Imin Chen
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Imin Chen has a total of 23 co-authors. Here are the top co-authors by number of shared publications:
- Luis Dieulefait
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain
9 shared publications - Nuno Freitas
Instituto de Ciencias Matemáticas, CSIC, Calle Nicolás Cabrera 13–15, 28049 Madrid, Spain
9 shared publications - Nicolas Billerey
Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne et CNRS, Campus universitaire des Cézeaux, 3, place Vasarely, 63178 Aubière Cedex, France
7 shared publications - Gleb Glebov
Simon Fraser University
3 shared publications David Sun
Simon Fraser Univ Burnaby, Dept Math, Burnaby, BC V5A 1S6, Canada
2 shared publications- Joshua Swidinsky
Simon Fraser University
2 shared publications
Latest Publications
- Open Access
- 27 October 2025
- Published by MDPI AG in Preprints.org
Cognitive Grounding for Visual Question Reasoning via Dynamic Knowledge Imagination
Visual question answering (VQA) represents a critical intersection of vision and language understanding, where models must perceive visual scenes and reason about their underlying semantics. However, human-like reasoning often extends beyond what is directly observable—requiring the invocation of prior knowledge, inference, and commonsense understanding. In this work, we reexamine the nature of external knowledge in multimodal reasoning and propose a unified framework named \textbf{Cognitive Grounded Imagination Network (COGINet)} that integrates dynamically generated commonsense imagination with vision-language understanding. Instead of merely retrieving symbolic facts from static knowledge bases, our framework leverages contextualized commonsense imagination synthesized through large-scale generative knowledge models, thereby grounding dynamic reasoning on both perceptual evidence and inferred world regularities. COGINet introduces a two-stage process: (1) a knowledge imagination module, which generates plausible contextual hypotheses from the interaction between visual regions and textual queries; and (2) a cross-modal reasoning transformer that fuses these contextualized inferences with multimodal embeddings through adaptive attention. We demonstrate that such cognitive grounding enables the model to reason about abstract or implied concepts, extending beyond explicit cues in the image or text. Extensive experiments conducted on OK-VQA and A-OKVQA benchmarks show that COGINet achieves consistent improvements over prior static-knowledge methods, providing both quantitative gains and qualitative interpretability. Further analysis reveals the model’s ability to discern when external knowledge is useful, selectively invoking imagined context only when required for reasoning. Our findings highlight the importance of dynamic, cognitively inspired commonsense integration for achieving genuine multimodal understanding.
- 27 October 2025
- Published by MDPI AG in Preprints.org
Open AccessCognitive Grounding for Visual Question Reasoning via Dynamic Knowledge Imagination
Visual question answering (VQA) represents a critical intersection of vision and language understanding, where models must perceive visual scenes and reason about their underlying semantics. However, human-like reasoning often extends beyond what is directly observable—requiring the invocation of prior knowledge, inference, and commonsense understanding. In this work, we reexamine the nature of external knowledge in multimodal reasoning and propose a unified framework named \textbf{Cognitive Grounded Imagination Network (COGINet)} that integrates dynamically generated commonsense imagination with vision-language understanding. Instead of merely retrieving symbolic facts from static knowledge bases, our framework leverages contextualized commonsense imagination synthesized through large-scale generative knowledge models, thereby grounding dynamic reasoning on both perceptual evidence and inferred world regularities. COGINet introduces a two-stage process: (1) a knowledge imagination module, which generates plausible contextual hypotheses from the interaction between visual regions and textual queries; and (2) a cross-modal reasoning transformer that fuses these contextualized inferences with multimodal embeddings through adaptive attention. We demonstrate that such cognitive grounding enables the model to reason about abstract or implied concepts, extending beyond explicit cues in the image or text. Extensive experiments conducted on OK-VQA and A-OKVQA benchmarks show that COGINet achieves consistent improvements over prior static-knowledge methods, providing both quantitative gains and qualitative interpretability. Further analysis reveals the model’s ability to discern when external knowledge is useful, selectively invoking imagined context only when required for reasoning. Our findings highlight the importance of dynamic, cognitively inspired commonsense integration for achieving genuine multimodal understanding.
- Open Access
Double-orientations on supersingular isogeny graphs
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.
Open AccessDouble-orientations on supersingular isogeny graphs
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.
- Open AccessRevisiting the Fermat-type equation $x^{13} + y^{13} = 3z^7$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.Open AccessRevisiting the Fermat-type equation $x^{13} + y^{13} = 3z^7$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.
- Open AccessA note on conductors of Frey representations at $2$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.Open AccessA note on conductors of Frey representations at $2$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.
- Open Access
- 10 September 2025
- Published by Canadian Mathematical Society in Canadian Mathematical Bulletin
- Vol. 69 (1) , 273-291
Improved bounds for Serre’s open image theorem
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 ${\bar {\rho }}_{E,\ell }$ 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 $\rho _{E,2}$ by Rouse and Zureick–Brown, we show in this article 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.- 10 September 2025
- Published by Canadian Mathematical Society in Canadian Mathematical Bulletin
- Vol. 69 (1) , 273-291
Open AccessImproved bounds for Serre’s open image theorem
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 ${\bar {\rho }}_{E,\ell }$ 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 $\rho _{E,2}$ by Rouse and Zureick–Brown, we show in this article 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. - 25 March 2025
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
On Darmon’s program for the generalized Fermat equation, I
In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of -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. As an application, for all integers , we give a resolution of the generalized Fermat equation for solutions such that satisfies certain 2- or 11-adic conditions. We are also able to reduce the problem of solving to a weaker version of 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.- 25 March 2025
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
On Darmon’s program for the generalized Fermat equation, I
In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of -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. As an application, for all integers , we give a resolution of the generalized Fermat equation for solutions such that satisfies certain 2- or 11-adic conditions. We are also able to reduce the problem of solving to a weaker version of 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.- Open Access
Improved bounds for Serre's open image theorem
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.Open AccessImproved bounds for Serre's open image theorem
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. - 6 September 2024
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 94 (354) , 1977-2003
On Darmon’s program for the generalized Fermat equation, II
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 overdue to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equation for all integers . 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 depends only on the Cartan case of Darmon’s big image conjecture. In the process, we solve the previous equation for solutions such that and satisfy certain - or -adic conditions and all . - 6 September 2024
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 94 (354) , 1977-2003
On Darmon’s program for the generalized Fermat equation, II
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 overdue to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equation for all integers . 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 depends only on the Cartan case of Darmon’s big image conjecture. In the process, we solve the previous equation for solutions such that and satisfy certain - or -adic conditions and all . - Open Access
On Darmon's program for the Generalized Fermat equation, II
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 $n \ge 2$.Open AccessOn Darmon's program for the Generalized Fermat equation, II
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 $n \ge 2$. - Open Access
On Darmon's program for the generalized Fermat equation, I
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.Open AccessOn Darmon's program for the generalized Fermat equation, I
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.