Timo Keller’s blog

February 26, 2024: Article on the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction

Let \(E/\mathbf{Q}\) be an elliptic curve of analytic rank \(0\) or \(1\) such that the mod-\(p\) Galois representation \(\rho_p\colon G_\mathbf{Q}\to \operatorname{Aut}(E[p](\overline{\mathbf{Q}}))\) is reducible (“Eisenstein prime”). In our recent preprint “On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction” with Mulun Yin (University of California Santa Barbara), we prove (a) the Iwasawa Main Conjecture and Perrin-Riou’s Heegner point Main Conjecture over a Heegner field in which the odd prime \(p\) of good or bad multiplicative reduction is split, as an application (b) a \(p\)-converse theorem (also for modular abelian varieties), and (c) the \(p\)-part of strong BSD if \(p > 2\) has good reduction.

Here are the elliptic curves \(E\) over \(\mathbf{Q}\) in the LMFDB with \(E(\mathbf{Q})_\mathrm{tors}\cong\mathbf{Z}/p\), rank \(0\) or \(1\), and \(p\) a prime of good reduction: \(p=3\), \(p=5\), and \(p = 7\).

As an application of (b), we obtain better proportions for the Goldfeld conjecture for families of elliptic curves having a \(3\)-isogeny where \(3\) is a prime of good or bad multiplicative reduction. This follows from a combination with results of Ari Shnidman et al. on the average \(3\)-Selmer rank.

Let us now show that even our results on the \(p\)-part of strong BSD are for elliptic curves only (the reason is that [Jetchev–Skinner–Wan] for \(\rho_p\) irreducible work over imaginary quadratic fields that do not satisfy the Heegner condition), we can nevertheless use them to treat many more cases for modular abelian varieties with the article “Complete verification of strong BSD for many modular abelian surfaces over \(\mathbf{Q}\) with Michael Stoll:

Consider the genus \(2\) curve \[X\colon y^2 = 5x^6 + 10x^5 - 13x^4 - 30x^3 + 9x^2 + 20x - 8\] with absolutely simple semistable RM Jacobian \(J\) of squarefree level \(5 \cdot 13\) and rank \(0\) associated to the newform 65.2.a.c. Note that this curves is not covered by op. cit. The algorithms from op. cit. prove the prime-to-\(3\) part of strong BSD for \(J/\mathbf{Q}\) with \(\#\mathrm{Ш}(J/\mathbf{Q})_\mathrm{an}= 2\). Our results allow us to also prove the \(3\)-part: Let the good ordinary prime \(3\) split as \(\mathfrak{p}\mathfrak{p}'\) in \(\mathbf{Z}[f]=\mathbf{Z}[\sqrt{3}]\) such that \(\rho_\mathfrak{p}\) is irreducible and \(\rho_{\mathfrak{p}'}\) is reducible with a trivial \(1\)-dimensional subrepresentation coming from the \(3\)-torsion. The results and algorithms of op. cit. prove that the Heegner index \(I_K\) for \(D_K = -131\) (so \(3\) is split in \(\mathcal{O}_K\)) equals \(\mathfrak{p}'\) times some ideal lying above \(2\), and that the Tamagawa product considered as an ideal in \(\mathbf{Z}[f]\) equals \(\mathfrak{p}'\). Hence [loc. cit., Theorem 5.2.2] proves that \(\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}] = 0\). Our IMC (a) shows that \(v_{\mathfrak{p}'}(\#\mathrm{Ш}(J/K)[\mathfrak{p}'^\infty]) = 2v_{\mathfrak{p}'}(I_K) - 2v_{\mathfrak{p}'}(\operatorname{Tam}(J/\mathbf{Q})) = 2 - 2 = 0\). Since \(\mathfrak{p}' \nmid 2\), we have \(\mathrm{Ш}(J/K)[\mathfrak{p}'] \simeq\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}'] \oplus \mathrm{Ш}(J^K/\mathbf{Q})[\mathfrak{p}']\), hence we get \(\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}'] = 0\), so strong BSD holds for \(J/\mathbf{Q}\) with \(\mathrm{Ш}(J/\mathbf{Q}) \cong\mathbf{Z}/2\).

