On the Satisfiability of Modular Arithmetic Formulae

菲茨杰拉德的《了不起的盖茨比》，摘录书中比较经典的句子：Chapter 11. 每当你觉得想要批评什么人的时候，你切要记着，这个世界上的人并非都具备你禀有的条件。

Whenever you feel like criticizing any one, just remember that all the people in this world haven’t had the advantages that you’ve had.2.人们的善恶感一生下来就有差异。

A sense of the fundamental decencies is parceled out unequally at birth.3.人们的品行有的好像建筑在坚硬的岩石上，有的好像建筑在泥沼里，不过超过一定的限度，我就不在乎它建在什么之上了。

Conduct may be founded on the hard rock or the wet marshes, but after a certain point I don’t care what it’s founded on.Chapter 2这时，天色已经暗了下来，我们这排高高地俯瞰着城市的灯火通明的窗户，一定让街头偶尔抬头眺望的人感到了，人类的秘密也有其一份在这里吧，我也是这样的一个过路人，举头望着诧异着。

我既在事内又在事外，几杯永无枯竭的五彩纷呈的生活所吸引，同时又被其排斥着。

Yet high over the city our line of yellow windows must have contributed their share of human secrecy to the casual watcher in the darkening streets, and I was him too, looking up and wondering. I was within and without, simultaneously enchanted and repelled by the inexhaustible variety of life.Chapter 31. 他理解体谅地笑了——这笑比理解和体谅有更多的含义。

When writing an essay in English about My Model,its important to consider the context in which the term model is being used.Here are a few different approaches you might take,depending on the specific meaning of model in your essay:1.A Role Model:Begin by introducing who your role model is and why they are important to you. Discuss the qualities and achievements of your role model that you admire. Explain how their actions or life story has influenced your own life or goals.Example Paragraph:My role model is Malala Yousafzai,a Pakistani activist for female education and the youngest Nobel Prize laureate.Her courage and determination to fight for girls education rights in the face of adversity have deeply inspired me.Malalas story has taught me the importance of standing up for what I believe in,even when it is difficult.2.A Fashion Model:Describe the physical attributes and style of the model.Discuss the impact they have had on the fashion industry or their unique contributions to it.Explain why you find their work or presence in the industry notable.Example Paragraph:Kendall Jenner is a fashion model who has made a significant impact on the industry with her unique style and presence.Her tall and slender physique,combined with her ability to carry off diverse looks,has made her a favorite among designers and fashion enthusiasts alike.I admire her for her versatility and the way she uses her platform to promote body positivity.3.A Model in Science or Technology:Introduce the model as a theoretical framework or a practical tool used in a specific field.Explain the principles behind the model and how it is applied.Discuss the benefits or limitations of the model and its implications in the real world.Example Paragraph:The Standard Model in physics is a theoretical framework that describes three of the four known fundamental forces excluding gravity and classifies all known elementary particles.It has been instrumental in understanding the behavior of subatomic particles and predicting the existence of new particles,such as the Higgs boson.However,the models inability to incorporate gravity or dark matter has led to ongoing research for amore comprehensive theory.4.A Model in Business or Economics:Introduce the business or economic model and its purpose.Explain how the model works and the strategies it employs.Discuss the success or challenges associated with the model and its potential for future growth.Example Paragraph:The subscriptionbased business model has become increasingly popular in recent years, particularly in the software panies like Adobe have transitioned from selling packaged software to offering services on a subscription basis,allowing for continuous revenue streams and a more predictable income.This model has been successful in fostering customer loyalty and providing a steady income,although it requires ongoing innovation to maintain customer interest.5.A Model in Art or Design:Describe the aesthetic or functional qualities of the model.Discuss the creative process or design principles that inform the model.Explain the cultural or historical significance of the model and its influence on contemporary art or design.Example Paragraph:The Eames Lounge Chair,designed by Charles and Ray Eames,is a model of modern furniture that has become an icon of midcentury design.Its elegant form,made from molded plywood and leather,exemplifies the designers commitment to blending comfort with aesthetics.The chairs timeless appeal has made it a staple in both residential and commercial settings,influencing countless furniture designs that followed. Remember to structure your essay with a clear introduction,body paragraphs that develop your points,and a conclusion that summarizes your main e specific examples and evidence to support your claims,and ensure your writing is clear,concise, and engaging.。

门捷列夫英语作文In the realm of scientific discovery, Dmitri Mendeleev's contribution stands as a monumental milestone. His innovative approach to organizing elements into a coherent system revolutionized the field of chemistry.Mendeleev's enduring legacy, the Periodic Table, was not just a compilation of facts but a testament to his analytical prowess and foresight. He meticulously arranged the elements, predicting their properties and even the existence of undiscovered ones.The Periodic Table, a simple yet profound tool, has guided generations of scientists. It is a universal language that transcends borders, connecting researchers in their quest for understanding the fundamental building blocks of our universe.Despite facing skepticism and resistance, Mendeleev's unwavering belief in his work paved the way for its eventual acceptance. His perseverance is a lesson in the importance of conviction in the face of doubt.Today, the Periodic Table is not just a staple in chemistry classrooms but a symbol of scientific progress. It serves as a reminder that the pursuit of knowledge is an ongoing journey, with each discovery a stepping stone to the next.Mendeleev's story is one of curiosity, ingenuity, and the relentless pursuit of truth. His work has left an indelible mark on the scientific community, inspiring countless minds to delve deeper into the mysteries of the natural world.In conclusion, Dmitri Mendeleev's life and work exemplify the spirit of scientific inquiry. His Periodic Table is more than just an educational tool; it is a beacon of human ingenuity, guiding us toward a deeper understanding of the elements that constitute our world.。

涌现优于权威英文原文"Emergence Trumps Authority"In today's rapidly changing and interconnected world, the concept of emergence is gaining increasing attention as a more effective way to tackle complex problems and drive innovation. Emergence refers to the phenomenon where new and unexpected patterns, properties, or behaviors emerge from the interactions of simpler elements within a system. This stands in stark contrast to the traditional top-down approach of authority, where decisions and solutions are handed down from a single source of power.The main advantage of emergence over authority is its ability to harness the collective intelligence and creativity of a group. Instead of relying on the expertise of a few individuals at the top, emergence draws on the diverse perspectives and experiences of many. This leads to more robust and innovative solutions, as well as greater buy-in and support from those involved in the process.Furthermore, emergence is better suited to navigate the complexities and uncertainties of modern challenges. With the pace of change accelerating and the interdependencies of various systems becoming more evident, no single authority figure can possibly possess all the knowledge and insight needed to address the diverse and evolving issues we face. In contrast, emergence allows for a more organic and adaptive approach, where solutions can emerge and evolve over time as new information and perspectives come to light.Additionally, emergence encourages participation and empowerment, as individuals feel a sense of ownership and responsibility for the outcomes of the collective efforts. This can lead to increased motivation, collaboration, and resilience within the group, as well as a greater sense of satisfaction and fulfillment for all involved.While authority certainly has its time and place, especially in situations requiring clear direction and decisive action, the benefits of emergence cannot be overlooked. By recognizing and harnessing the power of emergence, organizations, communities, and individuals can better adapt to the complexities and uncertainties of our modern world and drive more effective and sustainable solutions. Ultimately, "Emergence Trumps Authority."。

