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82
INTEGRATION OVER THE uPLANE IN DONALDSON THEORY
, 1997
"... We analyze the uplane contribution to Donaldson invariants of a fourmanifold X. For b + 2(X)> 1, this contribution vanishes, but for b + 2 =1, the Donaldson invariants must be written as the sum of a uplane integral and an SW contribution. The uplane integrals are quite intricate, but can be a ..."
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Cited by 62 (2 self)
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We analyze the uplane contribution to Donaldson invariants of a fourmanifold X. For b + 2(X)> 1, this contribution vanishes, but for b + 2 =1, the Donaldson invariants must be written as the sum of a uplane integral and an SW contribution. The uplane integrals are quite intricate, but can be analyzed in great detail and even calculated. By analyzing the uplane integrals, the relation of Donaldson theory to N = 2 supersymmetric YangMills theory can be described much more fully, the relation of Donaldson invariants to SW theory can be generalized to fourmanifolds not of simple type, and interesting formulas can be obtained for the class numbers of imaginary quadratic fields. We also show how the results generalize to extensions of Donaldson theory obtained by including hypermultiplet matter fields.
Geometry of the moduli of higher spin curves
 Internat. J. of Math
"... Abstract. This article treats various aspects of the geometry of the moduli S 1/r g of rspin curves and its compactification S 1/r g. Generalized spin curves, or rspin curves, are a natural generalization of 2spin curves (algebraic curves with a thetacharacteristic), and have been of interest la ..."
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Cited by 37 (5 self)
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Abstract. This article treats various aspects of the geometry of the moduli S 1/r g of rspin curves and its compactification S 1/r g. Generalized spin curves, or rspin curves, are a natural generalization of 2spin curves (algebraic curves with a thetacharacteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the GelfandDikii (KdVr) heirarchy. There is also a Walgebra conjecture for these spaces [16] generalizing the the Virasoro conjecture of quantum cohomology. For any line bundle K on the universal curve over the stack of stable curves, there is a smooth stack S 1/r g,n(K) of triples (X, L, b) of a smooth curve X, a line bundle L on X, and an isomorphism b: L⊗r ✲ K. In the special case that K = ω is the relative dualizing sheaf, then S 1/r g,n(K) is the stack S 1/r g,n of rspin curves. We construct a smooth compactification S 1/r g,n (K) of the stack S1/r g,n(K), describe the geometric meaning of its points, and prove that it is projective. We also prove that when r is odd and g> 1, the compactified stack of spin curves S 1/r g and its coarse moduli space S 1/r g are irreducible, and when r is even and g> 1, S 1/r g is the disjoint union of two irreducible components. We give similar results for npointed spin curves, as required for Witten’s conjecture, and also generalize to the npointed case the classical fact that when g = 1, S 1/r 1 is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important [15], and also in the study of the in the study of the Picard group of S 1/r
Computing Hilbert class polynomials with the Chinese Remainder Theorem
, 2010
"... We present a spaceefficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(D  1/2+ɛ log P) space and has an expected running time of O ..."
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Cited by 35 (3 self)
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We present a spaceefficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(D  1/2+ɛ log P) space and has an expected running time of O(D  1+ɛ). We describe practical optimizations that allow us to handle larger discriminants than other methods, with D  as large as 1013 and h(D) up to 106. We apply these results to construct pairingfriendly elliptic curves of prime order, using the CM method.
Computing the endomorphism ring of an ordinary elliptic curve over a finite field
 Journal of Number Theory
"... Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the se ..."
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Cited by 27 (12 self)
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Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second algorithm depends primarily on log DE, where DE is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed. 1.
Finiteness results for modular curves of genus at least 2
 Amer. J. Math
, 2005
"... Abstract. A curve X over Q is modular if it is dominated by X1(N) for some N; if in addition the image of its jacobian in J1(N) is contained in the new subvariety of J1(N), then X is called a new modular curve. We prove that for each g ≥ 2, the set of new modular curves over Q of genus g is finite a ..."
