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The algebraic parts of the central values of quadratic twists of modular L-functions modulo \(\ell \)

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Abstract

Let F be a newform of weight 2k on \(\Gamma _0(N)\) with an odd integer N and a positive integer k, and \(\ell \) be a prime larger than or equal to 5 with \((\ell ,N)=1\). For each fundamental discriminant D, let \(\chi _{D}\) be a quadratic character associated with quadratic field \(\mathbb {Q}(\sqrt{D})\). Assume that for each D, the \(\ell \)-adic valuation of the algebraic part of \(L(F\otimes \chi _{D},k)\) is non-negative. Let \(W_{\ell }^{+}\) (resp. \(W_{\ell }^{-}\)) be the set of positive (resp. negative) fundamental discriminants D with \((D,N)=1\) such that the \(\ell \)-adic valuation of the algebraic part of \(L(F\otimes \chi _{D},k)\) is zero. We prove that for each sign \(\epsilon \), if \(W_{\ell }^{\epsilon }\) is a non-empty finite set, then

$$\begin{aligned} W_{\ell }^{\epsilon } \subset \left\{ 1, (-1)^{\frac{\ell -1}{2}}\ell \right\} .\end{aligned}$$

By this result, we prove that if \(\epsilon \) is the sign of \((-1)^k\), then

$$\begin{aligned}k\ge \ell -1 \text { or } k=\frac{\ell -1}{2}. \end{aligned}$$

These are applied to obtain a lower bound for \(\#\{D\in W_{\ell }^{\epsilon } : |D|\le X \}\) and the indivisibility of the order of the Shafarevich–Tate group of an elliptic curve over \(\mathbb {Q}\). To prove these results, first we refine Waldspurger’s formula on the Shimura correspondence for general odd levels N. Next we study mod \(\ell \) modular forms of half-integral weight with few non-vanishing coefficients. To do this, we use the filtration of mod \(\ell \) modular forms and mod \(\ell \) Galois representations.

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References

  1. Ahlgren, S., Boylan, M.: Coefficients of half-integral weight modular forms modulo \(\ell ^j\). Math. Ann. 331(1), 219–239 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahlgren, S., Boylan, M.: Central critical values of modular \(L\)-functions and coefficients of half-integral weight modular forms modulo \(\ell \). Am. J. Math. 129(2), 429–454 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahlgren, S., Choi, D., Rouse, J.: Congruences for level four cusp forms. Math. Res. Lett. 16(4), 683–701 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atkin, A.O.L., Lehner, J.: Hecke operators on \(\Gamma _0(m)\). Math. Ann. 185, 134–160 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  5. Atkin, A.O.L., Li, W.: Twists of newforms and pseudo-eigenvalues of \(W\)-operators. Invent. Math. 48(3), 221–243 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bruinier, J.: Nonvanishing modulo \(\ell \) of Fourier coefficients of half-integral weight modular forms. Duke Math. J. 98(3), 595–611 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bruinier, J., Ono, K.: Coefficients of half-integral weight modular forms. J. Number Theory 99(1), 164–179 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carayol, H.: Sur les représentations \(\ell \)-adiques associées aux formes modularies de Hilbert. Ann. Sci. École Norm. Sup. (4) 19(3), 409–468 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Choi, D.: Modular forms of half-integral weight with few non-vanishing coefficients modulo \(\ell \). Proc. Am. Math. Soc. 136(8), 2683–2688 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Choi, D.: Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo \(\ell \). Trans. Am. Math. Soc. 361(7), 3817–3828 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Choi, D., Kilbourn, T.: The weight of half-integral weight modular forms with few non-vanishing coefficients mod \(\ell \). Acta Arith. 127(2), 193–197 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Flicker, Y.: Automorphic forms on covering groups of \(\text{ GL }(2)\). Invent. Math. 57(2), 119–182 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. James, K., Ono, K.: Selmer groups of quadratic twists of elliptic curves. Math. Ann. 314(1), 1–17 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Katz, N.: A result on modular forms in characteristic \(p\). In: Modular functions of one variable, V (Proceedings of Second International Conference, University of Bonn, Bonn, 1976). Lecture Notes in Mathematics, vol. 601, pp. 53–61. Springer, Berlin (1977)

  15. Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, vol. 97, 2nd edn. Springer, New York (1993)

    Book  MATH  Google Scholar 

  16. Kohnen, W.: Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271(2), 237–268 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kohnen, W., Ono, K.: Indivisibility of class numbers of imaginary quadratic fields and orders of Tate–Shafarevich groups of elliptic curves with complex multiplication. Invent. Math. 135(2), 387–398 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kohnen, W., Zagier, D.: Values of \(L\)-series of modular forms at the center of the critical strip. Invent. Math. 64(2), 175–198 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lamzouri, Y., Li, X., Soundararajan, K.: Conditional bounds for the least quadratic non-residue and related problems. Math. Comp. 84(295), 2391–2412 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mao, Z.: A generalized Shimura correspondence for newforms. J. Number Theory 128(1), 71–95 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and \(q\)-series, CBMS Regional Conference Series in Mathematics, 102. Published for the Conference Board of the Mathematical Sciences, Washington; by the American Mathematical Society, Providence (2004)

  22. Ono, K., Skinner, C.: Fourier coefficients of half-integral weight modular forms modulo \(\ell \). Ann. Math. (2) 147(2), 453–470 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pal, V.: Periods of quadratic twists of elliptic curves. Proc. Am. Math. Soc. 140(5), 1513–1525 (2012), With an appendix by Amod Agashe

  24. Serre, J.-P., Stark, H.: Modular forms of weight \(\frac{1}{2}\). In: Modular Functions of One Variable, VI (Proceedings of the Second International Conference, University of Bonn, Bonn, 1976). Lecture Notes in Mathematics, vol. 627, pp. 27–67. Springer, Berlin (1977)

  25. Shimura, G.: On the periods of modular forms. Math. Ann. 229(3), 211–221 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  26. Silverman, J.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106, 2nd edn. Springer, Dordrecht (2009)

    Book  Google Scholar 

  27. Sturm, J.: On the Congruence of Modular Forms, Number Theory (New York, 1984–1985). Lecture Notes in Mathematics, vol. 1240, pp. 275–280. Springer, Berlin (1987)

    Google Scholar 

  28. Vinogradov, I.M.: Sur la distribution des résidus et non résidus de puissances. Permski J. Phys. Isp. Ob. -wa 1, 18–28 and 94–98 (1918)

  29. Waldspurger, J.-L.: Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl. (9) 60(4), 375–484 (1981)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors appreciate Ken Ono for his kind and helpful comments. The authors also appreciate a referee for careful reading and useful comments. These comments improved the previous version of this paper. The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1A2C1007517). The second author was supported by a KIAS Individual Grant (MG086301) at Korea Institute for Advanced Study.

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Correspondence to Youngmin Lee.

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Choi, D., Lee, Y. The algebraic parts of the central values of quadratic twists of modular L-functions modulo \(\ell \). Res Math Sci 9, 65 (2022). https://doi.org/10.1007/s40687-022-00361-z

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