a r X i v :m a t h /0211394v 2 [m a t h .N T ] 22 D e c 2003FINITENESS RESULTS FOR MODULAR CURVES OF GENUS ATLEAST 2MATTHEW H.BAKER,ENRIQUE GONZ ´ALEZ-JIM ´ENEZ,JOSEP GONZ ´ALEZ,AND BJORN POONEN Abstract.A curve X over Q is modular if it is dominated by X 1(N )for some N ;if in addition the image of its jacobian in J 1(N )is contained in the new subvariety of J 1(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 Franchis-Severi Theorem.Similar finiteness results are proved for new modular curves of bounded gonality,for new modular curves whose jacobian is a quotient of J 0(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 2explicitly,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 genus-g curves over k dominated by a Fermat curve is finite and computable.1.Introduction Let X 1(N )be the usual modular curve over Q .(See Section 3.1for a definition.)A curve 1X over Q will be called modular if there exists a nonconstant morphism π:X 1(N )→X over Q .If X is modular,then X (Q )is nonempty,since it contains the image of the cusp ∞∈X 1(N )(Q ).The converse,namely that if X (Q )is nonempty then X is modular,holds if the genus g of X satisfies g ≤1[9].In particular,there are infinitely many modular curves over Q of genus 1.On the other hand,we propose the following:Conjecture 1.1.For each g ≥2,the set of modular curves over Q of genus g is finite.Remark 1.2.(i)When we speak of the finiteness of the set of curves over Q satisfying some condition,we mean the finiteness of the set of Q -isomorphism classes of such curves.(ii)For any fixed N ,the de Franchis-Severi Theorem (see Theorem 5.5)implies thefiniteness of the set of curves over Q dominated by X 1(N ).Conjecture 1.1can be2BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENthought of as a version that is uniform as one ascends the tower of modular curves X1(N),provided that onefixes the genus of the dominated curve.(iii)Conjecture1.1is true if one restricts the statement to quotients of X1(N)by sub-groups of its group of modular automorphisms.See Remark3.16for details.(iv)If X1(N)dominates a curve X,then the jacobian Jac X is a quotient2of J1(N):= Jac X1(N).The converse,namely that if X is a curve such that X(Q)is nonempty and Jac X is a quotient of J1(N)then X is dominated by X1(N),holds if the genusg of X is≤1,but can fail for g≥2.See Section8.2for other“pathologies.”(v)In contrast with Conjecture1.1,there exist infinitely many genus-two curves over Q whose jacobians are quotients of J1(N)for some N.See Proposition8.2(5).(vi)In Section9,we use a result of Aoki[3]to prove an analogue of Conjecture1.1in which X1(N)is replaced by the Fermat curve x N+y N=z N in P2.In fact,such an analogue can be proved over arbitraryfields of characteristic zero,not just Q.We prove many results towards Conjecture1.1in this paper.Given a variety X over afield k,letΩ=Ω1X/k denote the sheaf of regular1-forms.Call a modular curve X over Q newof level N if there exists a nonconstant morphismπ:X1(N)→X(defined over Q)such that π∗H0(X,Ω)is contained in the new subspace H0(X1(N),Ω)new,or equivalently if the image of the homomorphismπ∗:Jac X→J1(N)induced by Picard functoriality is contained in the new subvariety J1(N)new of J1(N).(See Section3.1for the definitions of H0(X1(N),Ω)new, J1(N)new,J1(N)new,and so on.)For example,it is known that every elliptic curve E over Q is a new modular curve of level N,where N is the conductor of E.Here the conductor cond(A)of an abelian variety A over Q is a positive integer p p f p,where each exponent f p is defined in terms of the action of an inertia subgroup of Gal(2Quotients or subvarieties of varieties,and morphisms between varieties,are implicitly assumed to be defined over the samefield as the original varieties.If X is a curve over Q,and we wish to discuss automor-phisms over C,for example,we will write Aut(X C).Quotients of abelian varieties are assumed to be abelian variety quotients.MODULAR CURVES OF GENUS AT LEAST23 If we drop the assumption that our modular curves are new,we can still prove results,but (so far)only if we impose restrictions on the level.Given m>0,let Sparse m denote the set of positive integers N such that if1=d1<d2<···<d t=N are the positive divisors of N, then d i+1/d i>m for i=1,...,t−1.Define a function B(g)on integers g≥2by B(2)=13, B(3)=17,B(4)=21,and B(g)=6g−5for g≥5.(For the origin of this function,see the proofs of Propositions2.1and2.8.)A positive integer N is called m-smooth if all primes p dividing N satisfy p≤m.Let Smooth m denote the set of m-smooth integers.Theorem1.5.Fix g≥2,and let S be a subset of{1,2,...}.The set of modular curves over Q of genus g and of level contained in S isfinite if any of the following hold:(i)S=Sparse B(g).(ii)S=Smooth m for some m>0.(iii)S is the set of prime powers.Remark1.6.Since Sparse B(g)∪Smooth B(g)contains all prime powers,parts(i)and(ii)of Theorem1.5imply(iii).Remark1.7.In contrast with Theorem1.3,we do not know,even in theory,how to compute thefinite sets of curves in Theorem1.5.The reason for this will be explained in Remark5.10. If X is a curve over afield k,and L is afield extension of k,let X L denote X×k L.The gonality G of a curve X over Q is the smallest possible degree of a nonconstant morphism X C→P1C.(There is also the notion of Q-gonality,where one only allows morphisms over Q.By defining gonality using morphisms over C instead of Q,we make the next theorem stronger.)In Section4.3,we combine Theorem1.3with a known lower bound on the gonality of X1(N)to prove the following:Theorem1.8.For each G≥2,the set of new modular curves over Q of genus at least2 and gonality at most G isfinite and computable.(We could similarly prove an analogue of Theorem1.5for curves of bounded gonality instead offixed genus.)Recall that a curve X of genus g over afield k is called hyperelliptic if g≥2and the canonical map X→P g−1is not a closed immersion:equivalently,g≥2and there exists a degree-2morphism X→Y where Y has genus zero.If moreover X(k)=∅then Y≃P1k, and if also k is not of characteristic2,then X is birational to a curve of the form y2=f(x) where f is a separable polynomial in k[x]of degree2g+1or2g+2.Recall that X(k)=∅is automatic if X is modular,because of the cusp∞.Taking G=2in Theorem1.8,wefind that the set of new modular hyperelliptic curves over Q isfinite and computable.We can say more:Theorem1.9.Let X be a new modular hyperelliptic curve over Q of genus g≥3and level N.Then(i)g≤16.(ii)If Jac X is a quotient of J0(N),then g≤10.If moreover3|N,then X is the genus-3 curve X0(39).(iii)If Jac X is not a quotient of J0(N),then either g is even or g≤9.Further information is given in Sections6.3and6.5,and in the appendix.See Section3.1 for the definitions of X0(N)and J0(N).4BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENAs we have already remarked,if we consider all genera g≥2together,there are infinitely many new modular curves.To obtainfiniteness results,so far we have needed to restrict either the genus or the gonality.The following theorem,proved in Section7,gives a different type of restriction that again impliesfiniteness.Theorem1.10.For each prime p,the set of new modular curves over Q of genus at least2 whose jacobian is a quotient of J0(N)new for some N divisible by p isfinite and computable. Question1.11.Does Theorem1.10remain true if J0(N)new is replaced by J1(N)new? Call a curve X over afield k of characteristic zero k-modular if there exists a nonconstant morphism X1(N)k→X(over k).Question1.12.Is it true that for everyfield k of characteristic zero,and every g≥2,the set of k-modular curves over k of genus g up to k-isomorphism isfinite?Remark 1.13.If X is a k-modular curve over k,and we define k0=k∩k,then X=X0×k0k for some k0-modular curve X0.This follows from the de Franchis-Severi Theorem.Remark1.14.If k and k′arefields of characteristic zero with[k′:k]finite,then a positive answer to Question1.12for k′implies a positive answer for k,since Galois cohomology and thefiniteness of automorphism group of curves of genus at least2show that for each X′over k′,there are at mostfinitely many curves X over k with X×k k′≃X′.But it is not clear, for instance,that a positive answer forMODULAR CURVES OF GENUS AT LEAST25 Corollary 2.4.Let X be a curve of genus g≥2over afield k of characteristic zero. Then the image X′of the canonical map X→P g−1is the common zero locus of the set ofhomogeneous polynomials of degree4that vanish on X′.Proof.We may assume that k is algebraically closed.If X is hyperelliptic of genus g,saybirational to y2=f(x)where f has distinct roots,then we may choose{x i dx/y:0≤i≤g−1}as basis of H0(X,Ω),and then the image of the canonical map is the rational normalcurve cut out by{t i t j−t i′t j′:i+j=i′+j′}where t0,...,t g−1are the homogeneous coordinates on P g−1.If X is nonhyperelliptic of genus3,its canonical model is a planequartic.In all other cases,we use Petri’s Theorem.