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Cited by 21 (6 self)
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Abstract. A curve X over Q is modular if it is dominated by X1(N) for some N; if in addition the image of its jacobian in J1(N) is contained in the new subvariety of J1(N), then X is called a new modular curve. We prove that for each g ≥ 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de FranchisSeveri Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of J0(N) new with N divisible by a prescribed prime, and for modular curves (new or not) with levels in a restricted set. We study new modular hyperelliptic curves in detail. In particular, we find all new modular curves of genus 2 explicitly, and construct what might be the complete list of all new modular hyperelliptic curves of all genera. Finally we prove that for each field k of characteristic zero and g ≥ 2, the set of genusg curves over k dominated by a Fermat curve is finite and computable. 1. Introduction. Let X1(N) be the usual modular curve over Q; see Section 3.1 for a definition. (All curves and varieties in this paper are smooth, projective, and geometrically integral, unless otherwise specified. When we write an affine equation for a curve, its smooth projective model is implied.) A curve X
Torsion anomalous points and families of elliptic curves
 C. R. Acad. Sci. Paris Ser. I
"... Abstract. We prove that there are at most finitely many complex λ ̸ = 0, 1 such that two points on the Legendre elliptic curve Y 2 = X(X − 1)(X − λ) with coordinates X = 2, 3 both have finite order. This is a very special case of some conjectures on unlikely intersections in semiabelian schemes. ..."
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Cited by 21 (1 self)
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Abstract. We prove that there are at most finitely many complex λ ̸ = 0, 1 such that two points on the Legendre elliptic curve Y 2 = X(X − 1)(X − λ) with coordinates X = 2, 3 both have finite order. This is a very special case of some conjectures on unlikely intersections in semiabelian schemes.
Euler’s concordant forms
 Acta Arith
, 1996
"... In [6] Euler asks for a classification of those pairs of distinct nonzero integers M and N for which there are integer solutions (x, y, t, z) with xy ̸ = 0 to (1) x 2 + My 2 = t 2 and x 2 + Ny 2 = z 2. This is known as Euler’s concordant forms problem, and when M = −N Euler’s problem ..."
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Cited by 18 (1 self)
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In [6] Euler asks for a classification of those pairs of distinct nonzero integers M and N for which there are integer solutions (x, y, t, z) with xy ̸ = 0 to (1) x 2 + My 2 = t 2 and x 2 + Ny 2 = z 2. This is known as Euler’s concordant forms problem, and when M = −N Euler’s problem
The rank of elliptic curves over real quadratic number fields of class number 1
 Math. Comp
, 1995
"... Abstract. In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and SwinnertonDyer for curves over Q. Several examples are included. 1. ..."
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Cited by 16 (4 self)
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Abstract. In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and SwinnertonDyer for curves over Q. Several examples are included. 1.
ON MORDELLWEIL GROUPS OF ELLIPTIC CURVES INDUCED BY DIOPHANTINE TRIPLES
"... Dedicated to Professor Sibe Mardeˇsić on the occasion of his 80th birthday Abstract. We study the possible structure of the groups of rational points on elliptic curves of the form y 2 = (ax + 1)(bx + 1)(cx + 1), where a, b, c are nonzero rationals such that the product of any two of them is one le ..."
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Cited by 14 (12 self)
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Dedicated to Professor Sibe Mardeˇsić on the occasion of his 80th birthday Abstract. We study the possible structure of the groups of rational points on elliptic curves of the form y 2 = (ax + 1)(bx + 1)(cx + 1), where a, b, c are nonzero rationals such that the product of any two of them is one less than a square. 1.
LandauSiegel zeroes and black hole entropy
, 1999
"... There has been some speculation about relations of Dbrane models of black holes to arithmetic. In this note we point out that some of these speculations have implications for a circle of questions related to the generalized Riemann hypothesis on the zeroes of Dirichlet Lfunctions. ..."
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Cited by 12 (5 self)
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There has been some speculation about relations of Dbrane models of black holes to arithmetic. In this note we point out that some of these speculations have implications for a circle of questions related to the generalized Riemann hypothesis on the zeroes of Dirichlet Lfunctions.