December 24, 2023: Article on the verification of strong BSD for many modular abelian surfaces over \(\mathbf{Q}\)

In our recent preprint “Complete verification of strong BSD for many modular abelian surfaces over \(\mathbf{Q}\) Michael Stoll and I verify the strong Birch–Swinnerton-Dyer (BSD) conjecture for the first time for abelian surfaces over \(\mathbf{Q}\) in cases where it cannot immediately be reduced to elliptic curves, i.e., such that the surface is absolutely simple. Recall the conjecture for \(A/\mathbf{Q}\) a principally polarized abelian variety: \[L^*(A/\mathbf{Q},1) :=\lim_{s \to 1} (s-1)^{-r} L(A/\mathbf{Q}, s) = \frac{\Omega_A \prod_p c_p(A) \cdot \mathrm{Reg}_{A/\mathbf{Q}} \#\mathrm{Ш}(A/\mathbf{Q})}{(\#A(\mathbf{Q})_\mathrm{tors})^2} \,.\]

Assume the abelian surface is the Jacobian \(J\) of a genus \(2\) curve, with real multiplication, absolutely simple, and that the analytic rank of one of the newforms associated with it is \(0\) or \(1\). Then our Magma code can do the following:

  1. Computation of the images of all residual Galois representations.

  2. Computation of Heegner points and Heegner indices.

  3. Formulas for the “analytic order of Sha”.

  4. Euler system computations.

  5. Isogeny descents.

  6. Computations of approximations of \(p\)-adic \(L\)-functions.

  7. Compute a finite set of primes for which one has to compute \(\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}^\infty]\) to verify strong BSD. In practice, this set is small and often empty.

Assuming that one can compute the Mordell–Weil group of the abelian variety and heights, most of the algorithms generalize to modular Jacobians of any dimension.

Let me illustrate our algorithm with an example, the Jacobian of the genus \(2\) curve 4225.a.274625.1 of level \(N = 65 = 5\cdot13\) and rank \(0\).

  1. All residual Galois representations at primes not above \(2\) or \(3\) are irreducible.

  2. For the Heegner discriminant \(D = -51\), one gets the Heegner index \(2^2\cdot3\).

  3. The analytic order of Sha is \(1\), as one can compute using modular symbols since the rank is \(0\).

  4. The Euler system of Kolyvagin–Logachëv in an explicit version says that \(\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}] = 0\) for \(\mathfrak{p}\nmid 2,3\): One always has to exclude the primes above \(2\), and additionally those dividing the Tamagawa product and the gcd of some Heegner indices, as well as those such that the residual Galois representation is reducible.

  5. For the torsion prime \(\mathfrak{p}\) above \(3\), an isogeny descent shows that \(\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}] = 0\).

  6. No \(p\)-adic \(L\)-function is needed. In fact, for the “LMFDB examples”, we only use it for the curve with \(N = 188\).

  7. One easily computes \(\mathrm{Ш}(J/\mathbf{Q})[2] = 0\) using existing algorithms in Magma. Hence \(\#\mathrm{Ш}(J/\mathbf{Q}) = 1 = \#\mathrm{Ш}(J/\mathbf{Q})_\mathrm{an}\).

I’m currently running our algorithms on all absolutely simple modular Jacobians associated with RM newforms with real quadratic coefficients up to level \(N \leq 1000\).

December 24, 2023: Introduction to arithmetic geometry: plane quadrics

This is my first blog entry for the mathematically interested public.

Basic questions in arithmetic geometry.

My research interests lie in the field of number theory, especially arithmetic geometry. One studies properties of the natural numbers \(\mathbf{N}= \{0,1,2,\ldots\}\). One basic question is to ask for integral or rational solutions to systems of equations \(f_i(X_1, \ldots, X_n) = 0\) with \(f_i \in \mathbf{Z}[X_1,\ldots,X_n]\) polynomials in several variables. These are called Diophantine equations after Diophantus of Alexandria (maybe 250 AD). They define geometric objects, called varieties or schemes. These algebraic objects are investigated using both algebraic and geometric methods.

