nachum Dershowitz and Schmuel Zaks,
                 *Enumerations of ordered trees*,
                 Discrete Mathematics (1980), 31: 9-28.
@@ -2763,6 +2784,10 @@ REFERENCES:
              "Nearly rigid analytic modular forms and their values at CM points",
              Ph.D. thesis, McGill University, 2011.
 
+.. [FLHS2015] Nuno Freitas, Bao Le Hung, and S. Siksek.
+              *Elliptic curves over Real Quadratic Fields are Modular.*
+              Invent. math. (2015) 201, pp. 159--206.
+
 .. [FRT1990] Faddeev, Reshetikhin and Takhtajan.
              *Quantization of Lie Groups and Lie Algebras*.
              Leningrad Math. J. vol. **1** (1990), no. 1.
@@ -6540,6 +6565,10 @@ REFERENCES:
              height on elliptic curves over number fields, Math. Comp. 79
              (2010), pages 2431-2449.
 
+.. [Thorne2016] Jack Thorne.
+                *Automorphy of some residually dihedral Galois representations.*
+                Mathematische Annalen 364 (2016), No. 1--2, pp. 589--648.
+
 .. [Tho2011] Anders Thorup. *ON THE INVARIANTS OF THE SPLITTING ALGEBRA*,
              2011, :arxiv:`1105.4478`
 
@@ -6810,6 +6839,10 @@ REFERENCES:
 .. [Wie2000] \B. Wieland. *A large dihedral symmetry of the set of
              alternating sign matrices*. Electron. J. Combin. 7 (2000).
 
+.. [Wiles1995] Andrew Wiles.
+               *Modular elliptic curves and Fermat's last theorem*.
+               Annals of mathematics, 141(3), 443-551.
+
 .. [Wilson2008] Steve Wilson. *Rose Window Graphs*. Ars Mathematica
                 Contemporanea 1(1):7-19, 2008.
                 :doi:`10.26493/1855-3974.13.5bb`

src/sage/rings/rational_field.py

@@ -599,6 +599,39 @@ def signature(self):
         """
         return (Integer(1), Integer(0))
 
+    def is_totally_real(self):
+        r"""
+        Return ``True``, since `\QQ` has no non-real complex embeddings.
+
+        EXAMPLES::
+
+            sage: QQ.is_totally_real()
+            True
+        """
+        return True
+
+    def is_CM(self):
+        r"""
+        Return ``False``, since  `\QQ` is not a CM field.
+
+        EXAMPLES::
+
+            sage: QQ.is_CM()
+            False
+        """
+        return False
+
+    def is_abelian(self):
+        r"""
+        Return ``True``, since  `\QQ` is an abelian Galois extension of itself.
+
+        EXAMPLES::
+
+            sage: QQ.is_abelian()
+            True
+        """
+        return True
+
     def embeddings(self, K):
         r"""
         Return the list containing the unique embedding of `\QQ` into `K`,

src/sage/schemes/elliptic_curves/ell_number_field.py

@@ -6,16 +6,21 @@
 by a Weierstrass equation whose coefficients lie in `K` or by
 using ``base_extend`` on an elliptic curve defined over a subfield.
 
-One major difference to elliptic curves over `\QQ` is that there
-might not exist a global minimal equation over `K`, when `K` does
-not have class number one.
-Another difference is the lack of understanding of modularity for
-general elliptic curves over general number fields.
-
-Currently Sage can obtain local information about `E/K_v` for finite places
-`v`, it has an interface to Denis Simon's script for 2-descent, it can compute
-the torsion subgroup of the Mordell-Weil group `E(K)`, and it can work with
-isogenies defined over `K`.
+One major difference to elliptic curves over `\QQ` is that there might
+not exist a global minimal equation over `K`, when `K` does not have
+class number one.  When a minimal model does exist the method
+:meth:`global_minimal_model()` will compute it, and otherwise compute
+a model which is minimal at all primes except one.  Another difference
+is the relative lack of understanding of modularity for elliptic
+curves over general number fields; the method :meth:`is_modular()`
+does implement recent methods to prove modularity of elliptic curves,
+over totally real and imaginary quadratic fields only.
+
+Currently Sage can obtain local information about `E/K_v` for finite
+places `v`, it has an interface to Denis Simon's script for 2-descent,
+it can compute the torsion subgroup of the Mordell-Weil group `E(K)`,
+and it can work with isogenies defined over `K`, including the
+determination of the complete isogeny class of any curve.
 
