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THE RANK OF THE JACOBIAN OF MODULARCURVES: ANALYTIC METHODSBY EMMANUEL KOWALSKIiiPrefaceThe interaction of analytic and algebraic methods in number theory is as old as Euler,and assumes many guises. Of course, the basic algebraic structures are ever present inany modern mathematical theory, and analytic number theory is no exception, but toalgebraic geometry in particular it is indebted for the tremendous advances in under-standing of exponential sums over finite fields, since Andr´e Weil’s proof of the Riemannhypothesis for curves and subsequent deduction of the optimal bound for Kloostermansums to prime moduli.On the other hand, algebraic number theory has often used input fromL-functions;notonlyasasourceofresults, althoughfewdeeptheoremsinthisareaareprovedwith-out some appeal to Tchebotarev’s density theorem, but also as a source of inspiration,ideas and problems.One particular subject in arithmetic algebraic geometry which is now expected tobenefit from analytic methods is the study of the rank of the Mordell-Weil group ofan elliptic curve, or more generally of an abelian variety, over a number field. Thebeautiful conjecture of Birch and Swinnerton-Dyer asserts that this deep arithmeticinvariant can be recovered from the order of vanishing of the L-function of the abelianvariety at the center of the critical strip.This conjecture naturally opens two lines of investigation: to try to prove it, buthere one is, in general, hampered by the necessary ...