On the vanishing of twists of L-functions of elliptic curves Chantal ...

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$MATH 208 - Department of Mathematics and Statistics - Concordia ...$

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$Mast 330 /Math 370 Midterm Test 27 October 2004$

$MATH 208 - Department of Mathematics and Statistics - Concordia ...$

$Mast 330 / Math 370 Sec A Final Exam December 2002$

Distribution of the integers n χCurve E11A E14A E15A E37BFactor 10 6 8 12Torsion 5 6 8 3n χ = 0 1152 4347 807 1899n χ = 1 1662 344 287 150n χ = 2 1117 440 229 136n χ = 3 2676 1379 414 1419n χ = 4 1328 336 660 171n χ = 5 1069 288 219 147n χ = 6 1711 2707 470 785n χ = 7 1827 390 327 188n χ = 8 1125 442 504 105n χ = 9 2578 2365 378 2853n χ = 10 631 293 174 88n χ = 11 1336 299 227 118n χ = 12 2183 2188 872 993n χ = 13 1607 365 274 149n χ = 14 1015 489 229 98n χ = 15 1625 1044 288 810Proposition Suppose that E is isogenous to acurve with rational 3-torsion. Then3 ν(f χ)−1 | n χ .15
LetB(f χ ) =⎧⎪⎨⎪⎩c E 3 ν(f χ)−1√fχif E(Q)[3] ≠ ∅;c E√fχotherwise.Then, Prob (|L E (1, χ)| = 0) is= Prob (|L E (1, χ)| < B(f χ ))=∫ B(fχ )0a E (−1/2) R(N) dx= a E (−1/2) R(N) B(f χ ).Summing the probabilitiesN E (X)∼⎧⎪⎨⎪⎩b E X 1/2 log 1/4 X E(Q)[3] = ∅;b E X 1/2 log 9/4 X E(Q)[3] ≠ ∅;16
Page 1 and 2: On the vanishing of twists of L-fun
Page 3 and 4: Quadratic charactersLet χ d be a q
Page 5 and 6: Higher order twistsLet k be an inte
Page 7 and 8: Random matricesPhilosophy for a fam
Page 9: MomentsLet k be a positive integer.
Page 12 and 13: Random matricesKeating and Snaith:
Page 14 and 15: When N → ∞R(N) → N 1/4 G 2 (1
Page 18: Empirical results for E11A and E14A

On the vanishing of twists of L-functions of elliptic curves Chantal ...

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