Representations of a ternary quadratic form, modular forms of wheight 3/2 and Eisenstein series

Question

Let $Q(x_1,...,x_n)=\sum_{i,j=1}^na_{ij}x_ix_j$ be a positive-definite quadratic form with $a_{ij}\in\Bbb{Z}$.

In another question I ask about the following claim:

There is some $L>0$ (depending only on $Q$) such that for every $m\in \Bbb{N}$ there are some $x_1,....,x_n\in\Bbb{Z}$ with $q(x_1,...,x_n)\in[m-L,m+L]$.

I thought this was true for $n\geq 2$, but then someone answer me saying that this is true only for $n\geq 3$ and that this has to do with modular forms of wheight $n/2$ and Eisenstein series, "whose coefficients depend only on congruence conditions at a finite number of primes".

I know nothing about modular forms and this seems too interesting to be left behind, so I've made this post to try to get this straight.

I've read this pdf by Johnathan Hanke that if $r_Q(m)$ is the number of representations for $m$ by $Q$, then one might want to study the function $$\Theta_Q(z)=\sum_{m\geq 0}r_Q(m)e^{2\pi imz}.$$

He says that it "turns out" that there is a congruence subgroup $\Gamma_0(N)$ and a character $\chi_Q$ on $(\Bbb{Z}/N\Bbb{Z})^\times$ such that $$\Theta_Q(\gamma z)=\chi_Q(d)(cz+d)^{n/2}\Theta_Q(z),\text{ for each }\gamma z=\frac{az+b}{cz+d}\in \Gamma_0(N)$$

Hanke says that we can write $\Theta_Q(z)$ as $\Theta_Q(z)=E(z)+f(z)$ where $E(z)$ is an Eisenstein series and $f(z)$ is a cusp form.

This is what the guy who answered me was talking about, right?

My questions are:

1. What makes $n=3$ special?

2. How do I find these "congruence conditions on a finite number of primes"?

3. If I have an explicit $Q$, is it possible to find these congruent conditions explicitly?

4. In that case, can I find $L$ excplicitly? An estimate, at least?

5. Is this kind of thing already done in some book or material?

I'm really trying to understand this in detail, but I still have no idea how to begin to work it out.

Answer 1

Begin with these, positive ternaries with only diagonal terms that are "regular." That means that the numbers represented are given by a finite set of congruences; usually we say the numbers not represented.

I don't expect you need modular forms; suggest you look at most items I put at

http://zakuski.math.utsa.edu/~kap/

In particular, I think you are asking about this, Duke and Schulze-Pillot:

Answer 2

Finished the form 4,7,7,2,4,4 (Schiemann reduced). It represents all positive integers but these: $$ 4n+2$$ $$ 8n+1,3,5$$ $$ 9^k(3n+2)$$

We use $$ (2x+y+z)^2 + 6 y^2 + 6 z^2 = 4 x^2 + 7 y^2 + 7 z^2 +2xy+4zx+4xy. $$ That is, this form represents every number that can be expressed as $$n = u^2 +6v^2 + 6w^2$$ with $u+v+w$ even, as such an $n$ can then be represented by your form with integers $$ x = \frac{u-v-w}{2} \; , \; \; \; y=v \; , \; \; z=w \; .$$

All $24n + 7$ and $24n + 4$ are represented

The page that lists your form, showing it is alone in its genus (therefore regular)

from my giant text file. See if you can duplicate these lists on your computer. As you can see, the form represents about 135 numbers from 1 to 576. Note that a positive ternary is anisotropic at an odd number of primes, in this case either $2$ or $3.$ A complete accounting of represented numbers can be calculated.

=====Discriminant  576  ==Genus Size==   1
   Discriminant   576
  Spinor genus misses     no exceptions
       576:    4     7          7      2    4    4 vs. s.g.   regular candidate
--------------------------size 1
The 150 smallest numbers represented by full genus
     4     7    12    15    16    24    28    31    36    39
    40    48    52    55    60    63    64    76    79    84
    87    88    96   100   103   108   111   112   120   124
   127   132   135   136   144   148   151   156   159   160
   168   172   175   183   184   192   196   199   204   208
   216   220   223   228   231   232   240   244   247   252
   255   256   264   268   271   276   279   280   292   295
   300   303   304   312   316   319   324   327   328   336
   340   343   348   351   352   360   364   367   372   375
   376   384   388   391   399   400   408   412   415   420
   424   432   436   439   444   447   448   456   460   463
   468   471   472   480   484   487   492   495   496   508
   511   516   519   520   528   532   535   540   543   544
   552   556   559   564   567   568   576   580   583   588
   591   592   600   604   607   615   616   624   628   631

The 150 smallest numbers NOT represented by full genus
     1     2     3     5     6     8     9    10    11    13
    14    17    18    19    20    21    22    23    25    26
    27    29    30    32    33    34    35    37    38    41
    42    43    44    45    46    47    49    50    51    53
    54    56    57    58    59    61    62    65    66    67
    68    69    70    71    72    73    74    75    77    78
    80    81    82    83    85    86    89    90    91    92
    93    94    95    97    98    99   101   102   104   105
   106   107   109   110   113   114   115   116   117   118
   119   121   122   123   125   126   128   129   130   131
   133   134   137   138   139   140   141   142   143   145
   146   147   149   150   152   153   154   155   157   158
   161   162   163   164   165   166   167   169   170   171
   173   174   176   177   178   179   180   181   182   185
   186   187   188   189   190   191   193   194   195   197

Disc: 576
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