Why is it interesting?

One of the reasons why we are interested in them is that finding their solutions is a easy-to-state problem, but solving them can be extremely hard and lead to significant and deep theoretical advances in number theory and related fields. An example for this is the solution of the 17th century Fermat equation \(a^n + b^n = c^n\), which required late 20th century mathematics. Number theory uses almost every other area of pure mathematics. Many mathematicians developed deep theories to tackle diophantine questions, and often one cannot easily see anymore what their connection to the original questions is. The abstract objects may even become more interesting that the originating equations.

Undecidable in general.

However, the question whether a diophantine equation has an integral solution is undecidable (Hilbert’s 10th problem, theorem of Matiyasevich–Robinson–Davis–Putnam 1970). (There is no algorithm that enumerates the set of parameters of a parameterized diophantine equation for which there exists a solution.) Restricting the class of diophantine equations considered, the problem can or might be solvable, though. For example, it is solvable for (at most) quadratic equations, which follows from the Hasse–Minkowski theorem, a local-global principle. The class of curves, i.e., equations in only two variables is still open.

Curves.

Aside from \(0\)-dimensional objects, which are algebraically also very interesting, the easiest objects are curves, i.e., \(1\)-dimensional objects. There is a discrete invariant of curves, called the genus. There are infinitely many “different” (i.e., non-isomorphic) curves of any genus \(g \geq 0\). In this blog entry, I will talk about the simplest ones, those of genus \(0\).

Pythagorean triples.

A classical example is the (projective) equation \[% \label{eq:PythagoreanTriples} X^2 + Y^2 = Z^2.\] By the Theorem of Pythagoras, its positive integer solutions correspond to right triangles with integral side lengths.

Excluding the trivial case \(Z = 0\) (because it implies \(X = Y = 0\) in integers) and dividing by \(Z\), we can equivalently determine the rational solutions to \(x^2 + y^2 = 1\) (with \(x = X/Z, y = Y/Z\)). Geometrically, the solutions to this equation form a circle:

image

The integral solutions to the projective equation correspond to rational solutions to the affine equation (with the exception of the points at infinity). The algebraic question of finding solutions to the defining equation translates into the geometric question of finding points on the curve.

Describing all solutions.

Now we do some geometry: There is one rational solution \((1,0)\), and if one draws a line through \((1,0)\) with slope \(t \in \mathbf{Q}\), this line intersects the circle in another point with rational coordinates (Bézout’s theorem—here we use that the equation has degree \(2\)), and all rational points can be obtained in this way. By calculating the coordinates of the intersection point, one gets an algebraic parameterization of the solutions: \[\begin{aligned} %\label{eq:Pythagorean parameterization} (x,y) = \Big(\frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2}\Big), \quad t \in \mathbf{Q} \end{aligned}\] Reverting the process of dividing by \(Z\) again yields a parameterization of the original equation \[(X,Y,Z) = k(m^2 - n^2, 2mn, m^2 + n^2)\] with \(m,n,k \in \mathbf{Z}\). For example, letting \(m = 2, n = 1, k = 1\) yields the solution and most well-known Pythagorean triple \(3^2 + 4^2 = 5^2\).

Plane quadrics.

More generally, plane quadrics (i.e., curves described by quadratic polynomials in two variables) \(ax^2 + by^2 = c\) belong to the simplest group of curves, those of genus \(0\). They have either no or infinitely many solutions, and there is a practical algorithm to decide what is the case. If there are infinitely many of them, one can parameterize them with one parameter \(t \in \mathbf{Q}\), as above.

An elliptic curve.

A more complicated example of a curve is the (affine) elliptic curve \[y^2 = x^3 - x\] with real picture:

image

It is a curve of genus \(1\). Its properties are better described by the complex points, which form a complex torus.

