arXiv subjects
Publications and source records attributed to Imin Chen.
We recall and define various kinds of supersingular $\ell$-isogeny graphs and precise graph isomorphism with a corresponding quaternion $\ell$-ideal graph. In particular, we introduce the notion of double-orientations on supersingular elliptic curves and study the structure of double-oriented supersingular $\ell$-isogeny graphs.
We solve the Fermat-type equation \[ x^{13} + y^{13} = 3 z^7, \qquad \gcd(x,y,z) = 1 \] combining a unit sieve, the multi-Frey modular method, level raising, computations of systems of eigenvalues modulo 7 over a totally real field, and results for reducibility of certain Galois representations.
In 2000, Darmon introduced the notion of Frey representations within the framework of the modular method for studying the generalized Fermat equation. A central step in this program is the computation of their conductors, with the case at the prime $2$ presenting particular challenges. In this article we study the conductor exponent at $2$ for Frey representations of signatures $(p,p,r)$, $(r,r,p)$, $(2,r,p)$, and $(3,5,p)$, all of which have hyperelliptic realizations. In particular we are able to determine the conductor at $2$ for even degree Frey representations of signature $(p,p,r)$ and $(3,5,p)$ and all rational parameters.
We give an overview of Darmon's program for resolving families of generalized Fermat equations with one varying exponent and survey what is currently known about this approach based on recent work of Billerey-Chen-Dieulefait-Freitas and Chen-Koutsianas. Additionally, we provide background material which is helpful to understand and apply the methods developed in these recent works. In particular, we explain the basic strategy for and simplified examples of each of the steps that is required in order to resolve a family of generalized Fermat equations.
In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of $\mathrm{GL}_2$-type over totally real fields. The original approach was based on hard open conjectures, which have made it difficult to apply in practice. In this paper, building on the progress surrounding the modular method from the last two decades, we analyze and expand the current limits of this program by developing all the necessary ingredients to use Frey abelian varieties for new Diophantine applications. In particular, we deal with all but the fifth and last step in the modular method for Fermat equations of signature $(r,r,p)$ in almost full generality. As an application, for all integers $n \geq 2$, we give a resolution of the generalized Fermat equation $x^{11} + y^{11} = z^n$ for solutions $(a,b,c)$ such that $a + b$ satisfies certain $2$- or $11$-adic conditions. Moreover, the tools developed can be viewed as an advance in addressing a difficulty not treated in Darmon's original program: even assuming `big image' conjectures about residual Galois representations, one still needs to find a method to eliminate Hilbert newforms at the Serre level which do not have complex multiplication. In fact, we are able to reduce the problem of solving $x^5 + y^5 = z^p$ to Darmon's `big image conjecture', thus completing a line of ideas suggested in his original program, and notably only needing the Cartan case of his conjecture.
In this paper we carry out the steps of Darmon's program for the generalized Fermat equation $$ x^n + y^n = z^5. $$ In particular, we develop the machinery necessary to prove an optimal bound on the exponent $n$ for solutions satisfying certain $2$-adic and $5$-adic conditions which are natural from the point of view of the method. We also reduce the problem of resolving this equation to a `big image conjecture', completing a line of ideas suggested in his original program. The above equation is an example of a generalized Fermat equation for which the predicted Frey abelian varieties have dimension $ > 1$ and thus it represents an interesting test case for Darmon's program.
Let $E$ be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\mathbb{Q}$ has open image, and in particular there is a minimal natural number $C_E$ such that the mod $\ell$ representation $\barρ_{E,\ell}$ is surjective for any prime $\ell > C_E$. Assuming the Generalized Riemann Hypothesis, Mayle-Wang gave explicit bounds for $C_E$ which are logarithmic in the conductor of $E$ and have explicit constants. The method is based on using effective forms of the Chebotarev density theorem together with the Faltings-Serre method, in particular, using the `deviation group' of the $2$-adic representations attached to two elliptic curves. By considering quotients of the deviation group and a characterization of the images of the $2$-adic representation $ρ_{E,2}$ by Rouse and Zureick-Brown, we show in this paper how to further reduce the constants in Mayle-Wang's results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.
In this paper, we give a resolution of the generalized Fermat equations $$x^5 + y^5 = 3 z^n \text{ and } x^{13} + y^{13} = 3 z^n,$$ for all integers $n \ge 2$, and all integers $n \ge 2$ which are not a multiple of $7$, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents $n$. We also give a number of results for the equations $x^5 + y^5 = d z^n$, where $d = 1, 2$, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat's Last Theorem, and which uses a new application of level raising at $p$ modulo $p$.
Motivated by their central role in fault-tolerant quantum computation, we study the sets of gates of the third-level of the Clifford hierarchy and their distinguished subsets of `nearly diagonal' semi-Clifford gates. The Clifford hierarchy gates can be implemented via gate teleportation given appropriate magic states. The vast quantity of these resource states required for achieving fault-tolerance is a significant bottleneck for the practical realisation of universal quantum computers. Semi-Clifford gates are important because they can be implemented with far more efficient use of these resource states. We prove that every third-level gate of up to two qudits is semi-Clifford. We thus generalise results of Zeng-Chen-Chuang (2008) in the qubit case and of the second author (2020) in the qutrit case to the case of qudits of arbitrary prime dimension $d$. Earlier results relied on exhaustive computations whereas our present work leverages tools of algebraic geometry. Specifically, we construct two schemes corresponding to the sets of third-level Clifford hierarchy gates and third-level semi-Clifford gates. We then show that the two algebraic sets resulting from reducing these schemes modulo $d$ share the same set of rational points.