 EXAMPLES::
 
@@ -77,6 +82,7 @@
 
 - [Sil2] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves. Graduate Texts in
   Mathematics, 151. Springer, 1994.
+
 """
 
 # ****************************************************************************
@@ -4158,3 +4164,212 @@ def rational_points(self, **kwds):
             E = E.change_ring(kwds['F'])
 
         return [E(pt) for pt in Curve(self).rational_points(**kwds)]
+
+    def is_modular(self, verbose=False):
+        r"""Return ``True`` if the base field is totally real or imaginary
+        quadratic and modularity of this curve can be proved,
+        otherwise ``False``.
+
+        INPUT:
+
+        - ``verbose`` (bool, default ``False``) -- if ``True``, outputs a reason for the conclusion.
+
+        .. NOTE::
+
+            When ``False`` is returned, it does not mean that the
+            curve is not modular!  Only that modularity cannot be
+            proved with current knowledge and implemented methods.  It
+            is expected that the return value will be ``True`` for most
+            curves defined over totally real fields, and for all
+            defined over imaginary quadratic fields over which the
+            modular curve `X_0(15)` (which is the elliptic curve with
+            label `15a1`) has rank zero.  Any curve defined over
+            totally real fields for which the return value is
+            ``False`` is of interest, as its mod `p` Galois
+            representations for the primes 3, 5 and 7 are
+            simultaneously small.
+
+            Note that for over half of all imaginary quadratic fields,
+            there exist infinitely many elliptic curves (even up to
+            twist) whose mod 3 and mod 5 Galois representations are
+            both reducible (in other words, which possess a rational
+            15-isogeny).  Such curves can be proved to be modular, but
+            not using the methods implemented here.
+
+            There are currently no theoretical results which allow
+            modularity to be proved over fields other than totally
+            real fields and imaginary quadratic fields.
+
+        ALGORITHM:
+
+        This is based on code provided by S. Siksek for the totally
+        real case, and relies on theorems in the following papers:
+        [Wiles1995]_, [Breuil2001]_, [Chen1996]_, [FLHS2015]_,
+        [Thorne2016]_, [CarNew2023]_, [DNS2020]_, [Box2022]_.
+
+        It relies on checking that the images of the mod-`p` Galois
+        representation attached to the elliptic curve for `p=3,5,7`
+        are not simultaneously small, except for the cases (base field
+        `\QQ`, real quadratic, totally real cubic, or totally real quartic
+        not containing `\sqrt{5}`) where theoretical results show this to
+        be impossible.
+
+        Over `\QQ`: all curves are modular with no further checks
+        needed, by [Wiles1995]_ and [Breuil2001]_.
+
+        Over real quadratic fields:  all curves are modular with no further checks
+        needed, by [FLHS2015]_.
+
+        Over totally real cubic fields:  all curves are modular with no further checks
+        needed, by [DNS2020]_.
+
+        Over totally real quartic fields not containing `\sqrt{5}`:
+        all curves are modular with no further checks needed, by
+        [Box2022]_.
+
+        Over other totally real fields: modularity is implied by
+        conditions on the mod-`\ell` representations for `\ell=3,5,7`
+        by [FLHS2015]_.  It is sufficient for one of the following to
+        hold: (1) the image of the mod-3 representation is not
+        contained in either a Borel or the normaliser of a split
+        Cartan subgroup; (2) image of the mod-5 representation is
+        irreducible, and either 5 is not square in the field, or the
+        image is not contained in the normaliser of either a split or
+        nonsplt Cartan subgroup; (3) the image of the mod-7
+        representation is be not contained in either a Borel or the
+        normaliser of a split or nonsplt Cartan subgroup.
+
+        Over imaginary quadratic fields: by [CarNew2023]_, it is
+        sufficient for the mod-3 or mod-5 images to satisfy conditions
+        (1) or (2) above, respectively.
+
+        EXAMPLES:
+
+        Set ``verbose=True`` to see a reason for the conclusion::
+
+            sage: E = EllipticCurve('11a1')
+            sage: E.is_modular(verbose=True)
+            All elliptic curves over QQ are modular
+            True
+            sage: K.<a> = QuadraticField(5)
+            sage: E = EllipticCurve([0,1,0,a,0])
+            sage: E.is_modular(verbose=True)
+            All elliptic curves over real quadratic fields are modular
+            True
+            sage: E = EllipticCurve([0,0,0,a,0])
+            sage: E.is_modular(verbose=True)
+            All elliptic curves over real quadratic fields are modular
+            True