One can show that the only rational solutions are \((0,0),(\pm1,0)\). There are also elliptic curves with infinitely many points. Elliptic curves are much more complicated than plane quadrics, and several extremely deep conjectures, most notably the Birch–Swinnerton-Dyer (BSD) conjecture, have been formulated about them.

There is also a method, motivated by geometry, to generate new solutions out of old solutions. However, there is no simple geometric parameterization of all solutions anymore.

Next time.

I will write more about elliptic curves in my next blog entry for the public.

May 14, 2023: New preprint on quadratic points on \(X_0(N)\)

The preprint Computing Quadratic Points on Modular Curves \(X_0(N)\) is joint work with Nikola Adžaga, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman, and Borna Vukorepa. We started to work on this at the 2022 MIT Modular curves workshop, whose aim was to extend the LMFDB with data on modular curves.

The \(\mathbf{Q}\)-rational points on \(X_1(N)\) and \(X_0(N)\) for all \(N\) have been computed by Mazur and Kenku. For the modular curve \(X_1(N)\) classifying (generalized) elliptic curves with a point of order \(N\), there is a bound \(N(d)\) by Merel and Kamienny such that \(X_1(N)(K)\) contains only cusps if \(N > N(d)\) and \(K\) runs through all number fields of degree \(d\). This has been used to compute all degree \(d\) points for \(d \leq 7\) by combined work of Kamienny–Kenku–Momose (\(d=2\)), Derickx–Etropolski–van Hoeij–Morrow–Zureick-Brown (\(d = 3\)), and for \(4\leq d\leq 7\) and prime values of \(N\) by Derickx–Kamienny–Stein–Stoll.

However, there is no \(d > 1\) such that one knows all degree-\(d\) points on \(X_0(N)\) (classifying elliptic curves with a cyclic \(N\)-isogeny) for all \(N\)! Conjecturally, the only quadratic points are cusps and CM points for \(N \gg 0\).

In our preprint, we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves \(X_0(N)\) of genus up to \(8\), and genus up to \(10\) with \(N\) prime, for which they were previously unknown. This extends previous work by Bruin–Najman, Ozman–Siksek, Box, and Najman–Vukorepa. Our methods apply to more than these curves; we restricted to these to make the computations feasible.

We use the following three methods:

  1. Going down: If \(f\colon X \to Y\) is a finite morphism of curves and \(Y(K)\) is known and finite, one can compute \(X(K)\) by taking fibers. Problems arise if there are infinitely many quadratic points on \(Y\). We use this for \(N \in \{58, 68, 76 \}\).

  2. Rank \(0\): If \(J_0(N)(\mathbf{Q}) = J_0(N)(\mathbf{Q})_\mathrm{tors}\) is known (we verify the generalized Ogg conjecture in several cases), one can compute the quadratic points on \(X_0(N)\) using a variant of the Mordell–Weil sieve. We use this for \(N \in \{ 80, 98, 100 \}\).

  3. Atkin–Lehner sieve: This is the most involved method and a variant of the Mordell–Weil sieve, which, if applicable, reduces the problem to considering fixed points of an Atkin–Lehner involution and the rational points on a given Atkin–Lehner quotient. We use this for \(N \in \{ 74, 85, 97, 103, 107, 109, 113, 121, 127 \}\).

We also give optimized algorithms to compute models of Atkin–Lehner quotients of \(X_0(N)\) and the \(j\)-invariant morphism.

Note that there can be infinitely many quadratic points on a curve \(X\); this happens if and only if \(X\) is hyperelliptic (plug in infinitely many values for \(x\) in \(y^2 = f(x)\)) or bielliptic with elliptic curve of rank \(> 0\).

We get evidence for the fact that non-cuspidal, non-CM points are rare.

Update (2023-06-22).

Here are my slides for a talk at a conference in Dubrovnik.

Update (2023-08-23).

The article has been accepted for publication in Mathematics of Computation.

February 13, 2023: Two articles accepted

Recently, two articles have been accepted:

Quadratic Chabauty for modular curve quotients.

In Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6 (with Nikola Adžaga, Vishal Arul, Lea Beneish, Mingjie Chen, Shiva Chidambaram, and Boya Wen), accepted for publication in Acta arithmetica, we apply the quadratic Chabauty method to determine all \(\mathbf{Q}\)-rational points of Atkin–Lehner quotients \(X_0(p)^+\) of prime level \(p\) such that the genus is in \(\{4,5,6\}\).

Since the non-cuspidal points of \(X_0(p)\) correspond to elliptic curves with a \(p\)-isogeny and since the Atkin–Lehner (or Fricke) involution \(w_p\) maps such an isogeny to its dual, our result classifies elliptic curves with an unordered pair of \(p\)-isogenies. Computing the points on \(X_0(p)^+ := X_0(p)/w_p\) has been called an “extremely interesting arithmetic question” by Mazur in his seminal Eisenstein paper. For composite levels \(N\), one can mod out (potentially) more Atkin–Lehner involutions and go down to the quotient \(X_0(N)^*\). The \(\mathbf{Q}\)-points on the hyperelliptic \(X_0(N)^*\) have been determined in our ANTS paper, see my blog post from August 12, 2022 below.

Our article proves a conjecture of Galbraith for those curves. The main difficulty in computing the set of rational points was to find suitable plane models of those modular curves such that the existing implementation of the Quadratic Chabauty algorithm determined the set of rational points. It was surprising to the experts that we could go up to (relatively high) genus \(6\). This article originated from a project started at the 2020 Arizona Winter School.

Those of smaller genus have been tackled before, for example in the article Quadratic Chabauty for modular curves: Algorithms and examples by Balakrishnan–Dogra–Müller–Tuitman–Vonk. Those of composite level and genus \(\leq 6\) are known known by work of Momose with \(X_0(125)^+\) being solved in Arul–Müller.

Specialization of Mordell–Weil groups.

In Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves, accepted for publication in International Journal of Number Theory, I prove the following surjectivity result for specialization morphisms: Recall that Silverman’s specialization theorem (with generalizations by Wazir) roughly says that for an abelian variety \(A\) over a function field \(K\) over a global field, outside a set of bounded height, the specialization theorem from the Mordell–Weil group \(A(K)\) to the Mordell–Weil group to closed points is injective modulo torsion, i.e., the rank does not drop.

It is a challenging question whether there exist infinitely many specializations that the rank does also not jump! For example, when applied to a family of elliptic curves over \(\mathbf{Q}(T)\) with generic rank \(r\) (such families exist for small \(r\)), a positive answer to this question would imply that there are infinitely many non-isomorphic elliptic curves over \(\mathbf{Q}\) with rank exactly equal to \(r\). The latter is known for \(r = 0,1\) (explicit constructions or the Gross–Zagier formula combined with Kolyvagin’s Heegner point Euler system), but open already for \(r = 2\).

My article does not solve this question, but shows that for an abelian scheme over a surface, infinitely many specializations to curves have the same rank.

The proof uses a recent result of Ambrosi on a specialization theorem of Néron–Severi ranks and the Shioda–Tate formula to go from Néron–Severi ranks to Mordell–Weil ranks. The result holds more generally over infinite finitely generated fields.

My original motivation to prove such a result was to prove the reduction of the analog of the Birch–Swinnerton-Dyer conjecture over higher-dimensional bases to that over curves, with the reduction to surfaces as a basis established in my Documenta article and the reduction to curves by my recent article.

August 26, 2022: My quadratic Chabauty conference

I have been back to Bayreuth to my quadratic Chabauty conference (it had to take place there because of my funding).

The teams.

We had three remote working groups, one in the US and two in Europe, working on ongoing projects. Five of us worked in person starting something new. We will continue working on our problems after the conference.

August 12, 2022: Last day of ANTS

ANTS-XV.

Here are the slides of my 25 minutes talk I gave at ANTS-XV (Fifteenth Algorithmic Number Theory Symposium), taking place at the University of Bristol this year. This is the largest international conference on this topic.