(The zero locus of a homogeneouspolynomial h of degree d<4equals the zero locus of the set of homogeneous polynomialsof degree4that are multiples of h.) Lemma2.5.Let X be a hyperelliptic curve of genus g over afield k of characteristic zero, and suppose P∈X(k).Let{ω1,...,ωg}be a basis of H0(X,Ω)such that ord P(ω1)<···< ord P(ωg).Then x:=ωg−1/ωg and y:=dx/ωg generate the functionfield k(X),and there is a unique polynomial F(x)of degree at most2g+2such that y2=F(x).Moreover,F is squarefree.If P is a Weierstrass point,then deg F=2g+1and ord P(ωi)=2i−2for all i;otherwise deg F=2g+2and ord P(ωi)=i−1for all i.Finally,it is possible to replace eachωi by a linear combination ofωi,ωi+1,...,ωg to makeωi=x g−i dx/y for1≤i≤g. Proof.This follows easily from Lemma3.6.1,Corollary3.6.3,and Theorem3.6.4of[21]. Proof of Proposition2.1.Suppose that X,P,q,and the w i are as in the statement of the proposition.Letωi be the corresponding elements of H0(X,Ω).We will show that X is determined by the w i when B=max{8g−7,6g+1}.Since B>8g−8,Lemma2.2implies that the w i determine the set of homogeneouspolynomial relations of degree4satisfied by theωi,so by Corollary2.4the w i determine theimage X′of the canonical map.In particular,the w i determine whether X is hyperelliptic,and they determine X if X is nonhyperelliptic.Therefore it remains to consider the case where X is hyperelliptic.Applying Gaussianelimination to the w i,we may assume0=ord q(w1)<···<ord q(w g)≤2g−2and that thefirst nonzero coefficient of each w i is1.We use Lemma2.5repeatedly in what follows.Thevalue of ord(w2)determines whether P is a Weierstrass point.Suppose that P is a Weierstrass point.Then w i=q2i−2(1+···+O(q B−2i+2)),where each“···”here and in the rest of this proof represents some known linear combination of positive powers of q up to but not including the power in the big-O term.(“Known”means “determined by the original w i.”)Define x=w g−1/w g=q−2(1+···+O(q B−2g+2)).Define y=dx/(w g dq)=−2q−(2g+1)(1+···+O(q B−2g+2)).Then y2=4q−(4g+2)(1+···+O(q B−2g+2)).Since B≥6g+1,we have −(4g+2)+(B−2g+2)>0,so there is a unique polynomial F(of degree2g+1)such that y2=F(x).A similar calculation shows that in the case where P is not a Weierstrass point,thenB≥3g+2is enough. Remark2.6.Let us show that if the hypotheses of Proposition2.1are satisfied except that thew i belong tok ,Ω),then the conclusion still holds.Let E be afinite Galois extension of k containing all the coefficients of the w i.The E-span of the w i must be stable under Gal(E/k)if they come6BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENfrom a curve over Q,and in this case,we can replace the w i by a k-rational basis of this span.Then Proposition2.1applies.Remark2.7.We can generalize Proposition2.1to the case where q is not a uniformizing parameter on X:Fix an integer g≥2,and let k be afield of characteristic zero.Let B>0be the integer appearing in the statement of Proposition2.1,and let e bea positive integer.Then if w1,...,w g are elements of k[[q]]/(q eB),then upto k-isomorphism,there exists at most one curve X over k such that thereexist P∈X(k),an analytic uniformizing parameter q′∈ˆO X,P and a relationq′=c e q e+c e+1q e+1+...with c e=0,such that w1dq,...,w g dq are theexpansions modulo q eB of some basis of H0(X,Ω).The proof of this statement is similar to the proof of Proposition2.1,and is left to the reader. The rest of this section is concerned with quantitative improvements to Proposition2.1, and is not needed for the generalfiniteness and computability results of Sections4and5. Proposition2.8.Proposition2.1holds with B=B(g),where B(2)=13,B(3)=17, B(4)=21,and B(g)=6g−5for g≥5.Moreover,if we are given that the curve X to be recovered is hyperelliptic,then we can use B(g)=4g+5or B(g)=2g+4,according as P is a Weierstrass point or not.Proof.For nonhyperelliptic curves of genus g≥4,we use Theorem2.3instead of Corol-lary2.4to see that B>6g−6can be used in place of B>8g−8.Now suppose that X is hyperelliptic.As before,assume ord q(w1)<···<ord q(w g)and that thefirst nonzero coefficient of each w i is1.The value of ord(w2)determines whether P is a Weierstrass point.Suppose that P is a Weierstrass point.Then w i=q2i−2(1+···+O(q B−2i+2)).(As in the proof of Proposition2.1,···means a linear combination of positive powers of q,whose coefficients are determined by the w i.)Define˜x=w g−1/w g=q−2(1+···+O(q B−2g+2)).For 1≤i≤g−2,the expression˜x g−i w g=q2i−2(1+···+O(q B−2g+2))is the initial expansion of w i+ g j=i+1c ij w j for some c ij∈k,and all the c ij are determined if2+(B−2g+2)>2g−2, that is,if B≥4g−5.Let w′i=w i+ g j=i+1c ij w j=q2i−2(1+···+O(q B−2i+2)).Define x=w′1/w′2=q−2(1+···+O(q B−2)).Define y=−2q−(2g+1)(1+···+O(q B−2))as the solution to w′1dq=x g−1dx/y.Then y2=4q−(4g+2)(1+···+O(q B−2)),and if−(4g+2)+B−2>0, we can recover the polynomial F of degree2g+1such that y2=F(x).Hence B≥4g+5 suffices.A similar proof shows that B≥2g+4suffices in the case that P is not a Weierstrass point.Hence max{6g−5,4g+5,2g+4}suffices for all types of curves,except that the6g−5 should be8g−7when g=3.This is the function B(g). Remark2.9.We show here that for each g≥2,the bound B=4g+5for the precision needed to recover a hyperelliptic curve is sharp.Let F(x)∈C[x]be a monic polynomial of degree2g+1such that X:y2=F(x)and X′:y2=F(x)+1are curves of genus g that are not birationally equivalent.Let q be the uniformizing parameter at the point at infinity on X such that x=q−2and y=q−(2g+1)+O(q−2g).Define q′similarly for X′.A calculation shows that the q-expansions of the differentials x i dx/y for0≤i≤g−1are even power series in q times dq,and modulo q4g+4dq they agree with the corresponding q′-expansionsMODULAR CURVES OF GENUS AT LEAST27 for X′except for the coefficient of q4g+2dq in x g−1dx/y.By a change of analytic parameter q=Q+αQ4g+3for someα∈C,on X only,we can make even that coefficient agree.A similar proof shows that in the case that P is not a Weierstrass point,the bound2g+4 cannot be improved.Remark2.10.When studying new modular curves of genus g,we can also use the multi-plicativity of Fourier coefficients of modular forms(see(3.7))to determine some coefficients from earlier ones.Hence we can sometimes get away with less than B(g)coefficients of each modular form.2.2.Descending morphisms.The next result will be used a number of times throughout this paper.In particular,it will be an important ingredient in the proof of Theorem1.9. Proposition2.11.Let X,Y,Z be curves over afield k of characteristic zero,and assume that the genus of Y is>1.Then:(i)Given nonconstant morphismsπ:X→Z andφ:X→Y such thatφ∗H0(Y,Ω)⊆π∗H0(Z,Ω),there exists a nonconstant morphism u′:Z→Y making the diagramXπuY u′Ycommute.Proof.The conclusion of(i)is equivalent to the inclusionφ∗k(Y)⊆π∗k(Z).It suffices to prove that every function inφ∗k(Y)is expressible as a ratio of pullbacks of meromorphic differentials on Z.If Y is nonhyperelliptic,then thefield k(Y)is generated by ratios of pairs of differentials in H0(Y,Ω),so the inclusion follows from the hypothesisφ∗H0(Y,Ω)⊆π∗H0(Z,Ω).When Y is hyperelliptic,we must modify this argument slightly.We have k(Y)=k(x,y),where y2=F(x)for some polynomial F(U)in k[U]without double roots. Thefield generated by ratios of differentials in H0(Y,Ω)is k(x),soφ∗k(x)⊆π∗k(Z).To show thatφ∗y∈π∗k(Z)too,write y=x dx/(x dx/y)and observe that x dx/y∈H0(Y,Ω). Now we prove(ii).The hypothesis on u∗lets us apply(i)withφ=π◦u to construct u′:Y→Y.Since Y has genus>1and k has characteristic zero,the Hurwitz formula implies that u′is an automorphism.Considering functionfields proves uniqueness. Remark2.12.Both parts of Proposition2.11can fail if the genus of Y is1.On the other hand,(ii)remains true under the additional assumption that X→Y is optimal in the sense that it does not factor nontrivally through any other genus1curve.Remark2.13.Proposition2.11remains true if k hasfinite characteristic,provided that one assumes that the morphisms are separable.8BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONEN3.Some facts about modular curves3.1.Basic facts about X1(N).We record facts about X1(N)that we will need for the proof offiniteness in Theorem1.3.See[55]for a detailed introduction.Let H={z∈C:Im z>0}.The group SL2(R)acts on H by a b c d z=az+bq for some g-dimensional C-subspaceS2(N)of q C[[q]].We will not define modular forms in general here,but S2(N)is known as the space of weight2cusp forms onΓ1(N).If M|N and d|NM .Similarly define the old subvariety J1(N)old of J1(N).The space S2(N)has a hermitian inner product called the Petersson inner product.Let S2(N)new denote the orthogonal complement to S2(N)old in S2(N).The identifications above also give us new and old subspaces of H0(X1(N)C,Ω)and H0(J1(N)C,Ω).Let J1(N)new=J1(N)/J1(N)old. There is also an abelian subvariety J1(N)new of J1(N)that can be characterized in two ways: either as the abelian subvariety such thatker(H0(J1(N)C,Ω)→H0((J1(N)new)C,Ω))=H0(J1(N)C,Ω)old,or as the abelian subvariety such that J1(N)=J1(N)old+J1(N)new with J1(N)old∩J1(N)new finite.(The latter description uniquely characterizes J1(N)new because of a theorem that no Q-simple quotient of J1(N)old is isogenous to a Q-simple quotient of J1(N)new;this theorem can be proved by comparing conductors,using[11].)The abelian varieties J1(N)new and J1(N)new are Q-isogenous.We define X0(N),J0(N),and J0(N)new similarly,starting withΓ0(N):= a b c d ∈SL2(Z) c≡0(mod N)instead ofΓ1(N).For n≥1,there exist well-known correspondences T n on X1(N),and they induce en-domorphisms of S2(N)and of J1(N)known as Hecke operators,also denoted T n.There is a unique basis New N of S2(N)new consisting of f=a1q+a2q2+a3q3+...such that a1=1and T n f=a n f for all n≥1.The elements of New N are called the newforms ofMODULAR CURVES OF GENUS AT LEAST29level N.