+            sage: K.<a> = CyclotomicField(5)
+            sage: E = EllipticCurve([0,1,0,a,0])
+            sage: E.is_modular(verbose=True)
+            Unable to determine modularity except over totally real and imaginary quadratic fields
+            False
+
+        Some examples from the LMFDB.  Over a totally real quintic field::
+
+            sage: R.<x> = PolynomialRing(QQ)
+            sage: K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
+            sage: E = EllipticCurve([K([3,0,-5,0,1]),K([-3,-7,9,2,-2]),K([1,0,0,0,0]),K([-22,-79,-92,14,21]),K([12,-87,-324,7,74])])
+            sage: E.is_modular(verbose=True)
+            Modular since the mod 3 Galois image is neither reducible nor split
+            True
+
+        Over imaginary quadratic fields `K` over which `X_0(15)` has
+        positive rank, there are many curves where current results do
+        not imply modularity::
+
+            sage: R.<x> = PolynomialRing(QQ)
+            sage: K.<a> = NumberField(R([2, -1, 1]))
+            sage: EllipticCurve('15a1').change_ring(K).rank()
+            1
+            sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([1,1]),K([-105,98]),K([-615,1188])])
+            sage: E.is_modular(verbose=True)
+            Modularity not established: this curve has small image at 3 and 5
+            False
+            sage: EllipticCurve('15a1').change_ring(K).rank() # long time
+            1
+
+        Nevertheless, over the same field, most curves pass::
+
+            sage: E = EllipticCurve([K([0,0]),K([0,1]),K([1,1]),K([-8,-3]),K([-5,-5])])
+            sage: E.is_modular(verbose=True)
+            Modular since the mod 3 Galois image is neither reducible nor split
+            True
+
+        Over a field which is neither totally real nor imaginary
+        quadratic, no conclusion is currently possible::
+
+            sage: R.<x> = PolynomialRing(QQ)
+            sage: K.<a> = NumberField(R([1, 0, -1, 1]))
+            sage: K.is_totally_real()
+            False
+            sage: E = EllipticCurve([K([0,0,1]),K([1,0,0]),K([0,0,1]),K([0,0,0]),K([0,0,0])])
+            sage: E.is_modular(verbose=True)
+            Unable to determine modularity except over totally real and imaginary quadratic fields
+            False
+
+        """
+        K = self.base_field()
+        d = K.degree()
+
+        if d == 1: # base field QQ: [Wiles1995], [Breuil2001]
+            if verbose:
+                print("All elliptic curves over QQ are modular")
+            return True
+
+        TR =  K.is_totally_real()
+
+        if TR and d <= 3: # base field real quadratic or totally real cubic
+            if verbose:
+                if d==2:
+                    print("All elliptic curves over real quadratic fields are modular")
+                else:
+                    print("All elliptic curves over totaly real cubic fields are modular")
+            return True
+
+        if TR and d==4 and not K(5).is_square():
+            if verbose:
+                print("All elliptic curves over totally real quartic fields not containing sqrt(5) are modular")
+
+        if d > 2 and not TR:
+            if verbose:
+                print("Unable to determine modularity except over totally real and imaginary quadratic fields")
+            return False # meaning 'unknown'
+
+        # from here either K is totally real of degree at least 4 (TR is True) or imaginary quadratic
+
+        if self.has_cm():
+            if verbose:
+                print("All CM elliptic curves over totally real and imaginary quadratic fields are modular")
+            return True
+
+        G = self.galois_representation()
+
+        if not (G.is_borel(3) or G.is_split_normaliser(3)):
+            if verbose:
+                print("Modular since the mod 3 Galois image is neither reducible nor split")
+            return True
+
+        if not G.is_borel(5):
+            if not (TR and K(5).is_square()):
+                if verbose:
+                    print("Modular since the mod 5 Galois image is irreducible, and 5 not square")
+                return True
+            if TR and not (G.is_split_normaliser(5) or G.is_nonsplit_normaliser(5)):
+                if verbose:
+                    print("Modular since the mod 5 Galois image is not reducible, split, or nonsplit")
+                return True
+
+        if TR and not (G.is_borel(7) or G.is_split_normaliser(7) or G.is_nonsplit_normaliser(7)):
+            if verbose:
+                print("Modular since the mod 7 Galois image is not being reducible, split, or nonsplit")
+            return True
+
+        if verbose:
+            if TR:
+                print("Modularity not established: this curve has small image at 3, 5 and 7")
+            else:
+                print("Modularity not established: this curve has small image at 3 and 5")
+
+        return False # We've run out of tricks!