My talk.

I talked about the completion of the determination of the \(\mathbf{Q}\)-points on the Atkin-Lehner quotients \(X_0(N)^*\) which are hyperelliptic. There are exactly \(64\) of them, and they were determined by Hasegawa in 1997. Here is our paper and our Magma code with log files.

Our methods.

To do this, we used a combination of well-established (Chabauty–Coleman method in its implementation by Balakrishnan–Tuitman, elliptic curve Chabauty in its implementation by Bars–González–Xarles) and recent developments in the quadratic Chabauty method by Balakrishnan–Dogra–Müller–Tuitman–Vonk.

The quadratic Chabauty condition.

Looking at the root number, you expect that most \(X_0(N)^*\) with \(N\) square-free have \(r = g\) (\(g\) the genus of the curve and \(r\) the Mordell-Weil rank of its Jacobian \(J\)) because the corresponding space of cusp forms is \(S_2(\Gamma_0(N))^{w_d = +1 : d \mid N, (d,N/d)=1}\), so the analytic ranks of the \(L(f,s)\) are odd and in fact should conjecturally be \(1\) in most of the times. In these cases, the Chabauty–Coleman method will not be applicable (except if the \(\mathbf{Z}_p\)-rank of the closure of \(J(\mathbf{Q})\) in \(J(\mathbf{Q}_p)\) will be less than \(g\)), and you need quadratic Chabauty.

Fake residue discs.

As David Harvey asked, we usually have many fake residue discs coming from the various Chabauty methods. These need to be ruled out using the Mordell-Weil sieve. For genus \(g > 2\), one would not expect fake residue discs from quadratic Chabauty (in fact, we didn’t have this problem in our genus \(4,5,6\) paper) because there are more Chabauty functions if \(g-1 = \operatorname{rk}\, \mathrm{NS}(J) - 1 \geq 2\). If there are, there should be a geometric reason explaining them.

Ongoing work.

We (Nikola Adžaga, Shiva Chidambaram, Oana Padurariu, and me) and others will work on non-hyperelliptic \(X_0(N)^*\) and on quotients of Shimura curves on my conference in two weeks.

July 30, 2022: First blog entry, PCMI

This is the first entry of my blog started as announced in my Marie Skłodowska-Curie fellowship starting in June 2023 with Steffen Müller in Groningen.

PCMI: collaboration.

It is the end of the second week at PCMI “Number Theory Informed by Computation” organized by the Princeton Institute of Advanced Study, which I enjoy very much. I could continue many collaborations and start new ones, for example with Ross Paterson and Carlo Pagano. When there are preprints available at arXiv, I will write a blog post here.

PCMI: my talk.

Here are the slides of my 25 minutes talk I gave in the Research Program. The talk was about the exact verification of the strong Birch–Swinnerton-Dyer (BSD) conjecture for some abelian surfaces with real multiplication over \(\mathbf{Q}\). There were several interesting questions by Levent Alpöge, Sam Frengley, Bjorn Poonen, and Akshay Venkatesh. For example, I hope I can prove strong BSD for an example of a Jacobian of a genus \(2\) curve of level \(3200\) provided by Sam, which has rank \(0\) and \(7\)-torsion in the Shafarevich–Tate group. (All examples I computed had order of Sha equal to \(1,2,4\).) Akshay suggested to formulate a refinement of an equivariant BSD formula. I did so over Heegner fields and I will check whether this formula holds in my examples. The question of Levent was about the Galois representations, and Bjorn asked for an heuristic explanation why the Shafarevich–Tate group is trivial for all Atkin-Lehner quotients with Jacobian absolutely simple and modular of dimension \(2\).

ANTS-XV.

After the third week, I’m at ANTS-XV, where I will give a talk about our paper on the determination of the \(\mathbf{Q}\)-points on hyperelliptic Atkin-Lehner quotients of modular curves accepted there.

Future blog entries.

As the audience of this blog is also the interested mathematical public, I will post some introductory entries on curves and abelian varieties besides the research topics.

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