(For us,newforms are always normalized:this means that a1=1.)Each a n is√an algebraic integer,bounded byσ0(n)k/k).The Galois group G Q acts on New N.For any quotient A of J1(N),let S2(A)denote the image of H0(A C,Ω)→H0(J1(N)C,Ω)≃S2(N)(the last isomorphism drops the dq/q);similarly for any nonconstant morphismπ:X1(N)→X of curves,define S2(X):=π∗H0(X C,Ω)q(3.1)Q-isogenyf→A f.Shimura proved that J1(N)is isogenous to a product of these A f,and K.Ribet[53]proved that A f is Q-simple.This explains the surjectivity of(3.1).The injectivity is well-known to experts,but we could notfind a suitable reference,so we will prove it,as part of Proposi-tion3.2.The subfield E f=Q(a2,a3,...)of C is a numberfield,and dim A f=[E f:Q].Moreover, End(A f)⊗Q can be canonically identified with E f,and under this identification the element λ∈End A f acts on f as multiplication byλ(considered as element of E f),and on each Galois conjugateσf by multiplication byσλ.(Shimura[57,Theorem1]constructed an injection End(A f)⊗Q֒→E f,and Ribet[53,Corollary4.2]proved that it was an isomorphism.)If A and B are abelian varieties over Q,let A Q∼B denote the statement that A and B are isogenous over Q.Proposition3.2.Suppose f∈New N and f′∈New N′.Then A f Q∼A f′if and only if N=N′and f=τf′for someτ∈G Q.Proof(K.Ribet).The“if”part is immediate from Shimura’s construction.Therefore it suffices to show that one can recover f,up to Galois conjugacy,from the isogeny class of A f. Letℓbe a prime.Let V be the Qℓ-Tate module Vℓ(A f)attached to A f.Let Qℓ. The proof of Proposition4.1of[53]shows thatQℓ[G Q]-module of dimension2over Qℓ.Moreover,for p∤ℓN,the trace of the p-Frobenius automorphism acting on Vσequalsσ(a p),where a p∈E f is the coefficient of q p in the Fourier expansion of f.If f′∈New′N is another weight2newform,and A f Q∼A f′,then(using′in the obvious way to denote objects associated with f′),we have isomorphisms of G Q-modules V≃V′and V′.Fixσ:E f֒→V′,whereσ′is some embedding E f′֒→3Earlier,in Theorem7.14of[55],Shimura had attached to f an abelian subvariety of J1(N).10BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENWe have parallel decompositionsS2(N)new= f∈G Q\New N τ:E f֒→C CτfJ1(N)new Q∼ f∈G Q\New N A fand parallel decompositionsS2(N)= M|N f∈G Q\New M d|Nc d f(q d)Mfor some M|N and f∈New M,where c d∈E f depends on f and d.Proof.By multiplying the quotient map J1(N)→A by a positive integer,we may assume that it factors through the isogenyJ1(N)→ M|N f∈G Q\New M A n f fof(3.4).We may also assume that A is Q-simple,and even that A is isomorphic to A fwith for some f,so that the quotient map J1(N)→A is the composition of J1(N)→A n ffa homomorphism A n ff→A.The latter is given by an n f-tuple c=(c d)of elements of End(A f),indexed by the divisors d of N/M.Underc→A,X1(N)֒→J1(N)→A n ffthe1-form on A C≃(A f)C corresponding to f pulls back to d|Nc d f(q d)Mfor some M|N and f∈New M,where c d∈E f depends on f and d.Proof.Apply Lemma3.5to the Albanese homomorphism J1(N)→Jac X.3.2.Automorphisms of X1(N).3.2.1.Diamonds.The action on H of a matrix a b c d ∈Γ0(N)induces an automorphism of X1(N)over Q depending only on(d mod N).This automorphism is called the diamond operator d .It induces an automorphism of S2(N).Letεbe a Dirichlet character modulo N,that is,a homomorphism(Z/N Z)∗→C∗.Let S2(N,ε)be the C-vector space{h∈S2(N):h| d =ε(d)h}.A form h∈S2(N,ε)is called a form of Nebentypusε.Every newform f∈New N is a form of some Nebentypus,and is therefore an eigenvector for all the diamond operators.Character theory gives a decompositionS2(N)= εS2(N,ε),whereεruns over all Dirichlet characters modulo N.Define S2(N,ε)new=S2(N,ε)∩S2(N)new.When we writeε=1,we mean thatεis the trivial Dirichlet character modulo N,that is,ε(n)= 1if(n,N)=10otherwise.A form of Nebentypusε=1is a form onΓ0(N).We recall some properties of a newform f= ∞n=1a n q n∈S2(N,ε).Let condεdenote the smallest integer M|N such thatεis a composition(Z/N Z)∗→(Z/M Z)∗→C∗. Throughout this paragraph,p denotes a prime,and(3.11)pv p(N)=1,ε=1=⇒f|W p=−a p f(3.12)(3.13)p∤N=⇒ε(p)a p.The equivalence(3.8)is trivial.For the remaining properties see[39]and[5].3.2.2.Involutions.For every integer M|N such that(M,N/M)=1,there is an automor-phism W M of X1(N)C inducing an isomorphism between S2(N,ε)new and S2(N,εM⊗f in S2(N,εM(p)a p if p∤M,a p if p|M.Then(3.14)f|W M=λM(f)(εN/M(−M)f,f|(W M′W M)=q.It is known thatS2(Γ(N,ε))=nk=1S2(N,εk),where n is the order of the Dirichlet characterε.The diamond operators and the Weil involution induce automorphisms of X(N,ε)C.If moreoverε=1,the curve X(N,1)is X0(N)and the automorphisms W M on X1(N)C induce involutions on X0(N)over Q that are usually called the Atkin-Lehner involutions.Remark 3.16.Define the modular automorphism group of X1(N)to be the subgroup of Aut(X1(N)Q ,which we continue to denote by d and W N respectively.Throughout the paper,D will denote the abelian subgroup of Aut(XQ )generated by Dand W N.If moreover X is hyperelliptic,and w is its hyperelliptic involution,then the group generated by D and W N.w will be denoted by D′N.Note that D is a subgroup of Aut(X),and the groups D N,D′N are G Q-stable by(3.15).If Jac X Q∼A f for some f with nontrivial Nebentypus,then D N is isomorphic to the dihedral group with2n elements,D2·n,where n is the order of the Nebentypus of f.For every nonconstant morphismπ:X1(N)→X of curves over Q such that S2(X)⊆⊕n i=1S2(N,εi)for some Nebentypusεof order n,there exists a nonconstant morphism π(ε):X(N,ε)→X over Q.This is clear when the genus of X is≤1,and follows from Proposition2.11(i)if the genus of X is>1.In particular,for a new modular curve X of genus>1,there exists a surjective morphism X0(N)→X if and only if D is the trivial group.More generally,we have the following result.Lemma3.17.Let X be a new modular curve of level N,and let G be a G Q-stable subgroup of Aut(XG:={φ∈Aut(X):φ∗ω=ωfor allω∈H0(X,Ω)G}.ThenG.Now supposeφ∈G⊆G as required.3.3.Supersingular points.We will use a lemma about curves with good reduction. Lemma3.19.Let R be a discrete valuation ring with fractionfield K.Suppose f:X→Y is afinite morphism of smooth,projective,geometrically integral curves over K,and X extends to a smooth projective model X over R(in this case we say that X has good reduction).If Y has genus≥1,then Y extends to a smooth projective model Y over R,and f extends to afinite morphism X→Y over R.Proof.This result is Corollary4.10in[40].See the discussion there also for references to earlier weaker versions. The next two lemmas are well-known(to coding theorists,for example),but we could not find explicit references,so we supply proofs.Lemma3.20.Let p be a prime.LetΓ⊆SL2(Z)denote a congruence subgroup of level N not divisible by p.Let XΓbe the corresponding integral smooth projective curve overF p-points on the reduction mapping to supersingular points on X(1)is at least(p−1)ψ/12.Proof.By[33],the curve X(N)admits a smooth model over Z[1/N],and has a rationalpoint(the cusp∞).Since p∤N and X(N)dominates XΓ,Lemma3.19implies that XΓhas good reduction at p,at least if XΓhas genus≥1.If XΓhas genus0,then the rational point on X(N)gives a rational point on XΓ,so XΓ≃P1,so XΓhas good reduction at p in any case.ReplacingΓby the group generated byΓand−id does not change XΓ,so withoutloss of generality,we may assume that−id∈Γ.Thenψ=(SL2(Z):Γ).If E is an elliptic curve,thenΓnaturally acts on thefinite set of ordered symplectic bases of E[N].The curve YΓ:=XΓ−{cusps}classifies isomorphism classes of pairs(E,L),where E is an elliptic curve and L is aΓ-orbit of symplectic bases of E[N].Fix E.Since SL2(Z)acts transitively on the symplectic bases of E[N],the number of Γ-orbits of symplectic bases is(SL2(Z):Γ)=ψ.Two such orbits L and L′correspond to the same point of XΓif and only if L′=αL for someα∈Aut(E).Thenψis the sum of the sizes of the orbits of Aut(E)acting on theΓ-orbits,soψ= (E,L)∈XΓ#Aut(E)F p,we obtainsupersingular points(E,L)∈XΓ(#Aut(E,L)=ψ supersingular E/#Aut(E)=(p−1)ψF p).Thereforethe number of supersingular points on XΓmust be at least2(p−1)ψ/24=(p−1)ψ/12. Lemma3.21.Let p be a prime.Given a supersingular elliptic curve E overF pand the p2-power Frobenius endomorphism of E′equals−p.Proof.Honda-Tate theory supplies an elliptic curve E over F p such that the characteristicpolynomial of the p-power Frobenius endomorphism Frob p satisfies Frob2p=−p.All super-singular elliptic curves over Fp→E.The inseparable part of this isogeny is a power of Frob p,so without loss of generality,we mayassume thatφis separable.The kernel K ofφis preserved by−p=Frob2p,so K is defined over F p2.Take E′=E Fp2/K. The following is a generalization of inequalities used in[48].Lemma3.22.Let X be a new modular hyperelliptic curve of level N and genus g over Q. If p is a prime not dividing N,then(p−1)(g−1)<2(p2+1).Proof.We may assume g≥2.Since p∤N,Lemma3.19implies that X1(N)and X have good reduction at p,and the morphismπ:X1(N)→X induces a corresponding morphism of curves over F p.By Proposition2.11(ii),the diamond automorphism −p of X1(N) induces an automorphism of X,which we also call −p .These automorphisms induces automorphisms of the corresponding curves over F p.For the rest of this proof,X1(N),X,π, −p represent these objects over F p.Also denote by −p the induced morphism P1→P1。

