gchar package ====================================================================== **/ g = gcharinit(x^2+23,1) g.cyc g = gcharinit(x^2+23,9) g.cyc chareval(g,[1,0,1],7) chareval(g,[1,0,1],idealprimedec(g.bnf,13)[1]) chareval(g,[1,0,1,3],idealprimedec(g.bnf,13)[1]) \\ weight = norm character pol = x^6 + 2854*x^4 + 2036329*x^2 + 513996528; g = gcharinit(pol,7*13); g.cyc gcharalgebraic(g) \\ no CM subfield g = gcharinit(polcyclo(7),7) g.cyc gcharalgebraic(g) /** Examples of Hecke Grossencharacters ====================================================================== We give examples of objects linked to Hecke grossencharacters by their L functions (in a lmfdb way). A modular form ---------------------------------------------------------------------- Consider the CM character $(a)\mapsto \bigl(\frac a{\abs{a}}\bigr)^4$ over $\Q(i)$ (considered here as a character on principal ideals). **/ g = gcharinit(x^2+1,1); chi = [1]; real(lfunan([g,[1]],20)) /** It becomes algebraic when multiplied by $\norm{z}^2$, hence the extra component. **/ real(lfunan([g,[1,2]],20)) /** We obtain $f(q) = q - 4 q^{2} + 16 q^{4} - 14 q^{5} - 64 q^{8} + 81 q^{9} + O(q^{10})$ We can recognize a modular form of $S_5(4)$. $f(z) = \eta(z)^{4}\eta(2z)^{2}\eta(4z)^{4}=q\prod_{n=1}^\infty(1 - q^{n})^{4}(1 - q^{2n})^{2}(1 - q^{4n})^{4}$ see https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/4/5/b/a/ **/ mf = mfsearch([4,5],[0,1,-4,0,16])[1] mfcoefs(mf,20) /** B CM-elliptic curve ---------------------------------------------------------------------- We consider the curve $y^2=x^3-x$ (32a3), which has CM by $\Q(i)$. **/ e = ellinit([-1,0]); ellan(e,20) /** We identify this series with a Hecke Grossencharacter. The conductor of the elliptic curve is $32$, so we need a modulus of norm $8$, hence $\gm=(1+i)^3$. **/ g = gcharinit(x^2+1,(1+x)^3); g.cyc \\ one single character lfunan([g,[1]],10) /** Again we adjust with a norm to make the series rational **/ lfunan([g,[1,1/2]],10) /** We confirm with the L function zeros **/ lfunzeros(e,5) lfunzeros([g,[1]],5) /** The curve of conductor $27$ has CM by $\Q(sqrt{-3})$, we identify it using gcharalgebraic **/ ellan(ellinit("27a1"),40) g = gcharinit(x^2+3,9); gcharalgebraic(g) lfunan([g,[0,0,1,1/2]],40) /** B CM abelian variety ---------------------------------------------------------------------- Consider the genus 2 curve $C: y^2-y=x^5$ it has CM by $K=\Q(\zeta_5)$, with $(x,y)\mapsto (\zeta x,y)$. Its L-function is **/ C = [x^5,-1]; lC = lfungenus2(C); lfunparams(lC) \\ = [3125, 2, [0, 0, 1, 1]] /** We identify this L function with that of a Hecke character over K Let $\pi=1-\zeta$ be the prime dividing $5$, then $3125=5^5=d_KN(\pi^2)$ **/ g = gcharinit(polcyclo(5),(1-x)^2); g.cyc \\ [0, 0, 0] chi = [0,1,0]; lchi = lfuncreate([g,chi]); lfunparams(lchi) \\ [3125, 1, [1/2, 1/2, 3/2, 3/2]] \\ YEP lfunzeros(lchi,10) lfunzeros(lC,10) /** Hecke characters are cheaper than curves **/ lfunzeros(lchi,100); lfunzeros(lC,100); /** Remark: this one was identified by trivial guessing **/ lfunparams([g,[1,0,0]]) \\ [3125, 1, [1, 0, 2, 1]] \\ wrong /** C More modular forms ---------------------------------------------------------------------- .. seealso:: :ref:hecke_cusp_forms Weil converse theorem - let $K=\Q(\sqrt{-d})$, $D=\disc(K)$ and its quadratic character $\kronecker{D}{\cdot}$ - let $N=\abs{D}N'$, and $\psi:(\Z/N\Z)^\times\to\C^\times$ a Dirichlet character - find Hecke characters $\A^\times\to\C^\times$ of weight $k$ and conductor of norm $N'$ such that $\prod_{v\mid p}\chi_v(\pi_v)=\psi(p)p^{k-1}$ for $p\nmid N$ - then (Weil's converse theorem) $\chi$ coincides with a modular form $f\in S_k(\Gamma_0(N))$ of character $\psi\kronecker{D}\cdot$. Example: - $K = \Q(\sqrt{-7})$ - $\gm = 3$ **/ g = gcharinit(x^2+7,3); g.cyc gcharparameters(g,[-1,2,1/2]) lfunzeros([g,[-1,2,1/2]],4) /** We exhibit the corresponding modular form **/ N = 63; k = 3; mf = mfinit([N,k,Mod(6,7)]); L = lfunmf(mf)[1][1]; lfunzeros(L,4) /** Another example /** D Maass Forms ---------------------------------------------------------------------- .. seealso:: :ref:heckesmallmaass A real quadratic field $K = \Q(\sqrt2)$, and the character $\chi(\alpha) = \bigl(\frac{\alpha_1}{\alpha_2}\bigr)^{it}$ where the first embedding is $(\sqrt2)_1<0$, and $t=-\frac{\pi}{\log(\epsilon}$ for the fundamental unit $\epsilon=1+\sqrt2$ (so that $\chi(\epsilon)=1$). **/ k = bnfinit(x^2-2); t = Pi/k.reg \\ 3.5644279563827382235488269059450986816 g = gcharinit(k,1); g[1] \\ [-t,t] lmaass = lfuncreate([g,[1]]); lfunzeros(lmaass,10) /** A real cubic field **/ g=gcharinit(x^3-3*x-1,1); g.cyc gcharparameters(g,[1,0]) \\ vector (phi_sigma) gcharparameters(g,[0,1]) g[1] \\ internal basis lfunparams([g,[1,0]]) /** E Jacobi sums (hypergeometric motives) ---------------------------------------------------------------------- .. seealso:: :cite:WatkinsJacobiSums2018, :ref:jacobi_sums_grossencharacters :: v = Jacan(3,[1,1],40) **/ v = [1, 0, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 5, 0, 0, 4, 0, 0, -7, 0, 0, 0, 0, 0, -5, 0, 0, 2, 0, 0, -4, 0, 0, 0, 0, 0, 11, 0, 0, 0] g=gcharinit(polcyclo(3),9); gcharalgebraic(g) lfunan([g,[0,0,1,1/2]],40)