We obtain additional Diophantine applications of the methods surrounding Darmon's program for the generalized Fermat equation developed in the first part of this series of papers. As a first application, we use a multi-Frey approach combining two Frey elliptic curves over totally real fields, a Frey hyperelliptic curve over $\mathbb{Q}$ due to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equation $$x^7 + y^7 = 3 z^n$$ for all integers $n \ge 2$. Moreover, we explain how the use of higher dimensional Frey abelian varieties allows a more efficient proof of this result due to additional structures that they afford, compared to using only Frey elliptic curves. As a second application, we use some of these additional structures that Frey abelian varieties possess to show that a full resolution of the generalized Fermat equation $x^7 + y^7 = z^n$ depends only on the Cartan case of Darmon's big image conjecture. In the process, we solve the previous equation for solutions $(a,b,c)$ such that $a$ and $b$ satisfy certain $2$ or $7$-adic conditions and all $n \ge 2$.
We give an exposition of the hidden subgroup problem for dihedral groups from the point of view of the standard hidden subgroup quantum algorithm for finite groups. In particular, we recall the obstructions for strong Fourier sampling to succeed, but at the same time, show how the standard algorithm can be modified to establish polynomial quantum query complexity. Finally, we explain a new connection between the dihedral coset problem and cloning of quantum states.
We prove Fermat's Last Theorem over ${\mathbb Q}(\sqrt{5})$ and ${\mathbb Q}(\sqrt{17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer-Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of ${\mathbb Q}(\sqrt{5})$ is a generalization to a real quadratic base field of the one used by Chen-Siksek. For the case of ${\mathbb Q}(\sqrt{17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett-Chen-Dahmen-Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.
We develop a uniform method to derive Chudnovsky-Ramanujan type formulae for triangle groups based on a generalization of a method of Chudnovsky and Chudnovsky; in particular, we carry out the method systematically for non-compact arithmetic triangle groups and one non-Fuchsian covering. As a result, we derive all rational Ramanujan type series given by Chan-Cooper for levels 1-4, as well as two additional rational series of a similar form prescribed by Chan-Cooper for these levels, but not found in the paper of Chan-Cooper. These two additional series were first found by Z.-W. Sun in a slightly different form. We also derive additional rational series of a similar form, but not found in the papers of Chan-Cooper nor Z.-W. Sun. As an ingredient in the method, we give an algorithm to rigorously confirm the singular values of normalized Eisenstein series of weight 2, which may be of independent interest.
In this paper, we recall an alternative proof of Merel's conjecture which asserts that a certain explicit correspondence gives the isogeny relation between the Jacobians associated to the normalizer of split and non-split Cartan subgroups. This alternative proof does not require extensive representation theory and can be formulated in terms of certain finite geometries modulo $\ell$. Secondly, we generalize these arguments to exhibit an explicit correspondence which gives the isogeny relation between the Jacobians associated to split and non-split Cartan subgroups. An interesting feature is that the required explicit correspondence is considerably more complicated but can expressed as a certain linear combination of double coset operators whose coefficients we are able to make explicit.
We apply the method established in our previous work to derive a Chudnovsky-Ramanujan type formula for the Legendre family of elliptic curves. As a result, we prove two identities for $1/π$ in terms of hypergeometric functions.
In a well-known 1914 paper, Ramanujan gave a number of rapidly converging series for $1/π$ which are derived using modular functions of higher level. D. V. and G. V. Chudnovsky in their 1988 paper derived an analogous series representing $1/π$ using the modular function $J$ of level 1, which results in highly convergent series for $1/π$, often used in practice. In this paper, we explain the Chudnovsky method in the context of elliptic curves, modular curves, and the Picard-Fuchs differential equation. In doing so, we also generalize their method to produce formulae which are valid around any singular point of the Picard-Fuchs differential equation. Applying the method to the family of elliptic curves parameterized by the absolute Klein invariant $J$ of level 1, we determine all Chudnovsky-Ramanujan type formulae which are valid around one of the three singular points: $0, 1, \infty$.
We prove a diophantine result on generalized Fermat equations of the form $x^p + y^p = z^r$ which for the first time requires the use of Frey abelian varieties of dimension $\geq 2$ in Darmon's program. For that, we provide an irreducibility criterion for the mod $\mathfrak{p}$ representations attached to certain abelian varieties of $\text{GL}_2$-type over totally real fields.
We consider the classical theta operator $θ$ on modular forms modulo $p^m$ and level $N$ prime to $p$ where $p$ is a prime greater than 3. Our main result is that $θ$ mod $p^m$ will map forms of weight $k$ to forms of weight $k+2+2p^{m-1}(p-1)$ and that this weight is optimal in certain cases when $m$ is at least 2. Thus, the natural expectation that $θ$ mod $p^m$ should map to weight $k+2+p^{m-1}(p-1)$ is shown to be false. The primary motivation for this study is that application of the $θ$ operator on eigenforms mod $p^m$ corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the $θ$-operator mod $p^m$ gives an explicit weight bound on the twist of a modular mod $p^m$ Galois representation by the cyclotomic character.