Curiosity has always been a driving force in human exploration and discovery.It is the very essence of our desire to understand the world around us and beyond.As we gaze up at the night sky,the twinkling stars beckon us with their silent call,igniting the spark of curiosity within us.This essay will delve into the concept of curiosity and its role in our pursuit of the stars.The human race has been captivated by the cosmos since time immemorial.Ancient civilizations studied the heavens,mapping constellations and predicting celestial events with remarkable accuracy.This innate curiosity led to the development of astronomy,a field that has expanded our understanding of the universe and our place within it.The invention of the telescope by Galileo Galilei in the early17th century marked a significant leap in our ability to observe and study celestial bodies,further fueling our curiosity about the stars.Curiosity is not merely a passive interest it is an active pursuit of knowledge.It propels us to ask questions,to challenge existing theories,and to seek answers that may lie beyond our current understanding.This inquisitive nature has led to numerous scientific breakthroughs and technological advancements.For instance,the curiosity about the nature of stars has led to the discovery of nuclear fusion,the process that powers our sun and other stars,and has opened up the possibility of harnessing this energy for human use.The quest for knowledge about the stars has also led to the development of space exploration.From the first human spaceflight by Yuri Gagarin in1961to the recent Mars missions,our curiosity has taken us beyond our own planet and into the vast expanse of space.This exploration has not only expanded our understanding of the universe but has also provided us with a new perspective on our own planet and its place in the cosmos. Moreover,curiosity about the stars has inspired countless individuals to pursue careers in science,technology,engineering,and mathematics STEM.It has motivated generations of students to learn about the cosmos,to dream of becoming astronauts,and to contribute to our collective understanding of the universe.This curiosity has also led to the creation of numerous educational programs and initiatives aimed at fostering interest in astronomy and space exploration.In conclusion,curiosity is the catalyst that drives our exploration of the stars.It is the spark that ignites our desire to learn,to discover,and to understand the universe.As we continue to gaze up at the night sky,let us remember that it is our curiosity that will lead us to new horizons and unlock the mysteries of the cosmos.Whether through the development of new technologies,the pursuit of scientific knowledge,or the inspirationof future generations,curiosity will always be the guiding force in our journey to explore the stars.。

