Periodic Reporting for period 1 - CurveArithmetic (Arithmetic of Curves and Jacobians)
Período documentado: 2023-01-01 hasta 2025-06-30
The are three overarching research goals:
1) Parameterize rational points on certain unitary Shimura curves.
Shimura varieties are some of the most important and symmetric algebraic varieties and our goal is to find the solutions with rational coordinates. Mazur handled a large class of Shimura curves (modular curves) in the 1970's, leading to the classification of rational torsion points on all elliptic curves. Our goal is to do this for certain Shimura curves, with applications towards rational torsion points on abelian surfaces and Jacobians of certain curves of genus 2 and 3.
2) Study the statistics of Selmer groups and rational points on elliptic curves.
Selmer groups are algebraic tools used as a proxy for rational points on elliptic curves. Our ultimate goal would be to prove the conjecture of Katz-Sarnak stating that half of all elliptic curves over the rational numbers have no interesting rational points and the other half have exactly one independent point of infinite order. Toward this, we hope to prove that the 2-Selmer groups of elliptic curves are distributed according to a certain random variable described by Poonen and Rains. We also hope to resolve Hilbert's tenth problem over number fields (on the undecideability of solving polynomial equations in general).
3) Study the Ceresa cycles of genus 3 Jacobians.
Abelian varieties are one of the most well-studied algebraic varieties. Playing a prominent role is a class of "special" abelian varieties, those with "complex multiplication" which have extra symmetries (e.g. the elliptic curves x^3 + y^3 + z^3 = 0 or y^2 = x^4 + y^4). These are the abelian varieties that behave most like number fields. On the one hand, they are easier to understand because of their extra symmetries, but they are dense in the moduli space and crucial for understanding abelian varieties in general. What is the notion of "special" for algebraic curves C (of a given genus g)? We formulate things in terms of the motive h(C). Indeed, whereas a curve of genus g > 1 cannot be an algebraic group, we can ask whether the motive h(C) looks like the motive of a group. More formally, we can ask whether there exists a Chow-Kunneth decomposition of h(C) that respects the intersection pairing. This is "group like" since if A is any abelian variety then h(A) has a canonical such decomposition. It turns out that h(C) has such a decomposition precisely if the Ceresa cycle [C] - [-C] is torsion in the Chow group of the Jacobian J = Jac(C). Thus curves with torsion Ceresa cycle are "special" in that their motive is group-like. Which curves have torsion Ceresa cycle? In the last few years, interesting examples were discovered beyond the well-known case of hyperelliptic curves y^2 = f(x). So the interesting question is whether we can find all of them and characterize them in a simple way. The first non-trivial case is degree 4 (genus 3) plane curves f(x,y) = 0. For example, we have shown that the curves y^3 = x^4 + ax^2 + bx + c have torsion Ceresa cycle modulo algebraic equivalence. I expect something analogous to Schneider's theorem on the transcendence of j-invariants: for algebraic t in the upper half plane, j(t) is transcendental if and only if t is a degree 2 irrationality. However, before we can begin to prove an analogous theorem in genus 3, we must understand (even conjecturally) what is playing the role of the degree 2 irrationalities in Schneider's theorem.
1) Parameterize rational points on certain unitary Shimura curves.
Shimura varieties are some of the most important and symmetric algebraic varieties and our goal is to find the solutions with rational coordinates. Mazur handled a large class of Shimura curves (modular curves) in the 1970's, leading to the classification of rational torsion points on all elliptic curves. Our goal is to do this for certain Shimura curves, with applications towards rational torsion points on abelian surfaces and Jacobians of certain curves of genus 2 and 3.
2) Study the statistics of Selmer groups and rational points on elliptic curves.
Selmer groups are algebraic tools used as a proxy for rational points on elliptic curves. Our ultimate goal would be to prove the conjecture of Katz-Sarnak stating that half of all elliptic curves over the rational numbers have no interesting rational points and the other half have exactly one independent point of infinite order. Toward this, we hope to prove that the 2-Selmer groups of elliptic curves are distributed according to a certain random variable described by Poonen and Rains. We also hope to resolve Hilbert's tenth problem over number fields (on the undecideability of solving polynomial equations in general).
3) Study the Ceresa cycles of genus 3 Jacobians.
Abelian varieties are one of the most well-studied algebraic varieties. Playing a prominent role is a class of "special" abelian varieties, those with "complex multiplication" which have extra symmetries (e.g. the elliptic curves x^3 + y^3 + z^3 = 0 or y^2 = x^4 + y^4). These are the abelian varieties that behave most like number fields. On the one hand, they are easier to understand because of their extra symmetries, but they are dense in the moduli space and crucial for understanding abelian varieties in general. What is the notion of "special" for algebraic curves C (of a given genus g)? We formulate things in terms of the motive h(C). Indeed, whereas a curve of genus g > 1 cannot be an algebraic group, we can ask whether the motive h(C) looks like the motive of a group. More formally, we can ask whether there exists a Chow-Kunneth decomposition of h(C) that respects the intersection pairing. This is "group like" since if A is any abelian variety then h(A) has a canonical such decomposition. It turns out that h(C) has such a decomposition precisely if the Ceresa cycle [C] - [-C] is torsion in the Chow group of the Jacobian J = Jac(C). Thus curves with torsion Ceresa cycle are "special" in that their motive is group-like. Which curves have torsion Ceresa cycle? In the last few years, interesting examples were discovered beyond the well-known case of hyperelliptic curves y^2 = f(x). So the interesting question is whether we can find all of them and characterize them in a simple way. The first non-trivial case is degree 4 (genus 3) plane curves f(x,y) = 0. For example, we have shown that the curves y^3 = x^4 + ax^2 + bx + c have torsion Ceresa cycle modulo algebraic equivalence. I expect something analogous to Schneider's theorem on the transcendence of j-invariants: for algebraic t in the upper half plane, j(t) is transcendental if and only if t is a degree 2 irrationality. However, before we can begin to prove an analogous theorem in genus 3, we must understand (even conjecturally) what is playing the role of the degree 2 irrationalities in Schneider's theorem.
