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Tautological Algebra of the Moduli Space of Semistable Bundles on an Elliptic Curve
In this paper, our aim is to find the relations amongst the cohomology classes of Brill-Noether subvarieties of the moduli space of semistable...
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The Torsion Subgroup of a Family of Elliptic Curves Over the Maximal Abelian Extension of ℚ
We determine explicitly the structure of the torsion group over the maximal abelian extension of ℚ and over the maximal p -cyclotomic extensions of ℚ...
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Pairing-Based Cryptography on Elliptic Curves
We give a brief overview of a recent branch of Public Key Cryptography, the so called Pairing-based Cryptography or Identity-based Cryptography. We...
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Torsion Groups of a Family of Elliptic Curves Over Number Fields
We compute the torsion group explicitly over quadratic fields and number fields of degree coprime to 6 for a family of elliptic curves of the form E : y
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Avoiding Side-Channel Attacks by Computing Isogenous and Isomorphic Elliptic Curves
Smart cards are being attacked increasingly more, due to their numerous uses and the valuable information stored inside. For this reason, efficient...
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Flat Curves
We characterize the sextic trefoil among plane curves of low degree: first as a complex curve with compact, flat geometry; and then as a curve with...
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The Poncelet Theorems in Interpretation of Rafaɫ Koɫodziej
We present some results of R. Koɫodziej related with Poncelet’s theorems and several proofs of the Great Poncelet Theorem.
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Ramification in the division fields of elliptic curves with potential supersingular reduction
Let d ≥1 be fixed. Let F be a number field of degree d , and let E / F be an elliptic curve. Let E ( F ) tors be the torsion subgroup of E ( F ). In 1996, Merel...
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Non-commutative digit expansions for arithmetic on supersingular elliptic curves
We study non-commutative digit expansions in quaternion rings that arise in the context of supersingular elliptic curves. These digit expansions can...
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On Zagier’s adele
PurposeDon Zagier suggested a natural construction, which associates a real number and p -adic numbers for all primes p to the cusp form g = Δ of...
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The projective translation equation and unramified 2-dimensional flows with rational vector fields
Let x = ( x , y ). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1− z ) ϕ (x) = ϕ ( ϕ (x z )(1− z )/...
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On some polynomial values of repdigit numbers
We study the equal values of repdigit numbers and the k -dimensional polygonal numbers. We state some effective finiteness theorems, and for small...
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Noncommutative Blowups of Elliptic Algebras
We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some...
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Tate conjecture for products of Fermat varieties over finite fields
We prove under some assumptions that the Tate conjecture holds for products of Fermat varieties of different degrees. The method is to use a...
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The Trefoil
Dixon’s elliptic functions parameterize the real sextic trefoil curve by arc length and the complex curve as an embedded Platonic surface with 18 (or...
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Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization
In this paper, we present several methods for the construction of elliptic curves with large torsion group and positive rank over number fields of...
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Hasse invariants for the Clausen elliptic curves
Gauss’s