2024年普通高等学校招生全国统一考试北京卷英语试卷养成良好的答题习惯，是决定成败的决定性因素之一。

做题前，要认真阅读题目要求、题干和选项，并对答案内容作出合理预测;答题时，切忌跟着感觉走，最好按照题目序号来做，不会的或存在疑问的，要做好标记，要善于发现，找到题目的题眼所在，规范答题，书写工整;答题完毕时，要认真检查，查漏补缺，纠正错误。

第一部分知识运用(共两节，30分)第一节(共10小题;每小题 1. 5分，共15分)阅读下面短文，掌握其大意，从每题所给的A、B、C、D 四个选项中，选出最佳选项。

I’d just arrived at school, ready for another school day. I was reading a book in the classroom when there was an 1 . “Today at 1: 10 there will be auditions (面试) for a musical.” My friends all jumped up in excitement and asked me, “Will you be going, Amy?” “Sure,” I said. I had no 2 in drama, but I’d try out because my friends were doing it.At 1:10, there was a 3 outside the drama room. Everyone looked energetic. I hadn’t expected I’d be standing there that morning. But now that I was doing it, I 4 felt nervous. What if I wasn’t any good?I entered the room and the teachers made me say some lines from the musical. They then 5 my singing skills and asked what role I wanted to play. The teachers were smiling and praising me. I felt like I had a 6 , so I said, “A big role.” They said they’d look into it. I started getting really nervous. What if I didn’t get a main role?Soon, the cast list was 7 . My friends checked and came back shouting, “Amy, you got the main role!” Sure enough, my name was at the top. I just stared at it and started to 8 . I was so happy.After two months we were all prepared and ready to go on stage. It was fun. And when people started 9 , that gave me a boost of confidence. It stayed with me and made me feel 10 . I realised that by trying something new, I can have fun — even if it means stepping out of my comfort zone.1．A．assignment B．initiative C．announcement D．interview2．A．hesitancy B．interest C．worry D．regret3．A．game B．show C．play D．line4．A．suddenly B．continuously C．originally D．generally5．A．advertised B．tested C．challenged D．polished6．A．demand B．credit C．dream D．chance7．A．traded B．posted C．questioned D．claimed8．A．well up B．roll in C．stand out D．go off9．A．whispering B．arguing C．clapping D．stretching10．A．funnier B．fairer C．cleverer D．braver第二节(共10小题;每小题1. 5分，共15分)A阅读下列短文，根据短文内容填空。

名人演讲：国会大厦告别演讲道格拉斯·麦克阿瑟，美国陆军五星上将。

出生于阿肯色州小石城的军人世家。

1899年中学毕业后考入西点军校，1903年以名列第一的优异成绩毕业，到工程兵部队任职，并赴菲律宾执勤。

麦克阿瑟有过50年的军事实践经验，被美国国民称之为“一代老兵”，而其自身的又曾是“美国最年轻的准将、西点军校最年轻的校长、美国陆军历史上最年轻的陆军参谋长”，凭借精妙的军事谋略和敢战敢胜的胆略，麦克阿瑟堪称美国战争史上的奇才。

提起这句话：“老兵永远不死，只会慢慢凋零”（Old soldiers never die，the just fade **），就不由得想起那个叼着玉米棒子烟斗的麦克阿瑟，和他在1951年4月19日被解职后在国会大厦发表的题为《老兵不死》著名演讲。

我即将结束五十二年的军旅生涯。

我从军是在本世纪开始之前，而这是我童年的希望与梦想的实现。

自从我在西点军校的教练场上宣誓以来，这个世界已经过多次变化，而我的希望与梦想早已消逝，但我仍记着当时最流行的一首军歌词，极为自豪地宣示“老兵永远不死，只会慢慢凋零”。

I am losing m 52 ears of militar servie。

When I joined the Arm， even before the turn of the entur， it as the fulfillment of all of m boish hopes and dreams。

The orld has turned over man times sine I took the oath on the plain at West Point， and the hopes and dreams have long sine vanished， but I still remember the refrain of one of the most popular barrak ballads of that da hih prolaimed most proudl that "old soldiers never die; thejust fade **。

