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40
Modular elliptic curves and Fermat’s Last Theorem
 ANNALS OF MATH
, 1995
"... When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n> 2 such that a n + b n = c n. T ..."
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Cited by 390 (0 self)
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When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n> 2 such that a n + b n = c n. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.
Complexity Limitations on Quantum Computation
 Journal of Computer and System Sciences
, 1997
"... We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP.  BQP is low for PP, i.e., PP BQP = PP.  There exists a relativized world where P = ..."
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Cited by 98 (3 self)
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We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP.  BQP is low for PP, i.e., PP BQP = PP.  There exists a relativized world where P = BQP and the polynomialtime hierarchy is infinite.  There exists a relativized world where BQP does not have complete sets.  There exists a relativized world where P = BQP but P 6= UP " coUP and oneway functions exist. This gives a relativized answer to an open question of Simon.
An oracle builder’s toolkit
, 2002
"... We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and ..."
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Cited by 47 (10 self)
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We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
Quantum computing, postselection, and probabilistic polynomialtime
, 2004
"... I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic PolynomialTime. Using this result, I show that several simple ..."
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Cited by 39 (12 self)
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I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic PolynomialTime. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PPcomplete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.
Direct and longrange action of a DPP morphogen gradient. Cell
, 1996
"... Postprint available at: ..."
One Complexity Theorist's View of Quantum Computing
 THEORETICAL COMPUTER SCIENCE
, 2000
"... The complexity of quantum computation remains poorly understood. While physicists attempt to find ways to create quantum computers, we still do not have much evidence one way or the other as to how useful these machines will be. The tools of computational complexity theory should come to bear on ..."
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Cited by 22 (0 self)
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The complexity of quantum computation remains poorly understood. While physicists attempt to find ways to create quantum computers, we still do not have much evidence one way or the other as to how useful these machines will be. The tools of computational complexity theory should come to bear on these important questions. Quantum Computing
The cubic moment of central values of automorphic Lfunctions
 Ann. of Math
, 2000
"... 2. A review of classical modular forms 3. A review of Maass forms 4. Hecke Lfunctions ..."
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Cited by 22 (1 self)
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2. A review of classical modular forms 3. A review of Maass forms 4. Hecke Lfunctions
RankinSelberg Lfunctions in the level aspect
 Duke Math. J
"... In this paper we calculate the asymptotics of various moments of the central values of RankinSelberg convolution Lfunctions of large level, thus generalizing the results and methods of W. Duke, J. Friedlander, and H. Iwaniec and of the authors. Consequences include convexitybreaking bounds, nonva ..."
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Cited by 19 (6 self)
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In this paper we calculate the asymptotics of various moments of the central values of RankinSelberg convolution Lfunctions of large level, thus generalizing the results and methods of W. Duke, J. Friedlander, and H. Iwaniec and of the authors. Consequences include convexitybreaking bounds, nonvanishing of a positive proportion of central values, and linear independence results for certain Hecke operators. Contents
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 17 (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
Dynamical dimension change in supercritical string theory,” hepth/0409071
"... We study treelevel solutions to heterotic string theory in which the number of spacetime dimensions 10+n changes dynamically due to closed string tachyon condensation. Taking the largen limit, we compute the amount by which the dilaton gradient Φ,X 0 changes during the condensation process. At lea ..."
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Cited by 15 (5 self)
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We study treelevel solutions to heterotic string theory in which the number of spacetime dimensions 10+n changes dynamically due to closed string tachyon condensation. Taking the largen limit, we compute the amount by which the dilaton gradient Φ,X 0 changes during the condensation process. At leading order in n we find that the change in the invariant magnitude of Φ,X 0 compensates the change in spacelike central charge on the worldsheet, so that the total worldsheet central charges are equal at early and late times. This result supports our interpretation of the worldsheet theory as a CFT of critical central charge which describes a dynamical reduction in the total number of spacetime dimensions. Contents 1