1) We have not yet achieved our overarching goal but we (in work with Laga) have completely resolved our intermediate goal of finding the torsion points on the Jacobians of the curves y^3 = x^4 + ax^2 + b. In subsequent work with Laga, Schembri, and Voight we give nearly optimal constraints on the rational torsion subgroup of *any* abelian surface over Q with (potential) quaternionic multiplication.
2) We have made substantial progress in our approach to the Poonen-Rains heuristics for p = 2 but this is ongoing and we have not released a pre-print yet. Our method allows us to deduce that for any fixed number field F, and for all but at most one quadratic extension K/F, there exists an elliptic curve E/F with positive rank and the same rank over K. Unfortunately, because of the one possible exception, this is not enough to deduce negative solution to Hilbert’s tenth problem over rings of integers. Thus, we found a different way to prove that result (also using Selmer groups, but of odd degree isogenies of CM hyperelliptic Jacbians) in joint work with Alpoge, Bhargava, Ho. There, we prove a variant of the preceding statement with “elliptic curves” replaced by “abelian varieties”, and where no exceptions are allowed.
3) We made much substantial progress in understanding the torsion properties of Ceresa cycles, especially in our two works with Laga (one appearing in Crelle's journal, the other still under review). Namely, we have given many interesting new examples of torsion Ceresa cycles and have shown that their orders in the Chow group are unbounded. A full characterization of torsion Ceresa cycles is still an ambitious goal and will require more work; it is now an even more interesting question, given the exotic examples we've found. Interestingly, these two papers came out of our previous work related to Project 1), which we did not realize would end up being related to the Ceresa cycles of Project 3). In separate work with Lilienfeldt, we have proved a Gross-Zagier type formula for generalized Heegner cycles, which has applications to the motive of Ceresa cycles and their higher dimensional analogues. In the future we hope to drop various assumptions imposed as well as to prove results of this type in a wider context beyond Jacobians of curves with a specific type of complex multiplication.
2) We have made substantial progress in our approach to the Poonen-Rains heuristics for p = 2 but this is ongoing and we have not released a pre-print yet. Our method allows us to deduce that for any fixed number field F, and for all but at most one quadratic extension K/F, there exists an elliptic curve E/F with positive rank and the same rank over K. Unfortunately, because of the one possible exception, this is not enough to deduce negative solution to Hilbert’s tenth problem over rings of integers. Thus, we found a different way to prove that result (also using Selmer groups, but of odd degree isogenies of CM hyperelliptic Jacbians) in joint work with Alpoge, Bhargava, Ho. There, we prove a variant of the preceding statement with “elliptic curves” replaced by “abelian varieties”, and where no exceptions are allowed.
3) We made much substantial progress in understanding the torsion properties of Ceresa cycles, especially in our two works with Laga (one appearing in Crelle's journal, the other still under review). Namely, we have given many interesting new examples of torsion Ceresa cycles and have shown that their orders in the Chow group are unbounded. A full characterization of torsion Ceresa cycles is still an ambitious goal and will require more work; it is now an even more interesting question, given the exotic examples we've found. Interestingly, these two papers came out of our previous work related to Project 1), which we did not realize would end up being related to the Ceresa cycles of Project 3). In separate work with Lilienfeldt, we have proved a Gross-Zagier type formula for generalized Heegner cycles, which has applications to the motive of Ceresa cycles and their higher dimensional analogues. In the future we hope to drop various assumptions imposed as well as to prove results of this type in a wider context beyond Jacobians of curves with a specific type of complex multiplication.
1) We have given the first interesting examples of torsion classification of abelian varieties parameterized by a Shimura variety, beyond the case of elliptic curves.
2) We have completely resolved an open problem (Hilbert's 10th problem over rings of integers of number fields) that mathematicians have worked on since the 1980’s and introduces new methods to solve similar problems.
3) Our work with Laga epresents a big leap in our field’s understanding of the vanishing locus of the Ceresa cycle, which is a key and interesting question in the field of algebraic cycles.
2) We have completely resolved an open problem (Hilbert's 10th problem over rings of integers of number fields) that mathematicians have worked on since the 1980’s and introduces new methods to solve similar problems.
3) Our work with Laga epresents a big leap in our field’s understanding of the vanishing locus of the Ceresa cycle, which is a key and interesting question in the field of algebraic cycles.