历年考研英语阅读真题及答案解析历年考研英语阅读真题及答案解析多做做历年来的考研英语阅读理解，让自己发现阅读的规律。

下面是店铺给大家整理的历年考研英语阅读真题及答案解析，供大家参阅!1985年考研英语阅读真题及答案解析Section III Reading ComprehensionEach sentence or passage below is followed by four statements. One of the statements is a suggestion which can be made from the information given in the original sentence or passage. Read them carefully and make your choice. Put your choice in the brackets on the left. (10 points)EXAMPLE:[A] You should get up when he comes in.[B] You should support him.[C] You shouldn't be afraid to argue with him.[D] You must be of the same height as he is.ANSWER: [B]26. Watch your step when your turn comes to have an interview with the general manager.[A] When you are asked to see the general manager, be sure not to step into his office without his permission.[B] Watch the steps when you go upstairs to see the general manager at his office.[C] Be sure to be careful when it is your turn to go to the general manager's office for an interview with him.[D] Watch out and don't step into the general manager's office until it is your turn to have an interview with him.27. Since no additional fund is available, the extension of thebuilding is out of the question.[A] The extension of the building is impossible because we are unable to get extra fund for the purpose.[B] There is some problem about the extension of the building owing to lack of fund.[C] Since no additional fund is available, we have to solve the problem regarding the extension of the building with our own resources.[D] We can undertake the extension of the building even without additional fund. It is no problem at all.28. All along he has been striving not to fall short of his parents' expectations.[A] He has been trying hard all the time to live up to what his parents expect of him.[B] His parents have been expecting him to work hard.[C] All the time he has been trying hard to balance himself so as not to fall down as his parents thought he would.[D] All the time, as his parents expect him to do, he has been trying hard to save and not to be short of money.29. The various canals which drain away the excessive water have turned this piece of land into a highly productive agricultural area.[A] The canals have been used to water the land.[B] The canals have been used to raise agricultural production.[C] Excessive water has been helpful to agricultural production.[D] The production has been mainly agricultural.30. The replacement of man by machines has not led to unemployment. On the contrary, the total numbers engaged inthe textile industry have continued to rise. The fact should not be ignored by those who maintain that unemployment and machinery are inseparable companions.[A] The belief that the use of machinery causes unemployment is unfounded.[B] The use of machinery results in a rise in production.[C] Many people lose their jobs when machines are introduced.[D] Contrary to general belief, machinery and unemployment are inseparable companions.答案解析Section III: Reading Comprehension (10 points)26.[C]27.[A]28.[A]29.[B]30.[A]1986年考研英语阅读真题及答案解析Section III Reading ComprehensionEach of the two passages below is followed by five questions. For each question there are four answers. Read the passages carefully and choose the best answer to each of the questions. Put your choice in the brackets on the left. (10 points) Text 1There are a great many careers in which the increasing emphasis is on specialization. You find these careers in engineering, in production, in statistical work, and in teaching. But there is an increasing demand for people who are able to take in great area at a glance, people who perhaps do not know too much about any one field. There is, in other words, a demand for people who are capable of seeing the forest rather than the trees, of making general judgments. We can call these people “generalists.” And these “generalists” are particularly needed for positions in administration, where it is their job to see thatother people do the work, where they have to plan for other people, to organize other people’s work, to begin it and judge it.The specialist understands one field; his concern is with technique and tools. He is a “trained” man; and his educationa l background is properly technical or professional. The generalist -- and especially the administrator -- deals with people; his concern is with leadership, with planning, and with direction giving. He is an “educated” man; and the humanities are his strongest foundation. Very rarely is a specialist capable of being an administrator. And very rarely is a good generalist also a good specialist in particular field. Any organization needs both kinds of people, though different organizations need them in different proportions. It is your task to find out, during your training period, into which of the two kinds of jobs you fit, and to plan your career accordingly.Your first job may turn out to be the right job for you -- but this is pure accident. Certainly you should not change jobs constantly or people will become suspicious of your ability to hold any job. At the same time you must not look upon the first job as the final job; it is primarily a training job, an opportunity to understand yourself and your fitness for being an employee.26. There is an increasing demand for ________.[A] all round people in their own fields[B] people whose job is to organize other people’s work[C] generalists whose educational background is either technical or professional[D] specialists whose chief concern is to provide administrative guidance to others27. The specialist is ________.[A] a man whose job is to train other people[B] a man who has been trained in more than one fields[C] a man who can see the forest rather than the trees[D] a man whose concern is mainly with technical or professional matters28. The administrator is ________.[A] a “trained” man who is more a specialist than a generalist[B] a man who sees the trees as well as the forest[C] a man who is very strong in the humanities[D] a man who is an “educated” specialist29. During your training period, it is important ________.[A] to try to be a generalist[B] to choose a profitable job[C] to find an organization which fits you[D] to decide whether you are fit to be a specialist or a generalist30. A man’s first job ________.[A] is never the right job for him[B] should not be regarded as his final job[C] should not be changed or people will become suspicious of his ability to hold any job[D] is primarily an opportunity to fit himself for his final jobText 2At the bottom of the world lies a mighty continent still wrapped in the Ice Age and, until recent times, unknown to man. It is a great land mass with mountain ranges whose extent and elevation are still uncertain. Much of the continent is a complete blank on our maps. Man has explored, on foot, less than one per cent of its area. Antarctica differs fundamentally from the Arcticregions. The Arctic is an ocean, covered with drifting packed ice and hemmed in by the land masses of Europe, Asia, and North America. The Antarctic is a continent almost as large as Europe and Australia combined, centered roughly on the South Pole and surrounded by the most unobstructed water areas of the world -- the Atlantic, Pacific, and Indian Oceans.The continental ice sheet is more than two miles high in its centre, thus, the air over the Antarctic is far more refrigerated than it is over the Arctic regions. This cold air current from the land is so forceful that it makes the nearby seas the stormiest in the world and renders unlivable those regions whose counterparts at the opposite end of the globe are inhabited. Thus, more than a million persons live within 2,000 miles of the North Pole in an area that includes most of Alaska, Siberia, and Scandinavia -- a region rich in forest and mining industries. Apart from a handful of weather stations, within the same distance of the South Pole there is not a single tree, industry, or settlement.31. The best title for this selection would be ________.[A] Iceland[B] Land of Opportunity[C] The Unknown Continent[D] Utopia at Last32. At the time this article was written, our knowledge of Antarctica was ________.[A] very limited[B] vast[C] fairly rich[D] nonexistent33. Antarctica is bordered by the ________.[A] Pacific Ocean[B] Indian Ocean[C] Atlantic Ocean[D] All three34. The Antarctic is made uninhabitable primarily by ________.[A] cold air[B] calm seas[C] ice[D] lack of knowledge about the continent35. According to this article ________.[A] 2,000 people live on the Antarctic Continent[B] a million people live within 2,000 miles of the South Pole[C] weather conditions within a 2,000 mile radius of the South Pole make settlements impractical[D] only a handful of natives inhabit Antarctica答案解析Section III: Reading Comprehension (10 points)26.[B]27.[D]28.[C]29.[D]30.[B]31.[C]32.[A]33.[D]34.[A]35.[C]1987年考研英语阅读真题及答案解析Section II Reading ComprehensionEach of three passages below is followed by five questions. For each question there are four answers, read the passages carefully and choose the best answer to each of the questions. Put your choice in the ANSWER SHEET. (15 points)Text 1For centuries men dreamed of achieving vertical flight. In 400 A.D. Chinese children played with a fan-like toy that spun upwards and fell back to earth as rotation ceased. Leonardo da Vinci conceive the first mechanical apparatus, called a “Helix,” which could carry man straight up, but was only a design and wasnever tested.The ancient-dream was finally realized in 1940 when a Russian engineer piloted a strange looking craft of steel tubing with a rotating fan on top. It rose awkwardly and vertically into the air from a standing start, hovered a few feet above the ground, went sideways and backwards, and then settled back to earth. The vehicle was called a helicopter.Imaginations were fired. Men dreamed of going to work in their own personal helicopters. People anticipate that vertical flight transports would carry millions of passengers as do the airliners of today. Such fantastic expectations were not fulfilled.The helicopter has now become an extremely useful machine. It excels in military missions, carrying troops, guns and strategic instruments where other aircraft cannot go. Corporations use them as airborne offices, many metropolitan areas use them in police work, construction and logging companies employ them in various advantageous ways, engineers use them for site selection and surveying, and oil companies use them as the best way to make offshore and remote work stations accessible to crews and supplies. Any urgent mission to a hard-to-get-to place is a likely task for a helicopter. Among their other multitude of used: deliver people across town, fly to and from airports, assist in rescue work, and aid in the search for missing or wanted persons.11. People expect that ________.[A] the airliners of today would eventually be replaced by helicopters[B] helicopters would someday be able to transport large number of people from place to place as airliners are now doing[C] the imaginations fired by the Russian engineer’sinvention would become a reality in the future[D] their fantastic expectations about helicopters could be fulfilled by airliners of today12. Helicopters work with the aid of ________.[A] a combination of rotating devices in front and on top[B] a rotating device topside[C] one rotating fan in the center of the aircraft and others at each end[D] a rotating fan underneath for lifting13. What is said about the development of the helicopter?[A] Helicopters have only been worked on by man since 1940.[B] Chinese children were the first to achieve flight in helicopters.[C] Helicopters were considered more dangerous than the early airplanes.[D] Some people thought they would become widely used by average individuals.14. How has the use of helicopters developed?[A] They have been widely used for various purposes.[B] They are taking the place of high-flying jets.[C] They are used for rescue work.[D] They are now used exclusively for commercial projects.15. Under what conditions are helicopters found to be absolutely essential?[A] For overseas passenger transportation.[B] For extremely high altitude flights.[C] For high-speed transportation.[D] For urgent mission to places inaccessible to other kinds of craft.Text 2In ancient Greece athletic festivals were very important and had strong religious associations. The Olympian athletic festival held every four years in honor of Zeus, king of the Olympian Gods, eventually lost its local character, became first a national event and then, after the rules against foreign competitors had been abolished, international. No one knows exactly how far back the Olympic Games go, but some official records date from 776 B.C. The games took place in August on the plain by Mount Olympus. Many thousands of spectators gathered from all parts of Greece, but no married woman was admitted even as a spectator. Slaves, women and dishonored persons were not allowed to compete. The exact sequence of events uncertain, but events included boy’s gymnastics, boxing, wrestling, horse racing and field events, though there were fewer sports involved than in the modern Olympic Games.On the last day of the Games, all the winners were honored by having a ring of holy olive leaves placed on their heads. So great was the honor that the winner of the foot race gave his name to the year of his victory. Although Olympic winners received no prize money, they were, in fact, richly rewarded by their state authorities. How their results compared with modern standards, we unfortunately have no means of telling.After an uninterrupted history of almost 1,200 years, the Games were suspended by the Romans in 394 A.D. They continued for such a long time because people believed in the philosophy behind the Olympics: the idea that a healthy body produced a healthy mind, and that the spirit of competition in sports and games was preferable to the competition that caused wars. It was over 1,500 years before another such international athletic gathering took place in Athens in 1896.Nowadays, the Games are held in different countries in turn. The host country provides vast facilities, including a stadium, swimming pools and living accommodation, but competing courtiers pay their own athletes’ expenses.The Olympics start with the arrival in the stadium of a torch, lighted on Mount Olympus by the sun’s rays. It is carried by a succession of runners to the stadium. The torch symbolized the continuation of the ancient Greek athletic ideals, and it burns throughout the Games until the closing ceremony. The well-known Olympic flag, however, is a modern conception: the five interlocking rings symbolize the uniting of all five continents participating in the Games.16. In ancient Greece, the Olympic Games ________.[A] were merely national athletic festivals[B] were in the nature of a national event with a strong religious colour[C] had rules which put foreign participants in a disadvantageous position[D] were primarily national events with few foreign participants17. In the early days of ancient Olympic Games ________.[A] only male Greek athletes were allowed to participate in the games[B] all Greeks, irrespective of sex, religion or social status, were allowed to take part[C] all Greeks, with the exception of women, were allowed to compete in Games[D] all male Greeks were qualified to compete in the Games18. The order of athletic events at the ancient Olympics ________.[A] has not definitely been established[B] varied according to the number of foreign competitors[C] was decided by Zeus, in whose honor the Games were held[D] was considered unimportant19. Modern athletes’ results cannot be compared with those of ancient runners because ________.[A] the Greeks had no means of recording the results[B] they are much better[C] details such as the time were not recorded in the past[D] they are much worse20. Nowadays, the athletes’ expenses are paid for ________.[A] out of the prize money of the winners[B] out of the funds raised by the competing nations[C] by the athletes themselves[D] by contributions焦点导航考研英语完型 | 考研英语真题 | 考研英语阅读 | 考研英语翻译 | 考研英语经验交流考研英语作文 | 考研常见问题 | 专家解读Text 3In science the meaning of the word “explain” suffers with civilization’s every step in search of reality. Science cannot really explain electricity, magnetism, and gravitation; their effects can be measured and predicted, but of their nature no more is known to the modern scientist than to Thales who first looked into the nature of the electrification of amber, a hard yellowish-brown gum. Most contemporary physicists reject the notion that man can ever discover what these mysterious forces “really” are. “Electricity,” Bertrand Russell says, “is not a thing, like St.Paul’s Cathe dral; it is a way in which things behave. When we have told how things behave when they are electrified, and under what circumstances they are electrified, we have told all there is to tell.” Until recently scientists would have disapproved of such an idea. Aristotle, for example, whose natural science dominated Western thought for two thousand years, believed that man could arrive at an understanding of reality by reasoning from self-evident principles. He felt, for example, that it is a self-evident principle that everything in the universe has its proper place, hence one can deduce that objects fall to the ground because that’s where they belong, and smoke goes up because that’s where it belongs. The goal of Aristotelian science was to explain why things happen. Modern science was born when Galileo began trying to explain how things happen and thus originated the method of controlled experiment which now forms the basis of scientific investigation.21. The aim of controlled scientific experiments is ________.[A] to explain why things happen[B] to explain how things happen[C] to describe self-evident principles[D] to support Aristotelian science22. What principles most influenced scientific thought for two thousand years?[A] the speculations of Thales[B] the forces of electricity, magnetism, and gravity[C] Aristotle’s natural science[D] Galileo’s discoveries23. Bertrand Russell’s notion about electricity is ________.[A] disapproved of by most modern scientists[B] in agreement with Aristotle’s theo ry of self-evidentprinciples[C] in agreement with scientific investigation directed toward “how” things happen[D] in agreement with scientific investigation directed toward “why” things happen24. The passage says that until recently scientists disagreed with the idea ________.[A] that there are mysterious forces in the universe[B] that man cannot discover what forces “really” are[C] that there are self-evident principles[D] that we can discover why things behave as they do25. Modern science came into being ________.[A] when the method of controlled experiment was first introduced[B] when Galileo succeeded in explaining how things happen[C] when Aristotelian scientist tried to explain why things happen[D] when scientists were able to acquire an understanding of reality of reasoning[C] grants[D] credits答案解析Section II: Reading Comprehension (15 points)11.[B]12.[B]13.[D]14.[A]15.[D]16.[B]17.[A]18.[A]19.[C]20.[B]21.[B]22.[C]23.[C]24.[B]25.[A]1988年考研英语阅读真题及答案解析Section II Reading ComprehensionEach of the three passages below is followed by some questions. For each question there are four answers. Read thepassages carefully and chose the best answer to each of the questions. Put your choice in the ANSWER SHEET. (20 points) Text 1It doesn’t com e as a surprise to you to realize that it makes no difference what you read or study if you can’t remember it. You just waste your valuable time. Maybe you have already discovered some clever ways to keep yourself from forgetting.One dependable aid that does help you remember what you study is to have a specific purpose or reason for reading. You remember better what you read when you know why you’re reading.Why does a clerk in a store go away and leave you when your reply to her offer to help is, “No, thank you. I’m just looking”? Both you and she know that if you aren’t sure what you want, you are not likely to find it. But suppose you say instead, “Yes, thank you. I want a pair of sun glasses.” She says, “Right this way, please.” And you and she are off -- both eager to look for exactly what you want.It’s quite the same with your studying. If you chose a book at random, “just looking” for nothing in particular, you are likely to get just that -- nothing. But if you do know what you want, and if you have the right book, you are almost sure to get it. Your reasons will vary; they will include reading or studying “to find out more about”, “to understand the reasons for”, “to find out how”. A good student has a clear purpose or reason for what he is doing.This is the way it works. Before you start to study, you say to yourself something like this, “I want to know why Stephen Vincent Benet happened to write about America. I’m reading this article to find out.” Or, “I’m going to skim this story tosee what lif e was like in medieval England.” Because you know why you are reading or studying, you relate the information to your purpose and remember it better.Reading is not one single activity. At least two important processes go on at the same time. As you read, you take in ideas rapidly and accurately. But at the same time you express your own ideas to yourself as you react to what you read. You have a kind of mental conversation with the author. If you expressed your ideas orally, they might sound like this: “Ye s, I agree. That’s my opinion too.” or “Ummmm, I thought that record was broken much earlier. I’d better check those dates,” or “But there are some other facts to be considered!” You don’t just sit there taking in ideas -- you do something else, and that something else is very important.This additional process of thinking about what you read includes evaluating it, relating it to what you already know, and using it for your own purposes. In other words, a good reader is a critical reader. One part of critical reading, as you have discovered, is distinguishing between facts and opinions. Facts can be checked by evidence. Opinions are one’s own personal reactions.Another part of critical reading is judging sources. Still another part is drawing accurate inferences.16. If you cannot remember what you read or study, ________.[A] it is no surprise[B] it means you have not really learned anything[C] it means you have not chosen the right book[D] you realize it is of no importance17. Before you start reading, it is important ________.[A] to make sure why you are reading[B] to relate the information to your purpose[C] to remember what you read[D] to choose an interesting book18. Reading activity involves ________.[A] only two simultaneous processes[B] primarily learning about ideas and evaluating them critically[C] merely distinguishing between facts and opinions[D] mainly drawing accurate inferences19. A good reader is one who ________.[A] relates what he reads to his own knowledge about the subject matter[B] does lots of thinking in his reading[C] takes a critical attitude in his reading[D] is able to check the facts presented against what he has already knownText 2If you live in a large city, you are quite familiar with some of the problems of noise, but because of some of its harmful effects, you may not be aware of the extent of its influence on human behavior. Although everyone more or less knows what noise is, i.e., it is sounds that one would rather not hear, it is perhaps best to define it more precisely for scientific purposes. One such definition is that noise is sounds that are unrelated to the task at hand. Thus stimuli that at one time might be considered relevant will at another time be considered noise, depending on what one is doing at the moment. In recent years there has been a great deal of interest in the effects of noise on human behavior, and concepts such as “noise pollution” have arisen, together with movements to reduce noise.Exposure to loud noises can definitely produce a partial or complete loss of hearing, depending on the intensity, duration, and frequency composition of the noise. Many jobs present noise hazards, such as working in factories and around jet aircraft, driving farm tractors, and working (or sitting) in music halls where rock bands are playing. In general, continuous exposure to sounds of over 80 decibels (a measure of the loudness of sound) can be considered dangerous. Decibel values correspond to various sounds. Sounds above about 85 decibels may, if exposure is for a sufficient period of time, produce significant hearing loss. Actual loss will depend upon the particular frequencies to which one is exposed, and whether the sound is continuous or intermittent.Noise can have unexpected harmful effects on performance of certain kinds of tasks, for instance, if one is performing a watch keeping task that requires vigilance, in which he is responsible for detecting weak signals of some kind (e.g., watching a radar screen for the appearance of aircraft).Communicating with other people is unfavorably affected by noise. If you have ridden in the rear of a jet transport, you may have noticed that it was difficult to carry on a conversation at first, and that, eventually, you adjusted the loudness of your speech to compensate for the effect. The problem is noise.20. Noise differs from sound in that ________.[A] it is sounds that interfere with the task being done[B] it is a special type of loud sound[C] it is usually unavoidable in big cities[D] it can be defined more precisely than the latter21. One of the harmful effects of noise on human performance is that ________.[A] it reduces one’s sensitivity[B] it renders the victim helpless[C] it deprives one of the enjoyment of music[D] it drowns out conversations at worksites22. The purpose of this passage is ________.[A] to define the effects of noise on human behavior[B] to warn people of the danger of noise pollution[C] to give advice as to how to prevent hearing loss[D] to tell the difference between noise and soundText 3The traditional belief that a woman’s place is in the home and that a woman ought not to go out to work can hardly be reasonably maintained in present conditions. It is said that it is a woman’s task to care for the children, but families today tend to be small and with a year or two between children. Thus a woman’s whole period of childbearing may occur within five years. Furthermore, with compulsory education from the age of five or six her role as chief educator of her children soon ceases. Thus, even if we agree that a woman should stay at home to look after her children before they are of school age, for many women, this period would extend only for about ten years.It might be argued that the house-proud woman would still find plenty to do about the home. That may be so, but it is certainly no longer necessary for a woman to spend her whole life cooking, cleaning, mending and sewing. Washing machines take the drudgery out of laundry, the latest models being entirely automatic and able to wash and dry a large quantity of clothes in a few minutes. Refrigerators have made it possible to store food for long periods and many pre-cooked foods are obtainable in tins. Shopping, instead of being a daily task, can be completed。